19778:
12863:
12142:
13524:
13512:
13500:
53:
12599:
11908:
5911:
13489:
15431:
12858:{\displaystyle F_{1}=\left({\begin{array}{ccc}0&0&0\\0&0&-1\\0&1&0\end{array}}\right),\quad F_{2}=\left({\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}}\right),\quad F_{3}=\left({\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&0\end{array}}\right)~.\quad }
20017:(for example, in the diffeomorphism group of the circle, there are diffeomorphisms arbitrarily close to the identity that are not in the image of the exponential map). Moreover, in terms of the existing notions of infinite-dimensional Lie groups, some infinite-dimensional Lie algebras do not come from any group.
12137:{\displaystyle X=\left({\begin{array}{ccc}0&1&0\\0&0&0\\0&0&0\end{array}}\right),\quad Y=\left({\begin{array}{ccc}0&0&0\\0&0&1\\0&0&0\end{array}}\right),\quad Z=\left({\begin{array}{ccc}0&0&1\\0&0&0\\0&0&0\end{array}}\right)~.\quad }
19334:
asserts that every finite-dimensional Lie algebra over a field of characteristic zero is a semidirect product of its solvable radical and a semisimple Lie algebra. Moreover, a semisimple Lie algebra in characteristic zero is a product of simple Lie algebras, as mentioned above. This focuses attention
16086:
The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically,
19998:
of Lie groups. For finite-dimensional representations, there is an equivalence of categories between representations of a real Lie algebra and representations of the corresponding simply connected Lie group. This simplifies the representation theory of Lie groups: it is often easier to classify the
15097:
One important aspect of the study of Lie algebras (especially semisimple Lie algebras, as defined below) is the study of their representations. Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra
5624:
11489:
13333:
17580:
16938:
16794:
1637:
In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in
16161:. Typically, the space of states is far from being irreducible under the pertinent operators, but one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the
4515:
8541:
8139:
15319:
15150:
says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The finite-dimensional irreducible representations are well understood from several points of view; see the
5244:
17177:
7295:
5042:
12315:
20917:
18265:
15001:
7951:
22100:
19201:
21726:
18676:
17460:
6761:
1880:
6161:
5906:{\displaystyle {\begin{aligned}\left&={\begin{bmatrix}ax&by\\cx&dy\\\end{bmatrix}}-{\begin{bmatrix}ax&bx\\cy&dy\\\end{bmatrix}}\\&={\begin{bmatrix}0&b(y-x)\\c(x-y)&0\end{bmatrix}}\end{aligned}}}
10476:
1368:
20590:
8890:
11351:
11161:
11386:
18851:
2168:
8672:
21306:
20214:
14221:
21654:
19724:
14766:
6617:
6571:
4666:
21839:
13484:{\displaystyle H=\left({\begin{array}{cc}1&0\\0&-1\end{array}}\right),\ E=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right),\ F=\left({\begin{array}{cc}0&0\\1&0\end{array}}\right).}
9021:
8780:
4710:
18596:
6384:
6078:
7666:
17707:
16997:
15279:
17495:
16853:
16643:
4305:
4268:
15701:
8309:
17046:
16043:
15965:
21126:
7529:
5629:
4797:
5368:
5306:
15876:
12454:
12190:
10763:
8035:
7993:
7602:
20326:
20282:
14448:
14343:
14265:
14125:
13984:
13928:
13577:
13324:
11293:
11058:
10221:
9826:
9654:
9354:
9252:
5106:
4029:
3943:
3839:
1682:
6705:
4620:
4383:
21985:
21915:
21539:
15426:{\displaystyle T({\mathfrak {g}})=F\oplus {\mathfrak {g}}\oplus ({\mathfrak {g}}\otimes {\mathfrak {g}})\oplus ({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})\oplus \cdots }
12233:
10677:
10632:
21592:
20397:
of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphism). A
10378:
10029:
9915:
9782:
9208:
9162:
9077:
3985:
3884:
9873:
9740:
8044:
850:
21460:
15063:
11685:
9310:
4361:
1142:
19502:
8414:
4915:
19445:
9504:
8149:(defined below) over a field of characteristic zero, every derivation is inner. This is related to the theorem that the outer automorphism group of a semisimple Lie group is finite.
7089:
3732:
19604:
19553:
19321:
16571:
15641:
12521:
12368:
11762:
7893:
4836:
19072:
19030:
18984:
18942:
18900:
18739:
18310:
17797:
17620:
14902:
6345:
6033:
5449:
4079:
3689:
3147:
1922:
11892:
7802:
1773:
20362:
20136:
18464:
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17957:
17921:
17864:
17828:
17651:
16199:
16125:
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15751:
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13213:
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13128:
12562:
11199:
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8244:
8208:
7127:
5945:
5564:
5528:
5489:
5407:
5141:
3621:
3191:
20010:, once the Lie algebra is known. For example, the real semisimple Lie algebras were classified by Cartan, and so the classification of semisimple Lie groups is well understood.
16076:
15812:
15555:
10177:
8953:
13612:
10713:
7834:
6904:
2077:
20715:
20682:
16351:
16319:
16159:
14299:
13092:
10581:
10516:
10058:
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9533:
9420:
9116:
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4116:
1095:
1058:
20833:
20512:
20389:
20238:
20166:
20105:
19924:
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19661:
19256:
19228:
19115:
18201:
18177:
18147:
18123:
18099:
18075:
18051:
18021:
17487:
17211:
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17082:
16845:
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16595:
16463:
16290:
16254:
15996:
15775:
15486:
15462:
15307:
15144:
15120:
15087:
14938:
14841:
14813:
14704:
11709:
11565:
11379:
11249:
11007:
10839:
10811:
10547:
10409:
10336:
9987:
9451:
9391:
8700:
8568:
8337:
7626:
7556:
7170:
6669:
6641:
6525:
6501:
6477:
6316:
6004:
5972:
5066:
4942:
4754:
4581:
4557:
3335:
3311:
3283:
3259:
3038:
2912:
2755:
2567:
2463:
1624:
1592:
1317:
20046:
12240:
11615:
9692:
7352:
6830:
21754:
21178:
20838:
14515:
11097:
21415:
20612:
20469:
19985:
19856:
19683:
19633:
18486:
18402:
17744:
14404:
14375:
13280:
12588:
10301:
10279:
10113:
7386:
7160:
has finite dimension as a vector space.) For this reason, spaces of derivations are a natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of
2012:
804:
1732:
2695:
15506:
12394:
11641:
11541:
10257:
6939:
21774:
21020:
18769:
18538:
18428:
18336:
18206:
17983:
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16424:
15233:
13057:
12994:
12931:
11225:
10907:
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19753:
15605:
14676:
8618:
3782:
3498:
1510:
16398:
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2270:
14789:
14558:
13788:
13697:
2519:
2357:
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20766:
18681:
The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field
18512:
13878:
13653:
10865:
6451:
5616:
4182:
3098:
3014:
2888:
2862:
2387:
16045:
is injective implies that every Lie algebra (possibly of infinite dimension) has a faithful representation (of infinite dimension), namely its representation on
13735:
10983:
10945:
4878:
3448:
21862:
20941:
20066:
19900:
19823:
19803:
16371:
11515:
8910:
8406:
8170:
5326:
5264:
5135:
4936:
4226:
3585:
3565:
3538:
3518:
3425:
3405:
3372:
2543:
2407:
2100:
1962:
1942:
1416:
1396:
21996:
20987:
19123:
16535:
16503:
2806:
1448:
21483:—without considering individual elements. (In this section, the field over which the algebra is defined is assumed to be of characteristic different from 2.)
18696:
if its adjoint representation is semisimple. Every reductive Lie algebra is isomorphic to the product of an abelian Lie algebra and a semisimple Lie algebra.
21662:
18601:
17229:
14858:
extended this result to finite-dimensional Lie algebras over a field of any characteristic. Equivalently, every finite-dimensional Lie algebra over a field
6710:
1781:
3149:. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group's name: for example, the Lie algebra of
1566:
In more detail: for any Lie group, the multiplication operation near the identity element 1 is commutative to first order. In other words, every Lie group
14854:
states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation on a finite-dimensional vector space.
10782:
10418:
1964:. The Lie bracket is a measure of the non-commutativity between two rotations. Since a rotation commutes with itself, one has the alternating property
1329:
20533:
8785:
16165:, for example, quantum mechanics textbooks classify (more or less explicitly) the finite-dimensional irreducible representations of the Lie algebra
15122:. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather, the goal is to understand all possible representations of
1626:
the structure of a Lie algebra. It is a remarkable fact that these second-order terms (the Lie algebra) completely determine the group structure of
15152:
11484:{\displaystyle X=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad Y=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right).}
470:
11300:
11113:
1378:
for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors
18777:
2108:
518:
19089:) gives conditions for a finite-dimensional Lie algebra of characteristic zero to be solvable or semisimple. It is expressed in terms of the
8623:
1598:
at the identity. To second order, the group operation may be non-commutative, and the second-order terms describing the non-commutativity of
1247:
21183:
20729:
involves showing that a simple Lie algebra over the complex numbers comes from a Lie algebra over the integers, and then (with more care) a
20171:
20020:
Lie theory also does not work so neatly for infinite-dimensional representations of a finite-dimensional group. Even for the additive group
14173:
6083:
13930:
is simple, and classify its finite-dimensional representations (defined below). In the terminology of quantum mechanics, one can think of
4802:
For the Lie algebra of a Lie group, the Lie bracket is a kind of infinitesimal commutator. As a result, for any Lie algebra, two elements
523:
21603:
19688:
14716:
6576:
6530:
4625:
22178:
21782:
15147:
513:
508:
8958:
8717:
19760:
15704:
4675:
19931:
19772:
1556:
328:
18543:
20068:
can usually not be differentiated to produce a representation of its Lie algebra on the same space, or vice versa. The theory of
19729:
In the years leading up to 2004, the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic
17575:{\displaystyle 0={\mathfrak {m}}_{0}\subseteq {\mathfrak {m}}_{1}\subseteq \cdots \subseteq {\mathfrak {m}}_{r}={\mathfrak {g}},}
16933:{\displaystyle 0={\mathfrak {a}}_{0}\subseteq {\mathfrak {a}}_{1}\subseteq \cdots \subseteq {\mathfrak {a}}_{r}={\mathfrak {g}},}
16789:{\displaystyle {\mathfrak {g}}\supseteq \supseteq ,{\mathfrak {g}}]\supseteq ,{\mathfrak {g}}],{\mathfrak {g}}]\supseteq \cdots }
14941:
592:
475:
19836:
The relationship between Lie groups and Lie algebras can be summarized as follows. Each Lie group determines a Lie algebra over
7631:
17656:
16946:
15241:
4276:
4239:
23236:
23206:
23104:
23003:
22973:
22888:
22812:
15646:
20645:-groups. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the
17799:
is its center, the 1-dimensional subspace spanned by the identity matrix. An example of a solvable Lie algebra is the space
14458:
8266:
6350:
6044:
20403:
is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique up to isomorphism.
17002:
16005:
15927:
704:
623:
23137:
23121:
21061:
7493:
4759:
14355:
The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over
5331:
5269:
4510:{\displaystyle \phi \colon {\mathfrak {g}}\to {\mathfrak {h}},\quad \phi ()=\ {\text{for all}}\ x,y\in {\mathfrak {g}}.}
23270:
22931:
15817:
12399:
12153:
10721:
7998:
7956:
7565:
20287:
20243:
14409:
14304:
14226:
14086:
13945:
13889:
13538:
13285:
11254:
11012:
10182:
9787:
9615:
9315:
9213:
5074:
3990:
3904:
3800:
1648:
23075:
23042:
22842:
19763:.) It turns out that there are many more simple Lie algebras in positive characteristic than in characteristic zero.
9695:
8134:{\displaystyle {\text{Out}}_{F}({\mathfrak {g}})={\text{Der}}_{F}({\mathfrak {g}})/{\text{Inn}}_{F}({\mathfrak {g}})}
6674:
4589:
1285:
1240:
1192:
21926:
21870:
21500:
12198:
10637:
10592:
22188:
21547:
17724:, nilpotent (respectively, solvable) Lie groups correspond to nilpotent (respectively, solvable) Lie algebras over
10341:
9992:
9878:
9745:
9171:
9125:
9040:
3948:
3847:
20411:
A Lie algebra may be equipped with additional structures that are compatible with the Lie bracket. For example, a
11711:
can be broken into abelian "pieces", meaning that it is solvable (though not nilpotent), in the terminology below.
9839:
9706:
2761:
Given a Lie group, the Jacobi identity for its Lie algebra follows from the associativity of the group operation.
814:
22122:
20420:
16078:. This also shows that every Lie algebra is contained in the Lie algebra associated to some associative algebra.
8536:{\displaystyle {\mathfrak {g}}=\{X=c'(0)\in M_{n}(\mathbb {R} ):{\text{ smooth }}c:\mathbb {R} \to G,\ c(0)=I\}.}
485:
23299:
21430:
16433:
A more general class of Lie algebras is defined by the vanishing of all commutators of given length. First, the
16209:
Lie algebras can be classified to some extent. This is a powerful approach to the classification of Lie groups.
15009:
11646:
9271:
4313:
22860:
6764:
1109:
20006:
central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the
19456:
4890:
23323:
22872:
22117:
19402:
9464:
7895:. (This is a derivation as a consequence of the Jacobi identity.) That gives a homomorphism of Lie algebras,
7002:
6787:
3694:
480:
460:
22344:
By the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.
19564:
19513:
19261:
16540:
15610:
12486:
12333:
11727:
7839:
5239:{\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}:\in S\ {\text{ for all}}\ s\in S\}}
4805:
20013:
For infinite-dimensional Lie algebras, Lie theory works less well. The exponential map need not be a local
19035:
18993:
18947:
18905:
18863:
18702:
18273:
17760:
17588:
15168:
14865:
7436:
5412:
4042:
3652:
3110:
2023:
1892:
1233:
1100:
425:
333:
31:
17:
21487:
11809:
7777:
1737:
23318:
22213:
22128:
20331:
20110:
18433:
18349:
17926:
17895:
17833:
17802:
17625:
16168:
16094:
15881:
15710:
14591:
14134:
13222:
13182:
13133:
13097:
12531:
11170:
9579:
8346:
8213:
8175:
7094:
5919:
5533:
5502:
5458:
5381:
4031:
describes the failure of commutativity for matrix multiplication, or equivalently for the composition of
3784:. This is a special case of the previous example; it is a key example of a Lie algebra. It is called the
3590:
3160:
1324:
950:
21132:
is a Lie ring, with addition given by the group multiplication (which is abelian on each quotient group
17172:{\displaystyle \operatorname {ad} (u):{\mathfrak {g}}\to {\mathfrak {g}},\quad \operatorname {ad} (u)v=}
16048:
15784:
15527:
10149:
8918:
7290:{\displaystyle (1+\epsilon D)(xy)\equiv (1+\epsilon D)(x)\cdot (1+\epsilon D)(y){\pmod {\epsilon ^{2}}}}
5037:{\displaystyle {\mathfrak {z}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}:=0\ {\text{ for all }}s\in S\}}
4366:
In the correspondence between Lie groups and Lie algebras, subgroups correspond to Lie subalgebras, and
23067:
23026:
22864:
22158:
19934:. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are
19339:
14574:
13585:
12310:{\displaystyle \left({\begin{array}{ccc}1&a&c\\0&1&b\\0&0&1\end{array}}\right)}
10686:
7807:
6841:
2057:
710:
465:
20912:{\displaystyle G=G_{1}\supseteq G_{2}\supseteq G_{3}\supseteq \cdots \supseteq G_{n}\supseteq \cdots }
20691:
20658:
19858:(concretely, the tangent space at the identity). Conversely, for every finite-dimensional Lie algebra
16327:
16295:
16130:
14270:
13068:
10552:
10487:
10034:
9924:
9545:
9509:
9396:
9092:
7478:. So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. An
7442:
7407:
6321:
6009:
4087:
1071:
1034:
725:
23224:
21480:
20771:
20493:
20370:
20219:
20147:
20086:
19991:
19962:
19905:
19861:
19829:
Although Lie algebras can be studied in their own right, historically they arose as a means to study
19642:
19237:
19209:
19096:
18686:
18182:
18158:
18128:
18104:
18080:
18056:
18032:
18002:
17468:
17192:
17063:
16826:
16802:
16607:
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16444:
16271:
16235:
16088:
15977:
15756:
15467:
15443:
15288:
15125:
15101:
15068:
14919:
14822:
14794:
14685:
11690:
11546:
11360:
11230:
10988:
10820:
10792:
10521:
10383:
10310:
10307:
times a skew-hermitian matrix is hermitian, rather than skew-hermitian.) Likewise, the unitary group
9961:
9539:
of a matrix. The orthogonal group has two connected components; the identity component is called the
9425:
9365:
8681:
8549:
8318:
7607:
7537:
7400:
gives a derivation of the space of smooth functions by differentiating functions in the direction of
6650:
6622:
6506:
6482:
6458:
6297:
6291:
5985:
5953:
5047:
4735:
4669:
4562:
4538:
3316:
3292:
3264:
3240:
3019:
2893:
2736:
2548:
2444:
1605:
1573:
1560:
1298:
20072:
is a more subtle relation between infinite-dimensional representations for groups and Lie algebras.
20023:
11570:
9659:
7321:
6803:
21734:
21135:
20472:
17717:
16597:, in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous to the
14485:
11067:
8142:
3206:
3041:
1645:
An elementary example (not directly coming from an associative algebra) is the 3-dimensional space
1185:
988:
938:
616:
100:
21313:
20595:
20452:
19968:
19839:
19666:
19616:
18902:, the 1-dimensional subspace spanned by the identity matrix. Since the special linear Lie algebra
18469:
18385:
17727:
17716:
Every finite-dimensional Lie algebra over a field has a unique maximal solvable ideal, called its
14387:
14358:
13263:
13094:
is given by the same formula in terms of the standard basis; so that Lie algebra is isomorphic to
12571:
10284:
10262:
10075:
9576:, consisting of the orthogonal matrices with determinant 1. Both groups have the same Lie algebra
7369:
3374:
endowed with the identically zero Lie bracket becomes a Lie algebra. Such a Lie algebra is called
1967:
787:
23313:
19990:
The correspondence between Lie algebras and Lie groups is used in several ways, including in the
19607:
10680:
10061:
9918:
6942:
1690:
1451:
997:
731:
690:
18685:
has characteristic zero, every finite-dimensional representation of a semisimple Lie algebra is
18260:{\displaystyle {\mathfrak {g}}\cong {\mathfrak {g}}_{1}\times \cdots \times {\mathfrak {g}}_{r}}
9422:
to itself that preserve the length of vectors. For example, rotations and reflections belong to
3378:. Every one-dimensional Lie algebra is abelian, by the alternating property of the Lie bracket.
22223:
22148:
22138:
19927:
18151:
17994:
17183:
15156:
10716:
9265:
8146:
3520:
is a Lie algebra. (The Jacobi identity follows from the associativity of the multiplication on
3210:
2587:
1552:
1154:
1005:
956:
737:
420:
383:
351:
338:
15491:
14996:{\displaystyle \operatorname {ad} \colon {\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})}
14531:, modulo only the relations coming from the definition of a Lie algebra. The free Lie algebra
12373:
11620:
11520:
10226:
7946:{\displaystyle \operatorname {ad} \colon {\mathfrak {g}}\to {\text{Der}}_{F}({\mathfrak {g}})}
6912:
23354:
23158:
23145:
22153:
21759:
21048:. (For the Lazard correspondence, one takes the filtration to be the lower central series of
20992:
20652:
20069:
20007:
19995:
19082:
18748:
18693:
18517:
18407:
18315:
17962:
17869:
16630:
16403:
15188:
14381:
13000:
12937:
12874:
11204:
10870:
10769:
10482:
5569:
3150:
452:
120:
20742:
Here is a construction of Lie rings arising from the study of abstract groups. For elements
19732:
16087:
they are the relations of the Lie algebra of the relevant symmetry group. An example is the
15560:
14631:
8573:
3737:
3453:
1465:
195:
185:
175:
165:
23280:
23246:
23216:
23187:
23114:
23085:
23052:
23013:
22983:
22949:
22898:
22852:
22822:
20523:
19345:
of characteristic zero were classified by
Killing and Cartan in the 1880s and 1890s, using
17215:
16635:
16376:
15999:
14049:
14009:
13794:
12565:
12459:
11771:
10118:
9035:
8152:
In contrast, an abelian Lie algebra has many outer derivations. Namely, for a vector space
7674:
7471:
6975:
6948:
6776:
6169:
4715:
3842:
3642:
2930:
2704:
2412:
2186:
1375:
878:
752:
80:
70:
14774:
14534:
13749:
13658:
2483:
2276:
8:
23349:
23141:
23129:
23125:
22095:{\displaystyle \circ (\otimes \mathrm {id} )\circ (\mathrm {id} +\sigma +\sigma ^{2})=0.}
21025:
20745:
20081:
19196:{\displaystyle K(u,v)=\operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v)),}
18491:
16598:
15521:
13842:
13617:
12787:
12704:
12621:
12249:
12069:
11996:
11923:
10844:
10143:
9165:
6389:
5595:
4143:
3047:
2993:
2867:
2811:
2366:
2080:
1520:
1455:
1160:
968:
919:
864:
758:
744:
672:
640:
609:
597:
438:
268:
13702:
10950:
10912:
4845:
3430:
23175:
22957:
22112:
21847:
21721:{\displaystyle \sigma =(\mathrm {id} \otimes \tau )\circ (\tau \otimes \mathrm {id} ),}
21471:
The definition of a Lie algebra can be reformulated more abstractly in the language of
20926:
20726:
20412:
20051:
19885:
19808:
19788:
19756:
19331:
19231:
18025:
17721:
17053:
16356:
16162:
14406:, with structure much like that of the finite-dimensional simple Lie algebras (such as
11722:
11500:
8895:
8703:
8675:
8391:
8155:
7152:. (This is literally true when the automorphism group is a Lie group, for example when
7145:
6644:
5311:
5249:
5120:
4921:
4229:
4187:
3570:
3550:
3523:
3503:
3410:
3390:
3357:
2528:
2392:
2085:
1947:
1927:
1401:
1381:
1173:
659:
369:
359:
20946:
20483:
The definition of a Lie algebra over a field extends to define a Lie algebra over any
18671:{\displaystyle {\mathfrak {so}}(4)\cong {\mathfrak {su}}(2)\times {\mathfrak {su}}(2)}
17455:{\displaystyle {\mathfrak {g}}\supseteq \supseteq ,]\supseteq ,],,]]\supseteq \cdots }
16508:
16468:
6756:{\displaystyle {\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}}
2767:
1875:{\displaystyle x\times (y\times z)+\ y\times (z\times x)+\ z\times (x\times y)\ =\ 0.}
1421:
23266:
23232:
23202:
23194:
23100:
23071:
23038:
22999:
22969:
22962:
22937:
22927:
22902:
22884:
22838:
22808:
22218:
20439:
20435:
13171:
5496:
5452:
3887:
3544:
2918:
1886:
1639:
1563:
of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra.
1214:
1011:
776:
717:
433:
396:
548:
286:
23167:
23059:
23030:
22919:
22876:
22798:
22248:
22183:
20722:
20484:
20416:
19951:
19825:
were the identity element of a Lie group, the tangent space would be a Lie algebra.
19636:
18203:
is semisimple if and only if it is isomorphic to a product of simple Lie algebras,
14855:
14851:
14519:
14465:
14128:
12193:
11900:
10587:
9360:
4532:
2103:
1524:
1271:
1220:
1206:
1020:
962:
925:
698:
684:
568:
248:
240:
232:
224:
216:
149:
130:
90:
13447:
13399:
13348:
11447:
11401:
11309:
11122:
10471:{\displaystyle i\mathbb {R} \subset \mathbb {C} ={\mathfrak {gl}}(1,\mathbb {C} )}
7534:
A Lie algebra can be viewed as a non-associative algebra, and so each Lie algebra
23276:
23242:
23212:
23183:
23110:
23081:
23048:
23009:
22991:
22979:
22945:
22894:
22848:
22818:
22802:
22228:
22198:
22193:
22143:
21472:
20447:
19939:
16222:
13939:
8386:
7355:
4367:
4082:
2031:
1544:
1371:
982:
932:
770:
553:
306:
291:
62:
19961:
Lie groups, there is a complete correspondence: taking the Lie algebra gives an
1570:
is (to first order) approximately a real vector space, namely the tangent space
1363:{\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}}
23262:
23153:
22203:
21424:
20718:
20585:{\displaystyle \colon {\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}}
20443:
20431:
20003:
18692:
A finite-dimensional Lie algebra over a field of characteristic zero is called
17220:
16226:
15509:
15437:
14168:
13580:
13167:
8885:{\displaystyle \exp(X)=I+X+{\tfrac {1}{2!}}X^{2}+{\tfrac {1}{3!}}X^{3}+\cdots }
6037:
1548:
1540:
1535:
for the Lie group.) Conversely, to any finite-dimensional Lie algebra over the
1026:
573:
391:
296:
22968:. Graduate Texts in Mathematics. Vol. 9 (2nd ed.). Springer-Verlag.
22923:
22880:
19086:
13215:
cannot be broken into pieces in the way that the previous examples can: it is
558:
23343:
23254:
23034:
22941:
22906:
22830:
22208:
22168:
22163:
20685:
20424:
20014:
16218:
14469:
14454:
10909:. (This determines the Lie bracket completely, because the axioms imply that
9956:
9119:
5492:
4531:
As with normal subgroups in groups, ideals in Lie algebras are precisely the
3044:
2, then anticommutativity implies the alternating property, since it implies
1685:
1532:
1528:
1167:
1063:
678:
281:
110:
16292:. In particular, the Lie algebra of an abelian Lie group (such as the group
14267:. As a result, it is common to analyze complex representations of the group
11346:{\displaystyle \left({\begin{array}{cc}c&d\\0&0\end{array}}\right).}
23259:
Group Theory and its
Application to the Quantum Mechanics of Atomic Spectra
22133:
20730:
20399:
19090:
15236:
14480:
11156:{\displaystyle \left({\begin{array}{cc}a&b\\0&1\end{array}}\right)}
11061:
10412:
8260:
7479:
7389:
6833:
2039:
1293:
1199:
974:
870:
578:
563:
364:
346:
276:
19965:
from simply connected Lie groups to Lie algebras of finite dimension over
1458:
gives rise to a Lie algebra, consisting of the same vector space with the
23092:
23064:
Representation Theory of
Semisimple Groups: an Overview Based on Examples
22918:. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer.
22916:
Lie groups, Lie
Algebras, and Representations: An Elementary Introduction
19346:
18846:{\displaystyle {\mathfrak {gl}}(n,F)\cong F\times {\mathfrak {sl}}(n,F),}
4884:
4137:
3898:
2163:{\displaystyle :{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}}
1536:
1261:
1179:
890:
764:
646:
404:
21475:. Namely, one can define a Lie algebra in terms of linear maps—that is,
21423:
For example, the Lie ring associated to the lower central series on the
20002:
Every connected Lie group is isomorphic to its universal cover modulo a
16353:) is abelian. Every finite-dimensional abelian Lie algebra over a field
14223:. The formulas for the Lie bracket are easier to analyze in the case of
1531:
at the identity. (In this case, the Lie bracket measures the failure of
23295:
23179:
22173:
21427:
of order 8 is the
Heisenberg Lie algebra of dimension 3 over the field
20646:
20393:
19206:
where tr denotes the trace of a linear operator. Namely: a Lie algebra
13995:
13326:
of 2 x 2 matrices of trace zero. A basis is given by the three matrices
8667:{\displaystyle {\mathfrak {g}}\subset {\mathfrak {gl}}(n,\mathbb {R} )}
5113:
4032:
3337:
is spanned (as a vector space) by all iterated brackets of elements of
2027:
1459:
944:
543:
409:
301:
21301:{\displaystyle G_{i}/G_{i+1}\times G_{j}/G_{j+1}\to G_{i+j}/G_{i+j+1}}
20209:{\displaystyle {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} }
19777:
14216:{\displaystyle {\mathfrak {so}}(3)\otimes _{\mathbb {R} }\mathbb {C} }
13219:, meaning that it is not abelian and its only ideals are 0 and all of
6156:{\displaystyle (x,x'),\,x\in {\mathfrak {g}},\ x'\in {\mathfrak {g'}}}
30:"Lie bracket" redirects here. For the operation on vector fields, see
23305:
22875:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
20423:
also comes with a differential, making the underlying vector space a
20140:
19830:
17056:, a Lie algebra over any field is nilpotent if and only if for every
15968:
9536:
4525:
3734:, is a Lie algebra with bracket given by the commutator of matrices:
1516:
904:
809:
40:
23171:
18077:. (In particular, a one-dimensional—necessarily abelian—Lie algebra
15146:. For a semisimple Lie algebra over a field of characteristic zero,
9393:
plays a basic role in geometry: it is the group of linear maps from
21476:
18101:
is by definition not simple, even though its only ideals are 0 and
9089:
matrices with determinant 1. This is the group of linear maps from
4232:
and algebras (and also for groups) has analogs for Lie algebras. A
898:
884:
23332:
21649:{\displaystyle \sigma :A\otimes A\otimes A\to A\otimes A\otimes A}
19719:{\displaystyle {\mathfrak {g}}\otimes _{\mathbb {R} }\mathbb {C} }
19613:
The classification of finite-dimensional simple Lie algebras over
14761:{\displaystyle \pi \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V).}
13614:. In particular, the Lie brackets of the vector fields shown are:
6612:{\displaystyle {\mathfrak {g}}/{\mathfrak {i}}\to {\mathfrak {g}}}
6566:{\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}}
4661:{\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}}
21834:{\displaystyle \sigma (x\otimes y\otimes z)=y\otimes z\otimes x.}
20626:
20615:
19639:
for an equivalent classification). One can analyze a Lie algebra
14384:
are a large class of infinite-dimensional Lie algebras, say over
13523:
13511:
13499:
8674:, one can recover the Lie group as the subgroup generated by the
3901:), where the group operation is matrix multiplication. Likewise,
3104:
1775:, and instead of associativity it satisfies the Jacobi identity:
782:
666:
500:
20592:
that satisfies the Jacobi identity. A Lie algebra over the ring
16229:, one can define abelian, nilpotent, and solvable Lie algebras.
9016:{\displaystyle \exp :M_{n}(\mathbb {C} )\to M_{n}(\mathbb {C} )}
8775:{\displaystyle \exp :M_{n}(\mathbb {R} )\to M_{n}(\mathbb {R} )}
21466:
20622:. (This is not directly related to the notion of a Lie group.)
19782:
19759:, Robert Lee Wilson, Alexander Premet, and Helmut Strade. (See
18342:
of characteristic zero (or just of characteristic not dividing
15464:, also called the free associative algebra on the vector space
10781:
Some Lie algebras of low dimension are described here. See the
2914:. Thus bilinearity and the alternating property together imply
21844:
With this notation, a Lie algebra can be defined as an object
17489:
is solvable if there is a finite sequence of Lie subalgebras,
4705:{\displaystyle \phi \colon {\mathfrak {g}}\to {\mathfrak {h}}}
3285:
such that any Lie subalgebra (as defined below) that contains
1277:
22247:
More generally, one has the notion of a Lie algebra over any
19947:
19943:
16322:
15643:; then the universal enveloping algebra is the quotient ring
14457:
is an infinite-dimensional Lie algebra that contains all the
13260:
Another simple Lie algebra of dimension 3, in this case over
13170:
angular-momentum component operators for spin-1 particles in
9268:
0. Similarly, one can define the analogous complex Lie group
4377:
is a linear map compatible with the respective Lie brackets:
1924:
may be pictured as an infinitesimal rotation around the axis
22125:(the integrability being the same as being a Lie subalgebra)
19761:
restricted Lie algebra#Classification of simple Lie algebras
18591:{\displaystyle {\mathfrak {so}}(3)\cong {\mathfrak {su}}(2)}
14527:. It is spanned by all iterated Lie brackets of elements of
4307:
is a linear subspace that satisfies the stronger condition:
1527:: every Lie group gives rise to a Lie algebra, which is the
23333:"An Elementary Introduction to Lie Algebras for Physicists"
21486:
For the category-theoretic definition of Lie algebras, two
19635:
is more complicated, but it was also solved by Cartan (see
16091:, whose commutation relations are those of the Lie algebra
10772:
are those listed above, along with variants over any field.
5974:
is an abelian Lie subalgebra, but it need not be an ideal.
52:
21864:
in the category of vector spaces together with a morphism
14624:
denote the Lie algebra consisting of all linear maps from
7661:{\displaystyle D\colon {\mathfrak {g}}\to {\mathfrak {g}}}
23301:
Course notes for MIT 18.745: Introduction to Lie
Algebras
20406:
17702:{\displaystyle {\mathfrak {m}}_{j}/{\mathfrak {m}}_{j-1}}
16992:{\displaystyle {\mathfrak {a}}_{j}/{\mathfrak {a}}_{j-1}}
15274:{\displaystyle {\mathfrak {g}}\mapsto U({\mathfrak {g}})}
11899:
It can be viewed as the Lie algebra of 3×3 strictly
2170:
called the Lie bracket, satisfying the following axioms:
19999:
representations of a Lie algebra, using linear algebra.
18689:(that is, a direct sum of irreducible representations).
16823:
is nilpotent if there is a finite sequence of ideals in
13886:
Using these formulas, one can show that the Lie algebra
11903:
matrices, with the commutator Lie bracket and the basis
10583:
consists of the skew-hermitian matrices with trace zero.
9026:
Here are some matrix Lie groups and their Lie algebras.
8315:
is matrix multiplication. The corresponding Lie algebra
4300:{\displaystyle {\mathfrak {i}}\subseteq {\mathfrak {g}}}
4263:{\displaystyle {\mathfrak {h}}\subseteq {\mathfrak {g}}}
4131:
3103:
It is customary to denote a Lie algebra by a lower-case
23095:(2010) , "Remarks on infinite-dimensional Lie groups",
19335:
on the problem of classifying the simple Lie algebras.
15696:{\displaystyle U({\mathfrak {g}})=T({\mathfrak {g}})/I}
14301:
by relating them to representations of the Lie algebra
5950:
Every one-dimensional linear subspace of a Lie algebra
4228:. Nonetheless, much of the terminology for associative
22964:
Introduction to Lie
Algebras and Representation Theory
17465:
becomes zero after finitely many steps. Equivalently,
16799:
becomes zero after finitely many steps. Equivalently,
16505:, meaning the linear subspace spanned by all brackets
16204:
8850:
8820:
8339:
is the space of matrices which are tangent vectors to
8304:{\displaystyle G\subset \mathrm {GL} (n,\mathbb {R} )}
7091:
is again a derivation. This operation makes the space
6379:{\displaystyle {\mathfrak {g}}\times {\mathfrak {g'}}}
6073:{\displaystyle {\mathfrak {g}}\times {\mathfrak {g'}}}
5838:
5780:
5729:
5681:
5642:
3897:
real matrices (or equivalently, matrices with nonzero
23136:
23120:
22296:
22284:
21999:
21929:
21873:
21850:
21785:
21762:
21737:
21665:
21606:
21550:
21503:
21433:
21316:
21186:
21138:
21064:
21028:
20995:
20949:
20929:
20841:
20774:
20748:
20694:
20661:
20649:
of two coset representatives; see the example below.
20598:
20536:
20496:
20455:
20373:
20334:
20290:
20246:
20222:
20174:
20150:
20113:
20089:
20054:
20026:
19971:
19908:
19888:
19864:
19842:
19811:
19791:
19735:
19691:
19669:
19645:
19619:
19567:
19516:
19459:
19405:
19264:
19240:
19212:
19126:
19099:
19038:
18996:
18950:
18908:
18866:
18780:
18751:
18705:
18604:
18546:
18520:
18494:
18472:
18436:
18410:
18388:
18352:
18318:
18276:
18209:
18185:
18161:
18131:
18107:
18083:
18059:
18035:
18005:
17965:
17929:
17898:
17892:. An example of a nilpotent Lie algebra is the space
17872:
17836:
17805:
17763:
17730:
17659:
17628:
17591:
17498:
17471:
17232:
17195:
17093:
17066:
17041:{\displaystyle {\mathfrak {g}}/{\mathfrak {a}}_{j-1}}
17005:
16949:
16856:
16829:
16805:
16646:
16610:
16579:
16543:
16511:
16471:
16447:
16406:
16379:
16359:
16330:
16298:
16274:
16238:
16171:
16133:
16097:
16051:
16038:{\displaystyle {\mathfrak {g}}\to U({\mathfrak {g}})}
16008:
16002:
over the universal enveloping algebra. The fact that
15980:
15960:{\displaystyle {\mathfrak {g}}\to U({\mathfrak {g}})}
15930:
15884:
15820:
15787:
15759:
15713:
15649:
15613:
15563:
15530:
15494:
15470:
15446:
15322:
15291:
15244:
15191:
15128:
15104:
15071:
15012:
14953:
14922:
14868:
14825:
14797:
14777:
14719:
14688:
14634:
14594:
14537:
14488:
14412:
14390:
14361:
14307:
14273:
14229:
14176:
14137:
14089:
14052:
14012:
13948:
13892:
13845:
13797:
13752:
13705:
13661:
13620:
13588:
13541:
13336:
13288:
13266:
13225:
13185:
13136:
13100:
13071:
13003:
12940:
12877:
12869:
The commutation relations among these generators are
12602:
12574:
12534:
12489:
12462:
12402:
12376:
12336:
12243:
12201:
12156:
11911:
11812:
11774:
11730:
11693:
11649:
11623:
11573:
11549:
11523:
11503:
11389:
11363:
11303:
11257:
11233:
11207:
11173:
11116:
11070:
11015:
10991:
10953:
10915:
10873:
10847:
10823:
10795:
10724:
10689:
10640:
10595:
10555:
10524:
10490:
10421:
10386:
10344:
10313:
10287:
10265:
10229:
10185:
10152:
10121:
10078:
10037:
9995:
9964:
9927:
9881:
9842:
9790:
9748:
9709:
9662:
9618:
9582:
9548:
9512:
9467:
9428:
9399:
9368:
9318:
9274:
9216:
9174:
9128:
9095:
9043:
8961:
8921:
8898:
8788:
8720:
8684:
8626:
8576:
8552:
8417:
8394:
8349:
8321:
8269:
8216:
8178:
8158:
8047:
8001:
7959:
7901:
7842:
7810:
7780:
7677:
7634:
7610:
7568:
7540:
7496:
7445:
7410:
7372:
7324:
7173:
7097:
7005:
6978:
6951:
6915:
6844:
6806:
6713:
6677:
6653:
6625:
6579:
6533:
6509:
6485:
6461:
6392:
6353:
6324:
6300:
6172:
6086:
6047:
6012:
5988:
5956:
5922:
5627:
5598:
5572:
5536:
5505:
5461:
5415:
5384:
5334:
5314:
5272:
5252:
5144:
5123:
5077:
5050:
4945:
4924:
4893:
4848:
4808:
4762:
4738:
4718:
4678:
4628:
4592:
4565:
4541:
4386:
4316:
4279:
4242:
4190:
4146:
4090:
4045:
3993:
3951:
3907:
3850:
3803:
3740:
3697:
3655:
3593:
3573:
3553:
3526:
3506:
3456:
3433:
3413:
3393:
3360:
3319:
3295:
3267:
3243:
3163:
3113:
3050:
3022:
2996:
2933:
2896:
2870:
2814:
2770:
2739:
2707:
2590:
2551:
2531:
2486:
2447:
2415:
2395:
2369:
2279:
2189:
2111:
2088:
2060:
2022:
Lie algebras were introduced to study the concept of
1970:
1950:
1930:
1895:
1784:
1740:
1693:
1651:
1608:
1576:
1468:
1424:
1404:
1384:
1332:
1301:
1286:
1112:
1074:
1037:
817:
790:
19338:
The simple Lie algebras of finite dimension over an
11687:
are abelian (because 1-dimensional). In this sense,
9696:
infinitesimal rotations with skew-symmetric matrices
8915:
The same comments apply to complex Lie subgroups of
1274:
23229:
Lie Groups, Lie
Algebras, and Their Representations
21121:{\displaystyle L=\bigoplus _{i\geq 1}G_{i}/G_{i+1}}
16260:if the Lie bracket vanishes; that is, = 0 for all
16212:
16081:
12568:is the space of skew-symmetric 3 x 3 matrices over
10783:
classification of low-dimensional real Lie algebras
10415:, and its Lie algebra (from this point of view) is
8381:: this consists of derivatives of smooth curves in
7524:{\displaystyle {\mathfrak {g}}\to {\text{Vect}}(X)}
4792:{\displaystyle {\mathfrak {g}}/{\text{ker}}(\phi )}
22961:
22262:-bilinear map that satisfies the Jacobi identity (
22094:
21979:
21909:
21856:
21833:
21768:
21748:
21720:
21648:
21586:
21533:
21454:
21409:
21300:
21172:
21120:
21040:
21014:
20981:
20935:
20911:
20827:
20760:
20709:
20676:
20606:
20584:
20506:
20463:
20383:
20356:
20320:
20276:
20232:
20208:
20160:
20130:
20099:
20060:
20040:
19979:
19918:
19894:
19874:
19850:
19817:
19797:
19747:
19718:
19677:
19655:
19627:
19598:
19547:
19496:
19439:
19315:
19250:
19222:
19195:
19109:
19066:
19024:
18978:
18936:
18894:
18845:
18763:
18733:
18670:
18590:
18532:
18506:
18480:
18458:
18422:
18396:
18374:
18330:
18304:
18259:
18195:
18171:
18141:
18117:
18093:
18069:
18045:
18015:
17977:
17951:
17915:
17884:
17858:
17822:
17791:
17738:
17701:
17645:
17614:
17574:
17481:
17454:
17205:
17171:
17076:
17040:
16991:
16932:
16839:
16815:
16788:
16620:
16589:
16565:
16529:
16497:
16457:
16418:
16392:
16365:
16345:
16313:
16284:
16248:
16193:
16153:
16119:
16070:
16037:
15990:
15959:
15916:
15870:
15806:
15769:
15745:
15695:
15635:
15599:
15549:
15500:
15480:
15456:
15425:
15301:
15273:
15227:
15138:
15114:
15081:
15057:
14995:
14932:
14896:
14835:
14807:
14783:
14760:
14698:
14670:
14616:
14552:
14509:
14475:The functor that takes a Lie algebra over a field
14442:
14398:
14369:
14337:
14293:
14259:
14215:
14159:
14119:
14070:
14030:
13978:
13922:
13872:
13830:
13782:
13729:
13691:
13647:
13606:
13571:
13483:
13318:
13274:
13247:
13207:
13158:
13122:
13086:
13051:
12988:
12925:
12857:
12582:
12556:
12515:
12475:
12448:
12388:
12362:
12309:
12227:
12184:
12136:
11886:
11792:
11768:is the three-dimensional Lie algebra with a basis
11756:
11703:
11679:
11635:
11609:
11559:
11535:
11509:
11483:
11373:
11345:
11287:
11243:
11219:
11193:
11155:
11091:
11052:
11009:can be viewed as the Lie algebra of the Lie group
11001:
10977:
10939:
10901:
10859:
10833:
10805:
10757:
10707:
10671:
10626:
10575:
10541:
10518:is the subgroup of matrices with determinant 1 in
10510:
10470:
10403:
10372:
10330:
10295:
10273:
10251:
10215:
10171:
10134:
10107:
10052:
10023:
9981:
9942:
9909:
9867:
9820:
9776:
9734:
9686:
9648:
9604:
9568:
9527:
9498:
9445:
9414:
9385:
9348:
9304:
9246:
9202:
9156:
9110:
9071:
9015:
8947:
8904:
8884:
8774:
8694:
8666:
8612:
8562:
8535:
8400:
8373:
8331:
8303:
8263:is a Lie group consisting of invertible matrices,
8238:
8202:
8164:
8133:
8029:
7987:
7945:
7887:
7828:
7796:
7758:
7660:
7620:
7596:
7550:
7523:
7462:
7427:
7380:
7346:
7289:
7121:
7083:
6991:
6964:
6933:
6898:
6824:
6755:
6699:
6663:
6635:
6611:
6565:
6519:
6495:
6471:
6445:
6378:
6339:
6310:
6279:
6155:
6072:
6027:
5998:
5966:
5939:
5905:
5610:
5584:
5558:
5522:
5483:
5443:
5401:
5363:{\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)}
5362:
5320:
5301:{\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)}
5300:
5258:
5238:
5129:
5100:
5060:
5036:
4930:
4909:
4872:
4830:
4791:
4748:
4724:
4704:
4660:
4614:
4575:
4551:
4509:
4355:
4299:
4262:
4220:
4176:
4110:
4073:
4023:
3979:
3937:
3878:
3833:
3776:
3726:
3683:
3615:
3579:
3559:
3532:
3512:
3492:
3442:
3419:
3399:
3366:
3329:
3305:
3277:
3253:
3185:
3141:
3092:
3032:
3008:
2978:
2906:
2882:
2856:
2800:
2749:
2725:
2689:
2561:
2537:
2513:
2457:
2433:
2401:
2381:
2351:
2264:
2162:
2094:
2071:
2006:
1956:
1936:
1916:
1874:
1767:
1726:
1676:
1618:
1586:
1504:
1442:
1410:
1390:
1362:
1311:
1136:
1089:
1052:
844:
798:
20075:
19230:is semisimple if and only if the Killing form is
16430:-dimensional vector space with Lie bracket zero.
15871:{\displaystyle e_{1}^{i_{1}}\cdots e_{n}^{i_{n}}}
15092:
14819:to itself, in such a way that the Lie bracket on
12449:{\displaystyle {\mathfrak {h}}_{3}(F)/(F\cdot Z)}
12185:{\displaystyle {\mathfrak {h}}_{3}(\mathbb {R} )}
10758:{\displaystyle {\mathfrak {sp}}(2n,\mathbb {R} )}
8030:{\displaystyle {\text{Der}}_{F}({\mathfrak {g}})}
7988:{\displaystyle {\text{Inn}}_{F}({\mathfrak {g}})}
7597:{\displaystyle {\text{Der}}_{F}({\mathfrak {g}})}
7140:Informally speaking, the space of derivations of
5977:
3450:, a Lie bracket may be defined by the commutator
23341:
20321:{\displaystyle {\mathfrak {sl}}(2,\mathbb {R} )}
20277:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
15153:representation theory of semisimple Lie algebras
14443:{\displaystyle {\mathfrak {sl}}(n,\mathbb {C} )}
14338:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
14260:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
14120:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
13979:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
13923:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
13572:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
13319:{\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}
11288:{\displaystyle {\mathfrak {gl}}(2,\mathbb {R} )}
11053:{\displaystyle G=\mathrm {Aff} (1,\mathbb {R} )}
10338:is a real Lie subgroup of the complex Lie group
10216:{\displaystyle {\mathfrak {gl}}(n,\mathbb {C} )}
9821:{\displaystyle {\mathfrak {so}}(n,\mathbb {C} )}
9649:{\displaystyle {\mathfrak {gl}}(n,\mathbb {R} )}
9349:{\displaystyle {\mathfrak {sl}}(n,\mathbb {C} )}
9247:{\displaystyle {\mathfrak {sl}}(n,\mathbb {R} )}
8714:is not connected.) Here the exponential mapping
5101:{\displaystyle {\mathfrak {z}}({\mathfrak {g}})}
4024:{\displaystyle {\mathfrak {gl}}(n,\mathbb {R} )}
3938:{\displaystyle {\mathfrak {gl}}(n,\mathbb {C} )}
3834:{\displaystyle {\mathfrak {gl}}(n,\mathbb {R} )}
1677:{\displaystyle {\mathfrak {g}}=\mathbb {R} ^{3}}
471:Representation theory of semisimple Lie algebras
22535:
22533:
20446:used differential graded Lie algebras over the
20240:. A real form need not be unique; for example,
19349:. Namely, every simple Lie algebra is of type A
15162:
6700:{\displaystyle {\mathfrak {g}}/{\mathfrak {i}}}
4615:{\displaystyle {\mathfrak {g}}/{\mathfrak {i}}}
1944:, with angular speed equal to the magnitude of
21980:{\displaystyle \circ (\mathrm {id} +\tau )=0,}
21910:{\displaystyle \colon A\otimes A\rightarrow A}
21534:{\displaystyle \tau :A\otimes A\to A\otimes A}
15173:The functor that takes an associative algebra
14843:corresponds to the commutator of linear maps.
12228:{\displaystyle \mathrm {H} _{3}(\mathbb {R} )}
10672:{\displaystyle \mathrm {GL} (2n,\mathbb {R} )}
10627:{\displaystyle \mathrm {Sp} (2n,\mathbb {R} )}
3381:
2808:and using the alternating property shows that
2049:
2030:in the 1870s, and independently discovered by
22829:
22611:
22500:
22461:
22449:
21587:{\displaystyle \tau (x\otimes y)=y\otimes x.}
19950:have isomorphic Lie algebras, but SU(2) is a
19942:. For instance, the special orthogonal group
11543:is an ideal in the 2-dimensional Lie algebra
11108:can be identified with the group of matrices
10373:{\displaystyle \mathrm {GL} (n,\mathbb {C} )}
10024:{\displaystyle \mathrm {GL} (n,\mathbb {C} )}
9910:{\displaystyle \mathrm {GL} (n,\mathbb {C} )}
9777:{\displaystyle \mathrm {SO} (n,\mathbb {C} )}
9612:, the subspace of skew-symmetric matrices in
9203:{\displaystyle \mathrm {GL} (n,\mathbb {R} )}
9157:{\displaystyle \mathrm {SL} (n,\mathbb {R} )}
9072:{\displaystyle \mathrm {SL} (n,\mathbb {R} )}
6941:. (The definition makes sense for a possibly
4672:holds for Lie algebras: for any homomorphism
3980:{\displaystyle \mathrm {GL} (n,\mathbb {C} )}
3879:{\displaystyle \mathrm {GL} (n,\mathbb {R} )}
1241:
617:
22859:
22722:
22718:
22716:
22530:
22413:
21467:Definition using category-theoretic notation
20048:, an infinite-dimensional representation of
18179:is 0. In characteristic zero, a Lie algebra
18029:if it is not abelian and the only ideals in
12483:. In the terminology below, it follows that
9868:{\displaystyle \mathrm {O} (n,\mathbb {C} )}
9735:{\displaystyle \mathrm {O} (n,\mathbb {C} )}
8527:
8428:
5233:
5173:
5031:
4974:
3945:is the Lie algebra of the complex Lie group
2764:Using bilinearity to expand the Lie bracket
1885:This is the Lie algebra of the Lie group of
845:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
23223:
22683:
22671:
22635:
22623:
22425:
22389:
21920:that satisfies the two morphism equalities
20655:are related to Lie algebras over the field
16573:. The commutator subalgebra is an ideal in
10179:consists of the skew-hermitian matrices in
7562:determines its Lie algebra of derivations,
6619:, as a homomorphism of Lie algebras), then
4622:is defined, with a surjective homomorphism
4270:which is closed under the Lie bracket. An
3225:. (They are "infinitesimal generators" for
3196:
23148:. MacTutor History of Mathematics Archive.
23132:. MacTutor History of Mathematics Archive.
22484:
22482:
22179:Particle physics and representation theory
21455:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
20943:, that is, a chain of subgroups such that
20625:Lie rings are used in the study of finite
15058:{\displaystyle \operatorname {ad} (x)(y)=}
11680:{\displaystyle {\mathfrak {g}}/(F\cdot Y)}
9828:are given by the same formulas applied to
9305:{\displaystyle {\rm {SL}}(n,\mathbb {C} )}
7490:determines a homomorphism of Lie algebras
7312:Example: the Lie algebra of vector fields.
4356:{\displaystyle \subseteq {\mathfrak {i}}.}
1248:
1234:
624:
610:
509:Particle physics and representation theory
51:
22956:
22804:Lie Groups and Lie Algebras: Chapters 1-3
22770:
22713:
22551:
22377:
22308:
21448:
21435:
20697:
20664:
20600:
20457:
20311:
20267:
20202:
20195:
20034:
19973:
19844:
19766:
19712:
19705:
19671:
19621:
18474:
18390:
17923:of strictly upper-triangular matrices in
17732:
16333:
16301:
15285:. To construct this: given a Lie algebra
14911:
14433:
14392:
14363:
14328:
14250:
14209:
14202:
14110:
13969:
13913:
13594:
13591:
13562:
13309:
13268:
13074:
12576:
12218:
12175:
11278:
11187:
11043:
10789:There is a unique nonabelian Lie algebra
10748:
10692:
10662:
10617:
10461:
10434:
10426:
10363:
10289:
10267:
10206:
10040:
10014:
9930:
9900:
9858:
9811:
9767:
9725:
9639:
9402:
9339:
9295:
9237:
9193:
9147:
9098:
9062:
9006:
8982:
8938:
8765:
8741:
8657:
8493:
8471:
8364:
8294:
7435:of vector fields into a Lie algebra (see
7374:
7164:. Indeed, writing out the condition that
6294:of Lie algebras. Note that the copies of
6110:
4014:
3970:
3928:
3869:
3824:
2126:
2119:
2115:
2061:
1904:
1664:
1137:{\displaystyle \mathbb {Z} (p^{\infty })}
1114:
1077:
1040:
838:
825:
792:
23330:
22990:
22797:
22659:
22647:
22599:
22587:
22575:
22563:
22524:
22365:
22353:
22332:
22320:
22263:
21756:is the identity morphism. Equivalently,
20637:. The lower central factors of a finite
19938:, and more strongly, they have the same
19776:
19497:{\displaystyle {\mathfrak {so}}(2n+1,F)}
18986:contains few ideals: only 0, the center
18540:. (There are "exceptional isomorphisms"
17988:
15924:natural numbers. In particular, the map
12590:. A basis is given by the three matrices
11227:. Its Lie algebra is the Lie subalgebra
10031:that preserves the length of vectors in
8570:is given by the commutator of matrices,
8041:is defined as the quotient Lie algebra,
4910:{\displaystyle S\subset {\mathfrak {g}}}
4081:can be viewed as the Lie algebra of the
3205:of a Lie algebra over a field means its
23152:
23020:
22746:
22734:
22479:
20367:Given a semisimple complex Lie algebra
19440:{\displaystyle {\mathfrak {sl}}(n+1,F)}
19393:. Here the simple Lie algebra of type A
17757:of characteristic zero, the radical of
9499:{\displaystyle A^{\mathrm {T} }=A^{-1}}
8172:with Lie bracket zero, the Lie algebra
7084:{\displaystyle :=D_{1}D_{2}-D_{2}D_{1}}
5451:is an abelian Lie subalgebra. (It is a
3727:{\displaystyle {\mathfrak {gl}}_{n}(F)}
3349:
2042:in the 1930s; in older texts, the term
1630:near the identity. They even determine
476:Representations of classical Lie groups
14:
23342:
23253:
23091:
22695:
22488:
20415:is a Lie algebra (or more generally a
20407:Lie algebra with additional structures
20284:has two real forms up to isomorphism,
19599:{\displaystyle {\mathfrak {so}}(2n,F)}
19548:{\displaystyle {\mathfrak {sp}}(2n,F)}
19316:{\displaystyle K({\mathfrak {g}},)=0.}
16566:{\displaystyle x,y\in {\mathfrak {g}}}
15636:{\displaystyle X,Y\in {\mathfrak {g}}}
14349:
12516:{\displaystyle {\mathfrak {h}}_{3}(F)}
12363:{\displaystyle {\mathfrak {h}}_{3}(F)}
11757:{\displaystyle {\mathfrak {h}}_{3}(F)}
10064:). Equivalently, this is the group of
8254:
7888:{\displaystyle \mathrm {ad} _{x}(y):=}
4917:is the set of elements commuting with
4831:{\displaystyle x,y\in {\mathfrak {g}}}
4535:of homomorphisms. Given a Lie algebra
4136:The Lie bracket is not required to be
3587:with the above Lie bracket is denoted
3229:, so to speak.) In mathematics, a set
1559:allows one to study the structure and
1374:. In other words, a Lie algebra is an
1319:together with an operation called the
23193:
23058:
22869:Representation theory. A first course
22782:
22758:
22707:
22512:
22401:
20438:form a graded Lie algebra, using the
19077:
19067:{\displaystyle {\mathfrak {gl}}(n,F)}
19025:{\displaystyle {\mathfrak {sl}}(n,F)}
18979:{\displaystyle {\mathfrak {gl}}(n,F)}
18937:{\displaystyle {\mathfrak {sl}}(n,F)}
18895:{\displaystyle {\mathfrak {gl}}(n,F)}
18734:{\displaystyle {\mathfrak {gl}}(n,F)}
18305:{\displaystyle {\mathfrak {sl}}(n,F)}
17792:{\displaystyle {\mathfrak {gl}}(n,F)}
17615:{\displaystyle {\mathfrak {m}}_{j-1}}
14897:{\displaystyle {\mathfrak {gl}}(n,F)}
14862:is isomorphic to a Lie subalgebra of
14479:to the underlying vector space has a
9453:. Equivalently, this is the group of
7531:. (An example is illustrated below.)
7300:(where 1 denotes the identity map on
5618:, this follows from the calculation:
5444:{\displaystyle {\mathfrak {gl}}(n,F)}
4132:Subalgebras, ideals and homomorphisms
4074:{\displaystyle {\mathfrak {gl}}(n,F)}
3684:{\displaystyle {\mathfrak {gl}}(n,F)}
3142:{\displaystyle {\mathfrak {g,h,b,n}}}
1917:{\displaystyle v\in \mathbb {R} ^{3}}
23156:(1969), "Rational homotopy theory",
22913:
22539:
22473:
22437:
19773:Lie group–Lie algebra correspondence
19685:by considering its complexification
18125:.) A finite-dimensional Lie algebra
17749:For example, for a positive integer
13942:. Indeed, for any representation of
11887:{\displaystyle =Z,\quad =0,\quad =0}
11357:In these terms, the basis above for
8955:and the complex matrix exponential,
8141:. (This is exactly analogous to the
7797:{\displaystyle x\in {\mathfrak {g}}}
5308:is the largest subalgebra such that
1768:{\displaystyle x\times y=-y\times x}
1515:Lie algebras are closely related to
705:Free product of associative algebras
329:Lie group–Lie algebra correspondence
27:Algebraic structure used in analysis
21308:given by commutators in the group:
20577:
20567:
20557:
20499:
20376:
20357:{\displaystyle {\mathfrak {su}}(2)}
20340:
20337:
20296:
20293:
20252:
20249:
20225:
20178:
20153:
20131:{\displaystyle {\mathfrak {g}}_{0}}
20117:
20092:
19911:
19867:
19694:
19648:
19573:
19570:
19522:
19519:
19465:
19462:
19411:
19408:
19296:
19286:
19273:
19243:
19215:
19102:
19044:
19041:
19002:
18999:
18956:
18953:
18914:
18911:
18872:
18869:
18820:
18817:
18786:
18783:
18711:
18708:
18654:
18651:
18632:
18629:
18610:
18607:
18574:
18571:
18552:
18549:
18459:{\displaystyle {\mathfrak {so}}(n)}
18442:
18439:
18375:{\displaystyle {\mathfrak {su}}(n)}
18358:
18355:
18282:
18279:
18246:
18223:
18212:
18188:
18164:
18134:
18110:
18086:
18062:
18038:
18008:
17952:{\displaystyle {\mathfrak {gl}}(n)}
17935:
17932:
17916:{\displaystyle {\mathfrak {u}}_{n}}
17902:
17859:{\displaystyle {\mathfrak {gl}}(n)}
17842:
17839:
17823:{\displaystyle {\mathfrak {b}}_{n}}
17809:
17769:
17766:
17682:
17663:
17646:{\displaystyle {\mathfrak {m}}_{j}}
17632:
17595:
17564:
17548:
17525:
17508:
17474:
17432:
17422:
17406:
17396:
17374:
17364:
17348:
17338:
17313:
17303:
17287:
17277:
17258:
17248:
17235:
17198:
17124:
17114:
17069:
17021:
17008:
16972:
16953:
16922:
16906:
16883:
16866:
16832:
16808:
16772:
16759:
16746:
16736:
16714:
16701:
16691:
16672:
16662:
16649:
16613:
16582:
16558:
16487:
16477:
16450:
16277:
16241:
16205:Structure theory and classification
16194:{\displaystyle {\mathfrak {so}}(3)}
16177:
16174:
16120:{\displaystyle {\mathfrak {so}}(3)}
16103:
16100:
16060:
16027:
16011:
15983:
15949:
15933:
15917:{\displaystyle i_{1},\ldots ,i_{n}}
15796:
15762:
15746:{\displaystyle e_{1},\ldots ,e_{n}}
15677:
15658:
15628:
15539:
15473:
15449:
15409:
15399:
15389:
15373:
15363:
15350:
15331:
15294:
15263:
15247:
15131:
15107:
15074:
14985:
14975:
14972:
14962:
14925:
14874:
14871:
14828:
14800:
14741:
14738:
14728:
14691:
14617:{\displaystyle {\mathfrak {gl}}(V)}
14600:
14597:
14418:
14415:
14313:
14310:
14235:
14232:
14182:
14179:
14160:{\displaystyle {\mathfrak {so}}(3)}
14143:
14140:
14095:
14092:
13954:
13951:
13898:
13895:
13547:
13544:
13294:
13291:
13248:{\displaystyle {\mathfrak {so}}(3)}
13231:
13228:
13208:{\displaystyle {\mathfrak {so}}(3)}
13191:
13188:
13159:{\displaystyle {\mathfrak {so}}(3)}
13142:
13139:
13123:{\displaystyle {\mathfrak {so}}(3)}
13106:
13103:
12557:{\displaystyle {\mathfrak {so}}(3)}
12540:
12537:
12493:
12406:
12340:
12160:
11734:
11715:
11696:
11652:
11552:
11366:
11263:
11260:
11236:
11194:{\displaystyle a,b\in \mathbb {R} }
10994:
10826:
10798:
10730:
10727:
10446:
10443:
10191:
10188:
10155:
9796:
9793:
9624:
9621:
9605:{\displaystyle {\mathfrak {so}}(n)}
9588:
9585:
9324:
9321:
9222:
9219:
9118:to itself that preserve volume and
8892:, which converges for every matrix
8687:
8642:
8639:
8629:
8555:
8420:
8374:{\displaystyle M_{n}(\mathbb {R} )}
8324:
8239:{\displaystyle {\mathfrak {gl}}(V)}
8222:
8219:
8203:{\displaystyle {\text{Out}}_{F}(V)}
8123:
8093:
8065:
8019:
7977:
7935:
7910:
7789:
7653:
7643:
7613:
7586:
7543:
7499:
7272:
7122:{\displaystyle {\text{Der}}_{k}(A)}
6748:
6738:
6726:
6716:
6692:
6680:
6656:
6628:
6604:
6594:
6582:
6558:
6546:
6536:
6512:
6488:
6464:
6367:
6356:
6328:
6303:
6144:
6119:
6061:
6050:
6016:
5991:
5959:
5940:{\displaystyle {\mathfrak {t}}_{2}}
5926:
5559:{\displaystyle {\mathfrak {gl}}(n)}
5542:
5539:
5523:{\displaystyle {\mathfrak {t}}_{n}}
5509:
5484:{\displaystyle {\mathfrak {gl}}(n)}
5467:
5464:
5421:
5418:
5402:{\displaystyle {\mathfrak {t}}_{n}}
5388:
5345:
5338:
5283:
5276:
5184:
5155:
5148:
5090:
5080:
5053:
4985:
4956:
4949:
4902:
4823:
4765:
4741:
4697:
4687:
4653:
4641:
4631:
4607:
4595:
4568:
4544:
4499:
4405:
4395:
4345:
4332:
4322:
4292:
4282:
4255:
4245:
4051:
4048:
3999:
3996:
3913:
3910:
3809:
3806:
3704:
3701:
3661:
3658:
3616:{\displaystyle {\mathfrak {gl}}(V)}
3599:
3596:
3322:
3298:
3270:
3246:
3186:{\displaystyle {\mathfrak {su}}(n)}
3169:
3166:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3025:
2899:
2742:
2554:
2450:
2155:
2145:
2135:
2064:
1654:
1611:
1579:
1355:
1345:
1335:
1304:
24:
23023:p-Automorphisms of Finite p-Groups
22060:
22057:
22043:
22040:
21955:
21952:
21742:
21739:
21708:
21705:
21679:
21676:
20768:of a group, define the commutator
19606:. The other five are known as the
18771:, it is isomorphic to the product
16138:
16135:
16071:{\displaystyle U({\mathfrak {g}})}
15807:{\displaystyle U({\mathfrak {g}})}
15550:{\displaystyle T({\mathfrak {g}})}
14568:
14278:
14275:
12523:is nilpotent (though not abelian).
12204:
11167:under matrix multiplication, with
11029:
11026:
11023:
10867:for which the bracket is given by
10645:
10642:
10600:
10597:
10560:
10557:
10526:
10495:
10492:
10388:
10349:
10346:
10315:
10172:{\displaystyle {\mathfrak {u}}(n)}
10000:
9997:
9966:
9886:
9883:
9844:
9753:
9750:
9711:
9669:
9553:
9550:
9519:
9474:
9461:orthogonal matrices, meaning that
9430:
9370:
9280:
9277:
9179:
9176:
9133:
9130:
9048:
9045:
8948:{\displaystyle GL(n,\mathbb {C} )}
8280:
8277:
7848:
7845:
7816:
7813:
7330:
7304:) gives exactly the definition of
4095:
4092:
3956:
3953:
3855:
3852:
3213:of the Lie algebra of a Lie group
1126:
25:
23366:
23288:
19882:, there is a connected Lie group
19325:
19093:, the symmetric bilinear form on
15814:is given by all ordered products
14628:to itself, with bracket given by
13986:, the relations above imply that
13607:{\displaystyle \mathbb {CP} ^{1}}
12235:, that is, the group of matrices
10776:
10708:{\displaystyle \mathbb {R} ^{2n}}
8702:. (To be precise, this gives the
7829:{\displaystyle \mathrm {ad} _{x}}
6899:{\displaystyle D(xy)=D(x)y+xD(y)}
3344:
2072:{\displaystyle \,{\mathfrak {g}}}
1634:globally, up to covering spaces.
1193:Noncommutative algebraic geometry
23099:, vol. 5, pp. 91–141,
22189:Orthogonal symmetric Lie algebra
20710:{\displaystyle \mathbb {Z} _{p}}
20677:{\displaystyle \mathbb {Q} _{p}}
19932:Baker–Campbell–Hausdorff formula
17830:of upper-triangular matrices in
16346:{\displaystyle \mathbb {T} ^{n}}
16314:{\displaystyle \mathbb {R} ^{n}}
16213:Abelian, nilpotent, and solvable
16154:{\displaystyle \mathrm {SO} (3)}
16082:Representation theory in physics
15781:-vector space, then a basis for
14579:
14294:{\displaystyle \mathrm {SO} (3)}
13522:
13510:
13498:
13087:{\displaystyle \mathbb {R} ^{3}}
13065:The cross product of vectors in
10576:{\displaystyle \mathrm {su} (n)}
10511:{\displaystyle \mathrm {SU} (n)}
10053:{\displaystyle \mathbb {C} ^{n}}
9943:{\displaystyle \mathbb {C} ^{n}}
9836:complex matrices. Equivalently,
9569:{\displaystyle \mathrm {SO} (n)}
9528:{\displaystyle A^{\mathrm {T} }}
9415:{\displaystyle \mathbb {R} ^{n}}
9111:{\displaystyle \mathbb {R} ^{n}}
7463:{\displaystyle {\text{Vect}}(X)}
7428:{\displaystyle {\text{Vect}}(X)}
6340:{\displaystyle {\mathfrak {g}}'}
6080:consisting of all ordered pairs
6041:Lie algebra is the vector space
6028:{\displaystyle {\mathfrak {g'}}}
4111:{\displaystyle \mathrm {GL} (n)}
2054:A Lie algebra is a vector space
1684:with Lie bracket defined by the
1270:
1090:{\displaystyle \mathbb {Q} _{p}}
1053:{\displaystyle \mathbb {Z} _{p}}
23294:
23261:. Translated by J. J. Griffin.
23097:Collected Papers of John Milnor
22776:
22764:
22752:
22740:
22728:
22710:, section III.3, Problem III.5.
22701:
22689:
22677:
22665:
22653:
22641:
22629:
22617:
22605:
22593:
22581:
22569:
22557:
22545:
22518:
22506:
22494:
22467:
22455:
22443:
22431:
22419:
22407:
22395:
22383:
22371:
22123:Frobenius integrability theorem
20828:{\displaystyle =x^{-1}y^{-1}xy}
20507:{\displaystyle {\mathfrak {g}}}
20421:differential graded Lie algebra
20419:) with a compatible grading. A
20384:{\displaystyle {\mathfrak {g}}}
20233:{\displaystyle {\mathfrak {g}}}
20161:{\displaystyle {\mathfrak {g}}}
20100:{\displaystyle {\mathfrak {g}}}
19919:{\displaystyle {\mathfrak {g}}}
19875:{\displaystyle {\mathfrak {g}}}
19656:{\displaystyle {\mathfrak {g}}}
19251:{\displaystyle {\mathfrak {g}}}
19223:{\displaystyle {\mathfrak {g}}}
19110:{\displaystyle {\mathfrak {g}}}
18196:{\displaystyle {\mathfrak {g}}}
18172:{\displaystyle {\mathfrak {g}}}
18142:{\displaystyle {\mathfrak {g}}}
18118:{\displaystyle {\mathfrak {g}}}
18094:{\displaystyle {\mathfrak {g}}}
18070:{\displaystyle {\mathfrak {g}}}
18046:{\displaystyle {\mathfrak {g}}}
18016:{\displaystyle {\mathfrak {g}}}
17482:{\displaystyle {\mathfrak {g}}}
17206:{\displaystyle {\mathfrak {g}}}
17132:
17077:{\displaystyle {\mathfrak {g}}}
16840:{\displaystyle {\mathfrak {g}}}
16816:{\displaystyle {\mathfrak {g}}}
16621:{\displaystyle {\mathfrak {g}}}
16590:{\displaystyle {\mathfrak {g}}}
16458:{\displaystyle {\mathfrak {g}}}
16285:{\displaystyle {\mathfrak {g}}}
16249:{\displaystyle {\mathfrak {g}}}
15991:{\displaystyle {\mathfrak {g}}}
15770:{\displaystyle {\mathfrak {g}}}
15481:{\displaystyle {\mathfrak {g}}}
15457:{\displaystyle {\mathfrak {g}}}
15302:{\displaystyle {\mathfrak {g}}}
15139:{\displaystyle {\mathfrak {g}}}
15115:{\displaystyle {\mathfrak {g}}}
15082:{\displaystyle {\mathfrak {g}}}
15065:. (This is a representation of
14933:{\displaystyle {\mathfrak {g}}}
14846:A representation is said to be
14836:{\displaystyle {\mathfrak {g}}}
14808:{\displaystyle {\mathfrak {g}}}
14699:{\displaystyle {\mathfrak {g}}}
12854:
12768:
12685:
12133:
12057:
11984:
11862:
11837:
11704:{\displaystyle {\mathfrak {g}}}
11560:{\displaystyle {\mathfrak {g}}}
11435:
11374:{\displaystyle {\mathfrak {g}}}
11244:{\displaystyle {\mathfrak {g}}}
11002:{\displaystyle {\mathfrak {g}}}
10834:{\displaystyle {\mathfrak {g}}}
10806:{\displaystyle {\mathfrak {g}}}
10542:{\displaystyle \mathrm {U} (n)}
10404:{\displaystyle \mathrm {U} (1)}
10331:{\displaystyle \mathrm {U} (n)}
9982:{\displaystyle \mathrm {U} (n)}
9446:{\displaystyle \mathrm {O} (n)}
9386:{\displaystyle \mathrm {O} (n)}
9023:(defined by the same formula).
8695:{\displaystyle {\mathfrak {g}}}
8563:{\displaystyle {\mathfrak {g}}}
8332:{\displaystyle {\mathfrak {g}}}
8311:, where the group operation of
7621:{\displaystyle {\mathfrak {g}}}
7551:{\displaystyle {\mathfrak {g}}}
7265:
6664:{\displaystyle {\mathfrak {i}}}
6636:{\displaystyle {\mathfrak {g}}}
6573:splits (i.e., admits a section
6520:{\displaystyle {\mathfrak {g}}}
6496:{\displaystyle {\mathfrak {i}}}
6472:{\displaystyle {\mathfrak {g}}}
6311:{\displaystyle {\mathfrak {g}}}
5999:{\displaystyle {\mathfrak {g}}}
5967:{\displaystyle {\mathfrak {g}}}
5061:{\displaystyle {\mathfrak {g}}}
4749:{\displaystyle {\mathfrak {h}}}
4576:{\displaystyle {\mathfrak {i}}}
4552:{\displaystyle {\mathfrak {g}}}
4413:
3427:with multiplication written as
3330:{\displaystyle {\mathfrak {g}}}
3306:{\displaystyle {\mathfrak {g}}}
3278:{\displaystyle {\mathfrak {g}}}
3254:{\displaystyle {\mathfrak {g}}}
3033:{\displaystyle {\mathfrak {g}}}
2907:{\displaystyle {\mathfrak {g}}}
2750:{\displaystyle {\mathfrak {g}}}
2562:{\displaystyle {\mathfrak {g}}}
2458:{\displaystyle {\mathfrak {g}}}
1619:{\displaystyle {\mathfrak {g}}}
1587:{\displaystyle {\mathfrak {g}}}
1450:. A Lie algebra is typically a
1312:{\displaystyle {\mathfrak {g}}}
22359:
22347:
22338:
22326:
22314:
22302:
22290:
22278:
22241:
22083:
22053:
22047:
22033:
22021:
22018:
22012:
22000:
21965:
21948:
21942:
21930:
21901:
21886:
21874:
21807:
21789:
21712:
21695:
21689:
21672:
21628:
21566:
21554:
21519:
21379:
21367:
21361:
21317:
21252:
20976:
20950:
20787:
20775:
20725:'s construction of the finite
20572:
20549:
20537:
20351:
20345:
20315:
20301:
20271:
20257:
20076:Real form and complexification
20041:{\displaystyle G=\mathbb {R} }
19946:and the special unitary group
19593:
19578:
19542:
19527:
19491:
19470:
19434:
19416:
19304:
19301:
19281:
19268:
19187:
19184:
19178:
19169:
19163:
19154:
19142:
19130:
19061:
19049:
19019:
19007:
18973:
18961:
18931:
18919:
18889:
18877:
18837:
18825:
18803:
18791:
18728:
18716:
18665:
18659:
18643:
18637:
18621:
18615:
18585:
18579:
18563:
18557:
18453:
18447:
18369:
18363:
18299:
18287:
18155:if the only solvable ideal in
17946:
17940:
17853:
17847:
17786:
17774:
17443:
17440:
17437:
17417:
17411:
17391:
17388:
17382:
17379:
17359:
17353:
17333:
17330:
17327:
17321:
17318:
17298:
17292:
17272:
17269:
17263:
17243:
17189:More generally, a Lie algebra
17166:
17154:
17145:
17139:
17119:
17106:
17100:
16777:
16764:
16751:
16731:
16728:
16725:
16719:
16706:
16686:
16683:
16677:
16657:
16524:
16512:
16492:
16472:
16188:
16182:
16148:
16142:
16114:
16108:
16065:
16055:
16032:
16022:
16016:
15954:
15944:
15938:
15801:
15791:
15705:Poincaré–Birkhoff–Witt theorem
15682:
15672:
15663:
15653:
15594:
15582:
15544:
15534:
15414:
15384:
15378:
15358:
15336:
15326:
15268:
15258:
15252:
15204:
15192:
15093:Goals of representation theory
15052:
15040:
15034:
15028:
15025:
15019:
14990:
14980:
14967:
14891:
14879:
14752:
14746:
14733:
14710:is a Lie algebra homomorphism
14647:
14635:
14611:
14605:
14547:
14541:
14504:
14498:
14492:
14437:
14423:
14332:
14318:
14288:
14282:
14254:
14240:
14193:
14187:
14154:
14148:
14114:
14100:
14065:
14053:
14025:
14013:
13973:
13959:
13940:raising and lowering operators
13917:
13903:
13858:
13846:
13810:
13798:
13765:
13753:
13718:
13706:
13674:
13662:
13633:
13621:
13566:
13552:
13313:
13299:
13242:
13236:
13202:
13196:
13153:
13147:
13117:
13111:
13030:
13004:
12967:
12941:
12904:
12878:
12551:
12545:
12510:
12504:
12443:
12431:
12423:
12417:
12357:
12351:
12222:
12214:
12179:
12171:
11875:
11863:
11850:
11838:
11825:
11813:
11751:
11745:
11674:
11662:
11610:{\displaystyle =Y\in F\cdot Y}
11586:
11574:
11282:
11268:
11074:
11047:
11033:
10966:
10954:
10928:
10916:
10813:of dimension 2 over any field
10752:
10735:
10666:
10649:
10621:
10604:
10570:
10564:
10536:
10530:
10505:
10499:
10465:
10451:
10398:
10392:
10367:
10353:
10325:
10319:
10259:). This is a Lie algebra over
10210:
10196:
10166:
10160:
10146:of a matrix). Its Lie algebra
10060:(with respect to the standard
10018:
10004:
9976:
9970:
9904:
9890:
9862:
9848:
9815:
9801:
9771:
9757:
9729:
9715:
9687:{\displaystyle X^{\rm {T}}=-X}
9643:
9629:
9599:
9593:
9563:
9557:
9440:
9434:
9380:
9374:
9343:
9329:
9299:
9285:
9241:
9227:
9197:
9183:
9151:
9137:
9066:
9052:
9010:
9002:
8989:
8986:
8978:
8942:
8928:
8801:
8795:
8769:
8761:
8748:
8745:
8737:
8661:
8647:
8589:
8577:
8518:
8512:
8497:
8475:
8467:
8451:
8445:
8368:
8360:
8298:
8284:
8233:
8227:
8197:
8191:
8128:
8118:
8098:
8088:
8070:
8060:
8024:
8014:
7982:
7972:
7940:
7930:
7915:
7882:
7870:
7864:
7858:
7753:
7750:
7744:
7732:
7726:
7717:
7711:
7705:
7699:
7696:
7684:
7681:
7648:
7591:
7581:
7518:
7512:
7504:
7457:
7451:
7422:
7416:
7347:{\displaystyle C^{\infty }(X)}
7341:
7335:
7283:
7266:
7261:
7255:
7252:
7237:
7231:
7225:
7222:
7207:
7201:
7192:
7189:
7174:
7116:
7110:
7032:
7006:
6893:
6887:
6872:
6866:
6857:
6848:
6825:{\displaystyle D\colon A\to A}
6816:
6770:
6765:semidirect sum of Lie algebras
6599:
6541:
6434:
6431:
6414:
6408:
6396:
6393:
6271:
6268:
6246:
6240:
6228:
6225:
6219:
6216:
6199:
6193:
6176:
6173:
6104:
6087:
5978:Product and semidirect product
5883:
5871:
5861:
5849:
5553:
5547:
5478:
5472:
5438:
5426:
5357:
5351:
5295:
5289:
5204:
5192:
5167:
5161:
5095:
5085:
5005:
4993:
4968:
4962:
4861:
4849:
4786:
4780:
4712:of Lie algebras, the image of
4692:
4636:
4471:
4468:
4462:
4453:
4447:
4441:
4435:
4432:
4420:
4417:
4400:
4337:
4317:
4215:
4212:
4200:
4191:
4171:
4162:
4150:
4147:
4126:
4105:
4099:
4068:
4056:
4018:
4004:
3974:
3960:
3932:
3918:
3873:
3859:
3828:
3814:
3753:
3741:
3721:
3715:
3678:
3666:
3610:
3604:
3469:
3457:
3180:
3174:
3084:
3072:
3063:
3051:
2967:
2955:
2946:
2934:
2845:
2833:
2827:
2815:
2795:
2771:
2675:
2672:
2660:
2651:
2645:
2642:
2630:
2621:
2615:
2612:
2600:
2591:
2499:
2487:
2346:
2334:
2325:
2313:
2304:
2280:
2256:
2244:
2235:
2223:
2214:
2190:
2150:
2127:
2112:
1983:
1971:
1857:
1845:
1830:
1818:
1803:
1791:
1706:
1694:
1481:
1469:
1437:
1425:
1350:
1131:
1118:
524:Galilean group representations
519:Poincaré group representations
13:
1:
23146:"Wilhelm Karl Joseph Killing"
22873:Graduate Texts in Mathematics
22297:O'Connor & Robertson 2005
22285:O'Connor & Robertson 2000
22272:
22118:Automorphism of a Lie algebra
21749:{\displaystyle \mathrm {id} }
21173:{\displaystyle G_{i}/G_{i+1}}
20736:
20442:. In a related construction,
18270:For example, the Lie algebra
17866:; this is not nilpotent when
14510:{\displaystyle V\mapsto L(V)}
13744:The Lie bracket is given by:
11517:, the 1-dimensional subspace
11092:{\displaystyle x\mapsto ax+b}
10072:unitary matrices (satisfying
9703:The complex orthogonal group
3209:. In physics, a vector space
3040:. If the field does not have
2024:infinitesimal transformations
1734:This is skew-symmetric since
514:Lorentz group representations
481:Theorem of the highest weight
22835:Introduction to Lie Algebras
22674:, Theorems 2.7.5 and 3.15.1.
22258:-module with an alternating
21410:{\displaystyle :=G_{i+j+1}.}
20607:{\displaystyle \mathbb {Z} }
20464:{\displaystyle \mathbb {Q} }
19992:classification of Lie groups
19980:{\displaystyle \mathbb {R} }
19851:{\displaystyle \mathbb {R} }
19678:{\displaystyle \mathbb {R} }
19628:{\displaystyle \mathbb {R} }
18745:of characteristic zero: for
18481:{\displaystyle \mathbb {R} }
18397:{\displaystyle \mathbb {R} }
17959:; this is not abelian when
17739:{\displaystyle \mathbb {R} }
15283:universal enveloping algebra
15169:Universal enveloping algebra
15163:Universal enveloping algebra
14560:is infinite-dimensional for
14399:{\displaystyle \mathbb {C} }
14370:{\displaystyle \mathbb {R} }
13275:{\displaystyle \mathbb {C} }
12583:{\displaystyle \mathbb {R} }
12321:under matrix multiplication.
10679:that preserves the standard
10296:{\displaystyle \mathbb {C} }
10274:{\displaystyle \mathbb {R} }
10108:{\displaystyle A^{*}=A^{-1}}
9917:that preserves the standard
9168:of the general linear group
7437:Lie bracket of vector fields
7381:{\displaystyle \mathbb {R} }
5108:. Similarly, for a subspace
2007:{\displaystyle =x\times x=0}
799:{\displaystyle \mathbb {Z} }
32:Lie bracket of vector fields
7:
23319:Encyclopedia of Mathematics
23199:Lie Algebras and Lie Groups
22214:Quasi-Frobenius Lie algebra
22105:
21599:cyclic-permutation braiding
20478:
19258:is solvable if and only if
12370:is the 1-dimensional ideal
11617:. Both of the Lie algebras
11295:consisting of all matrices
8249:
7604:. That is, a derivation of
6290:This is the product in the
4842:if their bracket vanishes:
3382:The Lie algebra of matrices
3207:dimension as a vector space
2050:Definition of a Lie algebra
1727:{\displaystyle =x\times y.}
1543:, there is a corresponding
951:Unique factorization domain
10:
23371:
23331:McKenzie, Douglas (2015).
23225:Varadarajan, Veeravalli S.
23201:(2nd ed.). Springer.
23068:Princeton University Press
23027:Cambridge University Press
22791:
22235:
22159:Lie algebra representation
20641:-group are finite abelian
19770:
19340:algebraically closed field
17992:
16089:angular momentum operators
15557:generated by the elements
15166:
14904:for some positive integer
14575:Lie algebra representation
14572:
12456:is abelian, isomorphic to
12192:is the Lie algebra of the
10985:.) Over the real numbers,
10817:, up to isomorphism. Here
7470:is the Lie algebra of the
7144:is the Lie algebra of the
5373:
3841:is the Lie algebra of the
3387:On an associative algebra
2017:
711:Tensor product of algebras
466:Lie algebra representation
29:
22924:10.1007/978-3-319-13467-3
22881:10.1007/978-1-4612-0979-9
22612:Erdmann & Wildon 2006
22501:Erdmann & Wildon 2006
22462:Erdmann & Wildon 2006
22450:Erdmann & Wildon 2006
22428:, section 2.10, Remark 2.
21481:category of vector spaces
20688:as well as over the ring
19963:equivalence of categories
17084:the adjoint endomorphism
15089:by the Jacobi identity.)
11381:is given by the matrices
10715:. Its Lie algebra is the
10681:alternating bilinear form
9742:, its identity component
8037:, and the Lie algebra of
6945:.) Given two derivations
6386:commute with each other:
4670:first isomorphism theorem
2690:{\displaystyle ]+]+]=0\ }
23035:10.1017/CBO9780511526008
22723:Fulton & Harris 1991
22414:Fulton & Harris 1991
21180:), and with Lie bracket
20490:. Namely, a Lie algebra
20473:rational homotopy theory
20168:if the complexification
19608:exceptional Lie algebras
15501:{\displaystyle \otimes }
14564:of dimension at least 2.
12389:{\displaystyle F\cdot Z}
11636:{\displaystyle F\cdot Y}
11536:{\displaystyle F\cdot Y}
10252:{\displaystyle X^{*}=-X}
9541:special orthogonal group
8343:inside the linear space
8143:outer automorphism group
7439:). Informally speaking,
7404:.) This makes the space
7156:is the real numbers and
6934:{\displaystyle x,y\in A}
5916:(which is not always in
5409:of diagonal matrices in
3197:Generators and dimension
1547:Lie group, unique up to
1325:alternating bilinear map
989:Formal power series ring
939:Integrally closed domain
461:Lie group representation
23021:Khukhro, E. I. (1998),
22914:Hall, Brian C. (2015).
22833:; Wildon, Mark (2006).
22761:, Part II, section V.1.
22698:, Warnings 1.6 and 8.5.
22404:, Part I, section VI.3.
22129:Gelfand–Fuks cohomology
21769:{\displaystyle \sigma }
21496:interchange isomorphism
21494:is a vector space, the
21015:{\displaystyle G_{i+j}}
19954:double cover of SO(3).
19781:The tangent space of a
18764:{\displaystyle n\geq 2}
18533:{\displaystyle n\geq 5}
18423:{\displaystyle n\geq 2}
18331:{\displaystyle n\geq 2}
17978:{\displaystyle n\geq 3}
17885:{\displaystyle n\geq 2}
16419:{\displaystyle n\geq 0}
15228:{\displaystyle :=XY-YX}
14850:if its kernel is zero.
13052:{\displaystyle =F_{2}.}
12989:{\displaystyle =F_{1},}
12926:{\displaystyle =F_{3},}
12150:Over the real numbers,
11220:{\displaystyle a\neq 0}
10902:{\displaystyle \left=Y}
10062:Hermitian inner product
9919:symmetric bilinear form
9030:For a positive integer
8210:can be identified with
7804:is the adjoint mapping
7362:. Then a derivation of
6943:non-associative algebra
6527:. If the canonical map
5585:{\displaystyle n\geq 2}
4887:subalgebra of a subset
4732:is a Lie subalgebra of
3630:and a positive integer
3217:may be called a set of
2034:in the 1880s. The name
1602:near the identity give
1452:non-associative algebra
998:Algebraic number theory
691:Total ring of fractions
486:Borel–Weil–Bott theorem
22224:Restricted Lie algebra
22149:Lie algebra cohomology
22139:Index of a Lie algebra
22096:
21981:
21911:
21858:
21835:
21770:
21750:
21722:
21650:
21588:
21535:
21456:
21411:
21302:
21174:
21122:
21042:
21016:
20983:
20937:
20913:
20829:
20762:
20711:
20678:
20618:is sometimes called a
20608:
20586:
20508:
20465:
20434:of a simply connected
20385:
20358:
20322:
20278:
20234:
20210:
20162:
20132:
20101:
20070:Harish-Chandra modules
20062:
20042:
19981:
19920:
19896:
19876:
19852:
19826:
19819:
19799:
19767:Relation to Lie groups
19749:
19748:{\displaystyle p>3}
19720:
19679:
19657:
19629:
19600:
19549:
19498:
19441:
19317:
19252:
19224:
19197:
19111:
19068:
19026:
18980:
18938:
18896:
18860:denotes the center of
18847:
18765:
18735:
18672:
18592:
18534:
18508:
18482:
18460:
18424:
18398:
18376:
18332:
18306:
18261:
18197:
18173:
18143:
18119:
18095:
18071:
18047:
18017:
17995:Semisimple Lie algebra
17979:
17953:
17917:
17886:
17860:
17824:
17793:
17740:
17703:
17647:
17616:
17576:
17483:
17456:
17207:
17173:
17078:
17042:
16993:
16934:
16841:
16817:
16790:
16622:
16591:
16567:
16531:
16499:
16459:
16420:
16394:
16367:
16347:
16321:under addition or the
16315:
16286:
16250:
16195:
16155:
16127:of the rotation group
16121:
16072:
16039:
15992:
15961:
15918:
15872:
15808:
15771:
15747:
15697:
15637:
15601:
15600:{\displaystyle XY-YX-}
15551:
15502:
15482:
15458:
15427:
15303:
15275:
15229:
15157:Weyl character formula
15140:
15116:
15083:
15059:
14997:
14944:is the representation
14942:adjoint representation
14934:
14912:Adjoint representation
14898:
14837:
14809:
14791:sends each element of
14785:
14762:
14700:
14672:
14671:{\displaystyle =XY-YX}
14618:
14554:
14511:
14459:classical Lie algebras
14444:
14400:
14371:
14339:
14295:
14261:
14217:
14161:
14121:
14072:
14032:
14002:(for a complex number
13980:
13924:
13874:
13832:
13784:
13731:
13693:
13649:
13608:
13573:
13485:
13320:
13276:
13249:
13209:
13160:
13124:
13088:
13053:
12990:
12927:
12859:
12584:
12558:
12517:
12477:
12450:
12390:
12364:
12311:
12229:
12186:
12138:
11888:
11794:
11758:
11705:
11681:
11637:
11611:
11561:
11537:
11511:
11485:
11375:
11347:
11289:
11245:
11221:
11195:
11157:
11093:
11062:affine transformations
11054:
11003:
10979:
10941:
10903:
10861:
10835:
10807:
10785:for further examples.
10770:classical Lie algebras
10759:
10717:symplectic Lie algebra
10709:
10673:
10628:
10577:
10543:
10512:
10472:
10405:
10374:
10332:
10297:
10275:
10253:
10217:
10173:
10136:
10109:
10054:
10025:
9983:
9944:
9911:
9869:
9822:
9784:, and the Lie algebra
9778:
9736:
9688:
9650:
9606:
9570:
9529:
9500:
9447:
9416:
9387:
9350:
9306:
9248:
9204:
9158:
9112:
9073:
9017:
8949:
8906:
8886:
8776:
8696:
8668:
8620:. Given a Lie algebra
8614:
8613:{\displaystyle =XY-YX}
8564:
8537:
8402:
8375:
8333:
8305:
8240:
8204:
8166:
8147:semisimple Lie algebra
8135:
8031:
7989:
7947:
7889:
7830:
7798:
7760:
7662:
7622:
7598:
7552:
7525:
7464:
7429:
7382:
7348:
7291:
7129:of all derivations of
7123:
7085:
6993:
6966:
6935:
6900:
6826:
6757:
6701:
6665:
6637:
6613:
6567:
6521:
6497:
6473:
6447:
6380:
6341:
6312:
6281:
6157:
6074:
6029:
6000:
5968:
5941:
5914:
5907:
5612:
5586:
5560:
5524:
5485:
5445:
5403:
5364:
5322:
5302:
5260:
5240:
5131:
5102:
5062:
5038:
4932:
4911:
4874:
4832:
4793:
4756:that is isomorphic to
4750:
4726:
4706:
4662:
4616:
4577:
4553:
4511:
4370:correspond to ideals.
4357:
4301:
4264:
4222:
4178:
4112:
4075:
4025:
3981:
3939:
3880:
3835:
3778:
3777:{\displaystyle =XY-YX}
3728:
3685:
3617:
3581:
3561:
3534:
3514:
3494:
3493:{\displaystyle =xy-yx}
3444:
3421:
3401:
3368:
3331:
3307:
3279:
3255:
3187:
3143:
3094:
3034:
3010:
2980:
2908:
2884:
2858:
2802:
2751:
2727:
2691:
2563:
2539:
2515:
2459:
2435:
2403:
2383:
2353:
2266:
2164:
2096:
2073:
2008:
1958:
1938:
1918:
1876:
1769:
1728:
1678:
1642:and particle physics.
1620:
1588:
1506:
1505:{\displaystyle =xy-yx}
1444:
1412:
1392:
1364:
1313:
1155:Noncommutative algebra
1138:
1091:
1054:
1006:Algebraic number field
957:Principal ideal domain
846:
800:
738:Frobenius endomorphism
384:Semisimple Lie algebra
339:Adjoint representation
23159:Annals of Mathematics
22785:, Part I, Chapter II.
22515:, Part I, Chapter IV.
22491:, Chapters 17 and 20.
22154:Lie algebra extension
22097:
21982:
21912:
21859:
21836:
21771:
21751:
21723:
21651:
21589:
21536:
21488:braiding isomorphisms
21457:
21412:
21303:
21175:
21123:
21043:
21017:
20984:
20938:
20914:
20830:
20763:
20712:
20679:
20635:Lazard correspondence
20609:
20587:
20509:
20466:
20386:
20359:
20323:
20279:
20235:
20211:
20163:
20133:
20107:, a real Lie algebra
20102:
20063:
20043:
19996:representation theory
19982:
19921:
19897:
19877:
19853:
19820:
19800:
19780:
19750:
19721:
19680:
19658:
19630:
19601:
19550:
19499:
19442:
19318:
19253:
19225:
19198:
19112:
19069:
19027:
18981:
18939:
18897:
18848:
18766:
18736:
18673:
18593:
18535:
18509:
18483:
18461:
18425:
18399:
18377:
18333:
18307:
18262:
18198:
18174:
18144:
18120:
18096:
18072:
18048:
18018:
17989:Simple and semisimple
17980:
17954:
17918:
17887:
17861:
17825:
17794:
17741:
17704:
17648:
17617:
17577:
17484:
17457:
17208:
17174:
17079:
17043:
16994:
16935:
16842:
16818:
16791:
16623:
16592:
16568:
16532:
16500:
16460:
16435:commutator subalgebra
16421:
16395:
16393:{\displaystyle F^{n}}
16368:
16348:
16316:
16287:
16251:
16196:
16156:
16122:
16073:
16040:
15993:
15962:
15919:
15873:
15809:
15772:
15748:
15698:
15638:
15602:
15552:
15503:
15483:
15459:
15428:
15304:
15276:
15230:
15185:as a Lie algebra (by
15141:
15117:
15084:
15060:
14998:
14935:
14899:
14838:
14815:to a linear map from
14810:
14786:
14763:
14701:
14673:
14619:
14584:Given a vector space
14555:
14512:
14445:
14401:
14372:
14340:
14296:
14262:
14218:
14162:
14127:is isomorphic to the
14122:
14073:
14071:{\displaystyle (c-2)}
14046:-eigenspace into the
14033:
14031:{\displaystyle (c+2)}
13981:
13925:
13875:
13833:
13831:{\displaystyle =-2F,}
13785:
13732:
13694:
13650:
13609:
13574:
13486:
13321:
13277:
13250:
13210:
13166:is equivalent to the
13161:
13125:
13089:
13054:
12991:
12928:
12860:
12585:
12559:
12518:
12478:
12476:{\displaystyle F^{2}}
12451:
12391:
12365:
12312:
12230:
12187:
12139:
11889:
11795:
11793:{\displaystyle X,Y,Z}
11759:
11706:
11682:
11638:
11612:
11562:
11538:
11512:
11486:
11376:
11348:
11290:
11246:
11222:
11196:
11158:
11094:
11055:
11004:
10980:
10942:
10904:
10862:
10836:
10808:
10760:
10710:
10674:
10629:
10578:
10544:
10513:
10483:special unitary group
10473:
10406:
10375:
10333:
10298:
10276:
10254:
10218:
10174:
10137:
10135:{\displaystyle A^{*}}
10110:
10055:
10026:
9984:
9945:
9912:
9870:
9823:
9779:
9737:
9689:
9651:
9607:
9571:
9530:
9501:
9448:
9417:
9388:
9351:
9307:
9254:consists of all real
9249:
9205:
9159:
9113:
9079:consists of all real
9074:
9018:
8950:
8907:
8887:
8777:
8697:
8669:
8615:
8565:
8538:
8403:
8376:
8334:
8306:
8241:
8205:
8167:
8136:
8032:
7990:
7948:
7890:
7831:
7799:
7761:
7759:{\displaystyle D()=+}
7663:
7623:
7599:
7553:
7526:
7465:
7430:
7383:
7358:on a smooth manifold
7349:
7292:
7124:
7086:
6994:
6992:{\displaystyle D_{2}}
6967:
6965:{\displaystyle D_{1}}
6936:
6901:
6827:
6758:
6702:
6666:
6638:
6614:
6568:
6522:
6498:
6479:be a Lie algebra and
6474:
6448:
6381:
6342:
6313:
6282:
6280:{\displaystyle =(,).}
6158:
6075:
6030:
6001:
5982:For two Lie algebras
5969:
5942:
5908:
5620:
5613:
5587:
5561:
5525:
5486:
5446:
5404:
5365:
5323:
5303:
5266:is a Lie subalgebra,
5261:
5241:
5132:
5103:
5063:
5044:. The centralizer of
5039:
4933:
4912:
4875:
4833:
4794:
4751:
4727:
4725:{\displaystyle \phi }
4707:
4668:of Lie algebras. The
4663:
4617:
4578:
4554:
4524:of Lie algebras is a
4512:
4358:
4302:
4265:
4236:is a linear subspace
4223:
4184:need not be equal to
4179:
4113:
4076:
4026:
3987:. The Lie bracket on
3982:
3940:
3881:
3836:
3797:is the real numbers,
3779:
3729:
3686:
3618:
3582:
3562:
3535:
3515:
3500:. With this bracket,
3495:
3445:
3422:
3402:
3369:
3332:
3308:
3280:
3256:
3188:
3144:
3095:
3035:
3011:
2981:
2979:{\displaystyle =-,\ }
2909:
2885:
2859:
2803:
2752:
2728:
2726:{\displaystyle x,y,z}
2692:
2564:
2540:
2516:
2460:
2436:
2434:{\displaystyle x,y,z}
2404:
2384:
2354:
2267:
2265:{\displaystyle =a+b,}
2165:
2097:
2074:
2009:
1959:
1939:
1919:
1877:
1770:
1729:
1679:
1621:
1589:
1507:
1445:
1413:
1393:
1370:, that satisfies the
1365:
1314:
1139:
1092:
1055:
847:
801:
453:Representation theory
21997:
21927:
21871:
21848:
21783:
21760:
21735:
21663:
21604:
21548:
21501:
21431:
21314:
21184:
21136:
21062:
21026:
20993:
20947:
20927:
20839:
20772:
20746:
20692:
20659:
20629:(for a prime number
20596:
20534:
20526:with an alternating
20494:
20475:in algebraic terms.
20453:
20371:
20332:
20288:
20244:
20220:
20172:
20148:
20111:
20087:
20052:
20024:
19969:
19906:
19886:
19862:
19840:
19809:
19789:
19733:
19689:
19667:
19643:
19617:
19565:
19514:
19457:
19403:
19262:
19238:
19210:
19124:
19097:
19036:
18994:
18948:
18906:
18864:
18778:
18749:
18703:
18602:
18544:
18518:
18492:
18470:
18434:
18408:
18404:is simple for every
18386:
18350:
18316:
18312:is simple for every
18274:
18207:
18183:
18159:
18129:
18105:
18081:
18057:
18033:
18003:
17963:
17927:
17896:
17870:
17834:
17803:
17761:
17728:
17657:
17626:
17589:
17496:
17469:
17230:
17193:
17091:
17064:
17003:
16947:
16854:
16827:
16803:
16644:
16636:lower central series
16608:
16577:
16541:
16509:
16469:
16445:
16404:
16377:
16357:
16328:
16296:
16272:
16236:
16169:
16131:
16095:
16049:
16006:
15978:
15928:
15882:
15818:
15785:
15757:
15711:
15647:
15611:
15561:
15528:
15516:-vector spaces. Let
15492:
15468:
15444:
15320:
15289:
15242:
15189:
15126:
15102:
15069:
15010:
14951:
14920:
14916:For any Lie algebra
14866:
14823:
14795:
14784:{\displaystyle \pi }
14775:
14717:
14686:
14632:
14592:
14553:{\displaystyle L(V)}
14535:
14486:
14410:
14388:
14359:
14305:
14271:
14227:
14174:
14135:
14087:
14050:
14010:
13946:
13890:
13843:
13795:
13783:{\displaystyle =2E,}
13750:
13703:
13692:{\displaystyle =-2F}
13659:
13618:
13586:
13539:
13334:
13286:
13264:
13223:
13183:
13134:
13098:
13069:
13001:
12938:
12875:
12600:
12572:
12566:rotation group SO(3)
12532:
12487:
12460:
12400:
12374:
12334:
12241:
12199:
12154:
11909:
11810:
11772:
11728:
11691:
11647:
11621:
11571:
11547:
11521:
11501:
11387:
11361:
11301:
11255:
11231:
11205:
11171:
11114:
11068:
11013:
10989:
10951:
10913:
10871:
10845:
10821:
10793:
10722:
10687:
10638:
10593:
10553:
10522:
10488:
10419:
10384:
10342:
10311:
10285:
10263:
10227:
10183:
10150:
10119:
10076:
10035:
9993:
9962:
9925:
9879:
9840:
9788:
9746:
9707:
9660:
9616:
9580:
9546:
9510:
9465:
9426:
9397:
9366:
9316:
9312:and its Lie algebra
9272:
9214:
9172:
9126:
9093:
9041:
9036:special linear group
8959:
8919:
8896:
8786:
8718:
8682:
8624:
8574:
8550:
8415:
8392:
8347:
8319:
8267:
8214:
8176:
8156:
8045:
7999:
7957:
7899:
7840:
7808:
7778:
7675:
7632:
7608:
7566:
7538:
7494:
7472:diffeomorphism group
7443:
7408:
7370:
7322:
7308:being a derivation.
7171:
7137:into a Lie algebra.
7095:
7003:
6976:
6949:
6913:
6842:
6804:
6711:
6675:
6651:
6623:
6577:
6531:
6507:
6483:
6459:
6390:
6351:
6322:
6298:
6170:
6084:
6045:
6010:
5986:
5954:
5920:
5625:
5596:
5592:. For example, when
5570:
5534:
5503:
5459:
5413:
5382:
5332:
5312:
5270:
5250:
5142:
5121:
5075:
5048:
4943:
4922:
4891:
4846:
4806:
4760:
4736:
4716:
4676:
4626:
4590:
4585:quotient Lie algebra
4563:
4539:
4384:
4314:
4277:
4240:
4188:
4144:
4088:
4043:
3991:
3949:
3905:
3848:
3843:general linear group
3801:
3738:
3695:
3653:
3591:
3571:
3551:
3524:
3504:
3454:
3431:
3411:
3391:
3358:
3350:Abelian Lie algebras
3317:
3293:
3265:
3241:
3161:
3111:
3048:
3020:
2994:
2931:
2894:
2868:
2812:
2768:
2737:
2705:
2588:
2549:
2529:
2514:{\displaystyle =0\ }
2484:
2445:
2413:
2393:
2367:
2352:{\displaystyle =a+b}
2277:
2187:
2109:
2086:
2058:
1968:
1948:
1928:
1893:
1782:
1738:
1691:
1649:
1606:
1574:
1466:
1422:
1402:
1382:
1376:algebra over a field
1330:
1299:
1161:Noncommutative rings
1110:
1072:
1035:
879:Non-associative ring
815:
788:
745:Algebraic structures
23130:"Marius Sophus Lie"
22958:Humphreys, James E.
22737:, Corollary II.6.2.
22416:, Proposition D.40.
21041:{\displaystyle i,j}
20761:{\displaystyle x,y}
20733:over the integers.
20082:complex Lie algebra
19928:Lie's third theorem
19755:were classified by
18507:{\displaystyle n=3}
18346:). The Lie algebra
16599:commutator subgroup
16441:) of a Lie algebra
15974:Representations of
15867:
15842:
15703:. It satisfies the
14350:Infinite dimensions
14038:-eigenspace, while
13873:{\displaystyle =H.}
13648:{\displaystyle =2E}
12396:, and the quotient
10860:{\displaystyle X,Y}
10634:is the subgroup of
10144:conjugate transpose
9989:is the subgroup of
9875:is the subgroup of
9166:commutator subgroup
9122:. More abstractly,
8546:The Lie bracket of
8255:Matrix Lie algebras
8145:of a group.) For a
7388:is equivalent to a
6999:, their commutator
6832:that satisfies the
6446:{\displaystyle =0.}
6163:, with Lie bracket
5611:{\displaystyle n=2}
5530:is not an ideal in
5019: for all
4177:{\displaystyle ,z]}
3093:{\displaystyle =-.}
3009:{\displaystyle x,y}
2883:{\displaystyle x,y}
2857:{\displaystyle +=0}
2382:{\displaystyle a,b}
2044:infinitesimal group
1553:Lie's third theorem
1456:associative algebra
920:Commutative algebra
759:Associative algebra
641:Algebraic structure
598:Table of Lie groups
439:Compact Lie algebra
23195:Serre, Jean-Pierre
22335:, §1.2. Example 2.
22323:, §1.2. Example 1.
22113:Affine Lie algebra
22092:
21977:
21907:
21854:
21831:
21766:
21746:
21718:
21646:
21584:
21531:
21452:
21407:
21298:
21170:
21118:
21086:
21038:
21012:
20979:
20933:
20909:
20825:
20758:
20727:groups of Lie type
20707:
20674:
20604:
20582:
20504:
20461:
20413:graded Lie algebra
20381:
20354:
20318:
20274:
20230:
20206:
20158:
20128:
20097:
20058:
20038:
19977:
19936:locally isomorphic
19916:
19892:
19872:
19848:
19827:
19815:
19795:
19757:Richard Earl Block
19745:
19716:
19675:
19653:
19625:
19596:
19545:
19494:
19437:
19332:Levi decomposition
19313:
19248:
19220:
19193:
19107:
19083:Cartan's criterion
19078:Cartan's criterion
19064:
19022:
18976:
18934:
18892:
18843:
18761:
18731:
18668:
18588:
18530:
18504:
18478:
18456:
18430:. The Lie algebra
18420:
18394:
18372:
18328:
18302:
18257:
18193:
18169:
18139:
18115:
18091:
18067:
18043:
18013:
17975:
17949:
17913:
17882:
17856:
17820:
17789:
17736:
17722:Lie correspondence
17699:
17643:
17612:
17572:
17479:
17452:
17203:
17169:
17074:
17038:
16989:
16930:
16837:
16813:
16786:
16618:
16587:
16563:
16527:
16495:
16455:
16439:derived subalgebra
16416:
16390:
16363:
16343:
16311:
16282:
16246:
16191:
16151:
16117:
16068:
16035:
15998:are equivalent to
15988:
15957:
15914:
15868:
15846:
15821:
15804:
15767:
15743:
15693:
15633:
15597:
15547:
15498:
15478:
15454:
15423:
15299:
15271:
15225:
15136:
15112:
15079:
15055:
14993:
14930:
14894:
14833:
14805:
14781:
14758:
14696:
14668:
14614:
14550:
14523:on a vector space
14507:
14440:
14396:
14382:Kac–Moody algebras
14367:
14335:
14291:
14257:
14213:
14157:
14117:
14068:
14028:
13976:
13920:
13870:
13828:
13780:
13730:{\displaystyle =H}
13727:
13689:
13645:
13604:
13569:
13481:
13472:
13424:
13376:
13316:
13272:
13245:
13205:
13156:
13120:
13084:
13049:
12986:
12923:
12855:
12842:
12759:
12676:
12580:
12554:
12513:
12473:
12446:
12386:
12360:
12307:
12301:
12225:
12182:
12134:
12121:
12048:
11975:
11884:
11790:
11754:
11723:Heisenberg algebra
11701:
11677:
11633:
11607:
11557:
11533:
11507:
11481:
11472:
11426:
11371:
11343:
11334:
11285:
11241:
11217:
11191:
11153:
11147:
11089:
11064:of the real line,
11050:
10999:
10978:{\displaystyle =0}
10975:
10940:{\displaystyle =0}
10937:
10899:
10857:
10831:
10803:
10755:
10705:
10669:
10624:
10573:
10549:. Its Lie algebra
10539:
10508:
10468:
10401:
10370:
10328:
10293:
10271:
10249:
10213:
10169:
10132:
10105:
10050:
10021:
9979:
9940:
9907:
9865:
9818:
9774:
9732:
9684:
9646:
9602:
9566:
9525:
9496:
9443:
9412:
9383:
9346:
9302:
9244:
9210:. Its Lie algebra
9200:
9154:
9108:
9069:
9013:
8945:
8902:
8882:
8864:
8834:
8772:
8704:identity component
8692:
8676:matrix exponential
8664:
8610:
8560:
8533:
8483: smooth
8398:
8371:
8329:
8301:
8236:
8200:
8162:
8131:
8027:
7985:
7943:
7885:
7826:
7794:
7774:associated to any
7756:
7658:
7618:
7594:
7548:
7521:
7460:
7425:
7396:. (A vector field
7378:
7344:
7287:
7146:automorphism group
7119:
7081:
6989:
6962:
6931:
6896:
6822:
6753:
6697:
6661:
6645:semidirect product
6633:
6609:
6563:
6517:
6493:
6469:
6443:
6376:
6337:
6308:
6277:
6153:
6070:
6025:
5996:
5964:
5937:
5903:
5901:
5893:
5817:
5766:
5706:
5667:
5608:
5582:
5556:
5520:
5497:compact Lie groups
5481:
5441:
5399:
5360:
5318:
5298:
5256:
5236:
5127:
5098:
5058:
5034:
4928:
4907:
4873:{\displaystyle =0}
4870:
4828:
4789:
4746:
4722:
4702:
4658:
4612:
4573:
4549:
4507:
4353:
4297:
4260:
4218:
4174:
4108:
4071:
4021:
3977:
3935:
3876:
3831:
3774:
3724:
3681:
3613:
3577:
3557:
3530:
3510:
3490:
3443:{\displaystyle xy}
3440:
3417:
3397:
3364:
3327:
3303:
3275:
3261:means a subset of
3251:
3237:for a Lie algebra
3183:
3139:
3090:
3030:
3006:
2976:
2904:
2880:
2854:
2798:
2747:
2723:
2687:
2559:
2535:
2511:
2455:
2431:
2399:
2379:
2349:
2262:
2160:
2092:
2069:
2004:
1954:
1934:
1914:
1889:, and each vector
1887:rotations of space
1872:
1765:
1724:
1674:
1616:
1584:
1502:
1440:
1408:
1388:
1360:
1309:
1174:Semiprimitive ring
1134:
1087:
1050:
858:Related structures
842:
796:
732:Inner automorphism
718:Ring homomorphisms
370:Affine Lie algebra
360:Simple Lie algebra
101:Special orthogonal
23238:978-0-387-90969-1
23208:978-3-540-55008-2
23106:978-0-8218-4876-0
23060:Knapp, Anthony W.
23005:978-0-486-63832-4
22975:978-0-387-90053-7
22890:978-0-387-97495-8
22814:978-3-540-64242-8
22799:Bourbaki, Nicolas
22626:, Theorem 3.16.3.
22219:Quasi-Lie algebra
21857:{\displaystyle A}
21071:
20936:{\displaystyle G}
20653:p-adic Lie groups
20548:
20542:
20440:Whitehead product
20436:topological space
20430:For example, the
20216:is isomorphic to
20061:{\displaystyle G}
19902:with Lie algebra
19895:{\displaystyle G}
19818:{\displaystyle x}
19798:{\displaystyle x}
18741:is reductive for
17709:abelian for each
16373:is isomorphic to
16366:{\displaystyle F}
15520:be the two-sided
14682:of a Lie algebra
13435:
13387:
13172:quantum mechanics
12850:
12129:
11567:, by the formula
11510:{\displaystyle F}
11104:The affine group
8905:{\displaystyle X}
8863:
8833:
8508:
8484:
8401:{\displaystyle I}
8183:
8165:{\displaystyle V}
8110:
8080:
8052:
8039:outer derivations
8006:
7964:
7922:
7573:
7510:
7449:
7414:
7102:
6129:
5495:in the theory of
5491:, analogous to a
5453:Cartan subalgebra
5321:{\displaystyle S}
5259:{\displaystyle S}
5223:
5219:
5215:
5130:{\displaystyle S}
5020:
5016:
4931:{\displaystyle S}
4778:
4484:
4480:
4476:
4221:{\displaystyle ]}
3580:{\displaystyle V}
3560:{\displaystyle F}
3545:endomorphism ring
3533:{\displaystyle A}
3513:{\displaystyle A}
3420:{\displaystyle F}
3400:{\displaystyle A}
3367:{\displaystyle V}
3354:Any vector space
2975:
2919:Anticommutativity
2686:
2538:{\displaystyle x}
2510:
2409:and all elements
2402:{\displaystyle F}
2095:{\displaystyle F}
1957:{\displaystyle v}
1937:{\displaystyle v}
1868:
1862:
1838:
1811:
1640:quantum mechanics
1454:. However, every
1411:{\displaystyle y}
1391:{\displaystyle x}
1258:
1257:
1215:Geometric algebra
926:Commutative rings
777:Category of rings
634:
633:
434:Split Lie algebra
397:Cartan subalgebra
259:
258:
150:Simple Lie groups
16:(Redirected from
23362:
23336:
23327:
23309:
23304:. Archived from
23284:
23250:
23220:
23190:
23149:
23133:
23117:
23088:
23055:
23017:
22992:Jacobson, Nathan
22987:
22967:
22953:
22910:
22856:
22826:
22786:
22780:
22774:
22768:
22762:
22756:
22750:
22744:
22738:
22732:
22726:
22720:
22711:
22705:
22699:
22693:
22687:
22684:Varadarajan 1984
22681:
22675:
22672:Varadarajan 1984
22669:
22663:
22657:
22651:
22645:
22639:
22636:Varadarajan 1984
22633:
22627:
22624:Varadarajan 1984
22621:
22615:
22609:
22603:
22597:
22591:
22585:
22579:
22573:
22567:
22561:
22555:
22549:
22543:
22537:
22528:
22522:
22516:
22510:
22504:
22498:
22492:
22486:
22477:
22471:
22465:
22464:, section 3.2.1.
22459:
22453:
22447:
22441:
22435:
22429:
22426:Varadarajan 1984
22423:
22417:
22411:
22405:
22399:
22393:
22390:Varadarajan 1984
22387:
22381:
22375:
22369:
22368:, section I.1.1.
22363:
22357:
22351:
22345:
22342:
22336:
22330:
22324:
22318:
22312:
22306:
22300:
22294:
22288:
22282:
22267:
22249:commutative ring
22245:
22184:Lie superalgebra
22101:
22099:
22098:
22093:
22082:
22081:
22063:
22046:
21986:
21984:
21983:
21978:
21958:
21916:
21914:
21913:
21908:
21863:
21861:
21860:
21855:
21840:
21838:
21837:
21832:
21775:
21773:
21772:
21767:
21755:
21753:
21752:
21747:
21745:
21727:
21725:
21724:
21719:
21711:
21682:
21655:
21653:
21652:
21647:
21593:
21591:
21590:
21585:
21540:
21538:
21537:
21532:
21493:
21461:
21459:
21458:
21453:
21451:
21443:
21438:
21416:
21414:
21413:
21408:
21403:
21402:
21360:
21359:
21338:
21337:
21307:
21305:
21304:
21299:
21297:
21296:
21275:
21270:
21269:
21251:
21250:
21235:
21230:
21229:
21217:
21216:
21201:
21196:
21195:
21179:
21177:
21176:
21171:
21169:
21168:
21153:
21148:
21147:
21127:
21125:
21124:
21119:
21117:
21116:
21101:
21096:
21095:
21085:
21047:
21045:
21044:
21039:
21021:
21019:
21018:
21013:
21011:
21010:
20989:is contained in
20988:
20986:
20985:
20982:{\displaystyle }
20980:
20975:
20974:
20962:
20961:
20942:
20940:
20939:
20934:
20918:
20916:
20915:
20910:
20902:
20901:
20883:
20882:
20870:
20869:
20857:
20856:
20834:
20832:
20831:
20826:
20818:
20817:
20805:
20804:
20767:
20765:
20764:
20759:
20723:Claude Chevalley
20716:
20714:
20713:
20708:
20706:
20705:
20700:
20683:
20681:
20680:
20675:
20673:
20672:
20667:
20613:
20611:
20610:
20605:
20603:
20591:
20589:
20588:
20583:
20581:
20580:
20571:
20570:
20561:
20560:
20546:
20540:
20513:
20511:
20510:
20505:
20503:
20502:
20485:commutative ring
20470:
20468:
20467:
20462:
20460:
20448:rational numbers
20417:Lie superalgebra
20390:
20388:
20387:
20382:
20380:
20379:
20363:
20361:
20360:
20355:
20344:
20343:
20327:
20325:
20324:
20319:
20314:
20300:
20299:
20283:
20281:
20280:
20275:
20270:
20256:
20255:
20239:
20237:
20236:
20231:
20229:
20228:
20215:
20213:
20212:
20207:
20205:
20200:
20199:
20198:
20188:
20187:
20182:
20181:
20167:
20165:
20164:
20159:
20157:
20156:
20138:is said to be a
20137:
20135:
20134:
20129:
20127:
20126:
20121:
20120:
20106:
20104:
20103:
20098:
20096:
20095:
20067:
20065:
20064:
20059:
20047:
20045:
20044:
20039:
20037:
19986:
19984:
19983:
19978:
19976:
19959:simply connected
19952:simply connected
19925:
19923:
19922:
19917:
19915:
19914:
19901:
19899:
19898:
19893:
19881:
19879:
19878:
19873:
19871:
19870:
19857:
19855:
19854:
19849:
19847:
19824:
19822:
19821:
19816:
19804:
19802:
19801:
19796:
19754:
19752:
19751:
19746:
19725:
19723:
19722:
19717:
19715:
19710:
19709:
19708:
19698:
19697:
19684:
19682:
19681:
19676:
19674:
19662:
19660:
19659:
19654:
19652:
19651:
19637:simple Lie group
19634:
19632:
19631:
19626:
19624:
19605:
19603:
19602:
19597:
19577:
19576:
19554:
19552:
19551:
19546:
19526:
19525:
19503:
19501:
19500:
19495:
19469:
19468:
19446:
19444:
19443:
19438:
19415:
19414:
19322:
19320:
19319:
19314:
19300:
19299:
19290:
19289:
19277:
19276:
19257:
19255:
19254:
19249:
19247:
19246:
19234:. A Lie algebra
19229:
19227:
19226:
19221:
19219:
19218:
19202:
19200:
19199:
19194:
19116:
19114:
19113:
19108:
19106:
19105:
19073:
19071:
19070:
19065:
19048:
19047:
19031:
19029:
19028:
19023:
19006:
19005:
18985:
18983:
18982:
18977:
18960:
18959:
18943:
18941:
18940:
18935:
18918:
18917:
18901:
18899:
18898:
18893:
18876:
18875:
18852:
18850:
18849:
18844:
18824:
18823:
18790:
18789:
18770:
18768:
18767:
18762:
18740:
18738:
18737:
18732:
18715:
18714:
18677:
18675:
18674:
18669:
18658:
18657:
18636:
18635:
18614:
18613:
18597:
18595:
18594:
18589:
18578:
18577:
18556:
18555:
18539:
18537:
18536:
18531:
18513:
18511:
18510:
18505:
18487:
18485:
18484:
18479:
18477:
18465:
18463:
18462:
18457:
18446:
18445:
18429:
18427:
18426:
18421:
18403:
18401:
18400:
18395:
18393:
18381:
18379:
18378:
18373:
18362:
18361:
18338:and every field
18337:
18335:
18334:
18329:
18311:
18309:
18308:
18303:
18286:
18285:
18266:
18264:
18263:
18258:
18256:
18255:
18250:
18249:
18233:
18232:
18227:
18226:
18216:
18215:
18202:
18200:
18199:
18194:
18192:
18191:
18178:
18176:
18175:
18170:
18168:
18167:
18148:
18146:
18145:
18140:
18138:
18137:
18124:
18122:
18121:
18116:
18114:
18113:
18100:
18098:
18097:
18092:
18090:
18089:
18076:
18074:
18073:
18068:
18066:
18065:
18052:
18050:
18049:
18044:
18042:
18041:
18022:
18020:
18019:
18014:
18012:
18011:
17984:
17982:
17981:
17976:
17958:
17956:
17955:
17950:
17939:
17938:
17922:
17920:
17919:
17914:
17912:
17911:
17906:
17905:
17891:
17889:
17888:
17883:
17865:
17863:
17862:
17857:
17846:
17845:
17829:
17827:
17826:
17821:
17819:
17818:
17813:
17812:
17798:
17796:
17795:
17790:
17773:
17772:
17745:
17743:
17742:
17737:
17735:
17708:
17706:
17705:
17700:
17698:
17697:
17686:
17685:
17678:
17673:
17672:
17667:
17666:
17652:
17650:
17649:
17644:
17642:
17641:
17636:
17635:
17621:
17619:
17618:
17613:
17611:
17610:
17599:
17598:
17581:
17579:
17578:
17573:
17568:
17567:
17558:
17557:
17552:
17551:
17535:
17534:
17529:
17528:
17518:
17517:
17512:
17511:
17488:
17486:
17485:
17480:
17478:
17477:
17461:
17459:
17458:
17453:
17436:
17435:
17426:
17425:
17410:
17409:
17400:
17399:
17378:
17377:
17368:
17367:
17352:
17351:
17342:
17341:
17317:
17316:
17307:
17306:
17291:
17290:
17281:
17280:
17262:
17261:
17252:
17251:
17239:
17238:
17212:
17210:
17209:
17204:
17202:
17201:
17178:
17176:
17175:
17170:
17128:
17127:
17118:
17117:
17083:
17081:
17080:
17075:
17073:
17072:
17047:
17045:
17044:
17039:
17037:
17036:
17025:
17024:
17017:
17012:
17011:
16998:
16996:
16995:
16990:
16988:
16987:
16976:
16975:
16968:
16963:
16962:
16957:
16956:
16939:
16937:
16936:
16931:
16926:
16925:
16916:
16915:
16910:
16909:
16893:
16892:
16887:
16886:
16876:
16875:
16870:
16869:
16846:
16844:
16843:
16838:
16836:
16835:
16822:
16820:
16819:
16814:
16812:
16811:
16795:
16793:
16792:
16787:
16776:
16775:
16763:
16762:
16750:
16749:
16740:
16739:
16718:
16717:
16705:
16704:
16695:
16694:
16676:
16675:
16666:
16665:
16653:
16652:
16627:
16625:
16624:
16619:
16617:
16616:
16596:
16594:
16593:
16588:
16586:
16585:
16572:
16570:
16569:
16564:
16562:
16561:
16536:
16534:
16533:
16530:{\displaystyle }
16528:
16504:
16502:
16501:
16498:{\displaystyle }
16496:
16491:
16490:
16481:
16480:
16464:
16462:
16461:
16456:
16454:
16453:
16425:
16423:
16422:
16417:
16399:
16397:
16396:
16391:
16389:
16388:
16372:
16370:
16369:
16364:
16352:
16350:
16349:
16344:
16342:
16341:
16336:
16320:
16318:
16317:
16312:
16310:
16309:
16304:
16291:
16289:
16288:
16283:
16281:
16280:
16255:
16253:
16252:
16247:
16245:
16244:
16200:
16198:
16197:
16192:
16181:
16180:
16160:
16158:
16157:
16152:
16141:
16126:
16124:
16123:
16118:
16107:
16106:
16077:
16075:
16074:
16069:
16064:
16063:
16044:
16042:
16041:
16036:
16031:
16030:
16015:
16014:
15997:
15995:
15994:
15989:
15987:
15986:
15966:
15964:
15963:
15958:
15953:
15952:
15937:
15936:
15923:
15921:
15920:
15915:
15913:
15912:
15894:
15893:
15877:
15875:
15874:
15869:
15866:
15865:
15864:
15854:
15841:
15840:
15839:
15829:
15813:
15811:
15810:
15805:
15800:
15799:
15776:
15774:
15773:
15768:
15766:
15765:
15752:
15750:
15749:
15744:
15742:
15741:
15723:
15722:
15702:
15700:
15699:
15694:
15689:
15681:
15680:
15662:
15661:
15642:
15640:
15639:
15634:
15632:
15631:
15606:
15604:
15603:
15598:
15556:
15554:
15553:
15548:
15543:
15542:
15507:
15505:
15504:
15499:
15487:
15485:
15484:
15479:
15477:
15476:
15463:
15461:
15460:
15455:
15453:
15452:
15432:
15430:
15429:
15424:
15413:
15412:
15403:
15402:
15393:
15392:
15377:
15376:
15367:
15366:
15354:
15353:
15335:
15334:
15308:
15306:
15305:
15300:
15298:
15297:
15280:
15278:
15277:
15272:
15267:
15266:
15251:
15250:
15234:
15232:
15231:
15226:
15145:
15143:
15142:
15137:
15135:
15134:
15121:
15119:
15118:
15113:
15111:
15110:
15088:
15086:
15085:
15080:
15078:
15077:
15064:
15062:
15061:
15056:
15002:
15000:
14999:
14994:
14989:
14988:
14979:
14978:
14966:
14965:
14939:
14937:
14936:
14931:
14929:
14928:
14903:
14901:
14900:
14895:
14878:
14877:
14856:Kenkichi Iwasawa
14842:
14840:
14839:
14834:
14832:
14831:
14814:
14812:
14811:
14806:
14804:
14803:
14790:
14788:
14787:
14782:
14767:
14765:
14764:
14759:
14745:
14744:
14732:
14731:
14705:
14703:
14702:
14697:
14695:
14694:
14677:
14675:
14674:
14669:
14623:
14621:
14620:
14615:
14604:
14603:
14559:
14557:
14556:
14551:
14520:free Lie algebra
14516:
14514:
14513:
14508:
14468:is important in
14466:Virasoro algebra
14449:
14447:
14446:
14441:
14436:
14422:
14421:
14405:
14403:
14402:
14397:
14395:
14376:
14374:
14373:
14368:
14366:
14344:
14342:
14341:
14336:
14331:
14317:
14316:
14300:
14298:
14297:
14292:
14281:
14266:
14264:
14263:
14258:
14253:
14239:
14238:
14222:
14220:
14219:
14214:
14212:
14207:
14206:
14205:
14186:
14185:
14166:
14164:
14163:
14158:
14147:
14146:
14129:complexification
14126:
14124:
14123:
14118:
14113:
14099:
14098:
14083:The Lie algebra
14077:
14075:
14074:
14069:
14037:
14035:
14034:
14029:
13985:
13983:
13982:
13977:
13972:
13958:
13957:
13929:
13927:
13926:
13921:
13916:
13902:
13901:
13879:
13877:
13876:
13871:
13837:
13835:
13834:
13829:
13789:
13787:
13786:
13781:
13736:
13734:
13733:
13728:
13698:
13696:
13695:
13690:
13654:
13652:
13651:
13646:
13613:
13611:
13610:
13605:
13603:
13602:
13597:
13578:
13576:
13575:
13570:
13565:
13551:
13550:
13526:
13514:
13502:
13490:
13488:
13487:
13482:
13477:
13473:
13433:
13429:
13425:
13385:
13381:
13377:
13325:
13323:
13322:
13317:
13312:
13298:
13297:
13281:
13279:
13278:
13273:
13271:
13254:
13252:
13251:
13246:
13235:
13234:
13214:
13212:
13211:
13206:
13195:
13194:
13179:The Lie algebra
13165:
13163:
13162:
13157:
13146:
13145:
13129:
13127:
13126:
13121:
13110:
13109:
13093:
13091:
13090:
13085:
13083:
13082:
13077:
13058:
13056:
13055:
13050:
13045:
13044:
13029:
13028:
13016:
13015:
12995:
12993:
12992:
12987:
12982:
12981:
12966:
12965:
12953:
12952:
12932:
12930:
12929:
12924:
12919:
12918:
12903:
12902:
12890:
12889:
12864:
12862:
12861:
12856:
12848:
12847:
12843:
12778:
12777:
12764:
12760:
12695:
12694:
12681:
12677:
12612:
12611:
12589:
12587:
12586:
12581:
12579:
12563:
12561:
12560:
12555:
12544:
12543:
12528:The Lie algebra
12522:
12520:
12519:
12514:
12503:
12502:
12497:
12496:
12482:
12480:
12479:
12474:
12472:
12471:
12455:
12453:
12452:
12447:
12430:
12416:
12415:
12410:
12409:
12395:
12393:
12392:
12387:
12369:
12367:
12366:
12361:
12350:
12349:
12344:
12343:
12330:, the center of
12316:
12314:
12313:
12308:
12306:
12302:
12234:
12232:
12231:
12226:
12221:
12213:
12212:
12207:
12194:Heisenberg group
12191:
12189:
12188:
12183:
12178:
12170:
12169:
12164:
12163:
12143:
12141:
12140:
12135:
12127:
12126:
12122:
12053:
12049:
11980:
11976:
11901:upper-triangular
11893:
11891:
11890:
11885:
11799:
11797:
11796:
11791:
11763:
11761:
11760:
11755:
11744:
11743:
11738:
11737:
11716:Three dimensions
11710:
11708:
11707:
11702:
11700:
11699:
11686:
11684:
11683:
11678:
11661:
11656:
11655:
11642:
11640:
11639:
11634:
11616:
11614:
11613:
11608:
11566:
11564:
11563:
11558:
11556:
11555:
11542:
11540:
11539:
11534:
11516:
11514:
11513:
11508:
11490:
11488:
11487:
11482:
11477:
11473:
11431:
11427:
11380:
11378:
11377:
11372:
11370:
11369:
11352:
11350:
11349:
11344:
11339:
11335:
11294:
11292:
11291:
11286:
11281:
11267:
11266:
11250:
11248:
11247:
11242:
11240:
11239:
11226:
11224:
11223:
11218:
11200:
11198:
11197:
11192:
11190:
11162:
11160:
11159:
11154:
11152:
11148:
11098:
11096:
11095:
11090:
11059:
11057:
11056:
11051:
11046:
11032:
11008:
11006:
11005:
11000:
10998:
10997:
10984:
10982:
10981:
10976:
10946:
10944:
10943:
10938:
10908:
10906:
10905:
10900:
10892:
10888:
10866:
10864:
10863:
10858:
10840:
10838:
10837:
10832:
10830:
10829:
10812:
10810:
10809:
10804:
10802:
10801:
10764:
10762:
10761:
10756:
10751:
10734:
10733:
10714:
10712:
10711:
10706:
10704:
10703:
10695:
10678:
10676:
10675:
10670:
10665:
10648:
10633:
10631:
10630:
10625:
10620:
10603:
10588:symplectic group
10582:
10580:
10579:
10574:
10563:
10548:
10546:
10545:
10540:
10529:
10517:
10515:
10514:
10509:
10498:
10477:
10475:
10474:
10469:
10464:
10450:
10449:
10437:
10429:
10410:
10408:
10407:
10402:
10391:
10379:
10377:
10376:
10371:
10366:
10352:
10337:
10335:
10334:
10329:
10318:
10302:
10300:
10299:
10294:
10292:
10280:
10278:
10277:
10272:
10270:
10258:
10256:
10255:
10250:
10239:
10238:
10222:
10220:
10219:
10214:
10209:
10195:
10194:
10178:
10176:
10175:
10170:
10159:
10158:
10141:
10139:
10138:
10133:
10131:
10130:
10114:
10112:
10111:
10106:
10104:
10103:
10088:
10087:
10059:
10057:
10056:
10051:
10049:
10048:
10043:
10030:
10028:
10027:
10022:
10017:
10003:
9988:
9986:
9985:
9980:
9969:
9949:
9947:
9946:
9941:
9939:
9938:
9933:
9916:
9914:
9913:
9908:
9903:
9889:
9874:
9872:
9871:
9866:
9861:
9847:
9827:
9825:
9824:
9819:
9814:
9800:
9799:
9783:
9781:
9780:
9775:
9770:
9756:
9741:
9739:
9738:
9733:
9728:
9714:
9693:
9691:
9690:
9685:
9674:
9673:
9672:
9655:
9653:
9652:
9647:
9642:
9628:
9627:
9611:
9609:
9608:
9603:
9592:
9591:
9575:
9573:
9572:
9567:
9556:
9534:
9532:
9531:
9526:
9524:
9523:
9522:
9505:
9503:
9502:
9497:
9495:
9494:
9479:
9478:
9477:
9452:
9450:
9449:
9444:
9433:
9421:
9419:
9418:
9413:
9411:
9410:
9405:
9392:
9390:
9389:
9384:
9373:
9361:orthogonal group
9355:
9353:
9352:
9347:
9342:
9328:
9327:
9311:
9309:
9308:
9303:
9298:
9284:
9283:
9263:
9253:
9251:
9250:
9245:
9240:
9226:
9225:
9209:
9207:
9206:
9201:
9196:
9182:
9163:
9161:
9160:
9155:
9150:
9136:
9117:
9115:
9114:
9109:
9107:
9106:
9101:
9088:
9078:
9076:
9075:
9070:
9065:
9051:
9022:
9020:
9019:
9014:
9009:
9001:
9000:
8985:
8977:
8976:
8954:
8952:
8951:
8946:
8941:
8911:
8909:
8908:
8903:
8891:
8889:
8888:
8883:
8875:
8874:
8865:
8862:
8851:
8845:
8844:
8835:
8832:
8821:
8781:
8779:
8778:
8773:
8768:
8760:
8759:
8744:
8736:
8735:
8701:
8699:
8698:
8693:
8691:
8690:
8673:
8671:
8670:
8665:
8660:
8646:
8645:
8633:
8632:
8619:
8617:
8616:
8611:
8569:
8567:
8566:
8561:
8559:
8558:
8542:
8540:
8539:
8534:
8506:
8496:
8485:
8482:
8474:
8466:
8465:
8444:
8424:
8423:
8407:
8405:
8404:
8399:
8380:
8378:
8377:
8372:
8367:
8359:
8358:
8338:
8336:
8335:
8330:
8328:
8327:
8310:
8308:
8307:
8302:
8297:
8283:
8245:
8243:
8242:
8237:
8226:
8225:
8209:
8207:
8206:
8201:
8190:
8189:
8184:
8181:
8171:
8169:
8168:
8163:
8140:
8138:
8137:
8132:
8127:
8126:
8117:
8116:
8111:
8108:
8105:
8097:
8096:
8087:
8086:
8081:
8078:
8069:
8068:
8059:
8058:
8053:
8050:
8036:
8034:
8033:
8028:
8023:
8022:
8013:
8012:
8007:
8004:
7994:
7992:
7991:
7986:
7981:
7980:
7971:
7970:
7965:
7962:
7952:
7950:
7949:
7944:
7939:
7938:
7929:
7928:
7923:
7920:
7914:
7913:
7894:
7892:
7891:
7886:
7857:
7856:
7851:
7835:
7833:
7832:
7827:
7825:
7824:
7819:
7803:
7801:
7800:
7795:
7793:
7792:
7772:inner derivation
7765:
7763:
7762:
7757:
7667:
7665:
7664:
7659:
7657:
7656:
7647:
7646:
7628:is a linear map
7627:
7625:
7624:
7619:
7617:
7616:
7603:
7601:
7600:
7595:
7590:
7589:
7580:
7579:
7574:
7571:
7557:
7555:
7554:
7549:
7547:
7546:
7530:
7528:
7527:
7522:
7511:
7508:
7503:
7502:
7469:
7467:
7466:
7461:
7450:
7447:
7434:
7432:
7431:
7426:
7415:
7412:
7387:
7385:
7384:
7379:
7377:
7356:smooth functions
7353:
7351:
7350:
7345:
7334:
7333:
7296:
7294:
7293:
7288:
7286:
7282:
7281:
7128:
7126:
7125:
7120:
7109:
7108:
7103:
7100:
7090:
7088:
7087:
7082:
7080:
7079:
7070:
7069:
7057:
7056:
7047:
7046:
7031:
7030:
7018:
7017:
6998:
6996:
6995:
6990:
6988:
6987:
6971:
6969:
6968:
6963:
6961:
6960:
6940:
6938:
6937:
6932:
6905:
6903:
6902:
6897:
6831:
6829:
6828:
6823:
6800:is a linear map
6762:
6760:
6759:
6754:
6752:
6751:
6742:
6741:
6735:
6730:
6729:
6720:
6719:
6706:
6704:
6703:
6698:
6696:
6695:
6689:
6684:
6683:
6670:
6668:
6667:
6662:
6660:
6659:
6643:is said to be a
6642:
6640:
6639:
6634:
6632:
6631:
6618:
6616:
6615:
6610:
6608:
6607:
6598:
6597:
6591:
6586:
6585:
6572:
6570:
6569:
6564:
6562:
6561:
6555:
6550:
6549:
6540:
6539:
6526:
6524:
6523:
6518:
6516:
6515:
6502:
6500:
6499:
6494:
6492:
6491:
6478:
6476:
6475:
6470:
6468:
6467:
6452:
6450:
6449:
6444:
6430:
6385:
6383:
6382:
6377:
6375:
6374:
6373:
6360:
6359:
6346:
6344:
6343:
6338:
6336:
6332:
6331:
6317:
6315:
6314:
6309:
6307:
6306:
6286:
6284:
6283:
6278:
6267:
6256:
6215:
6192:
6162:
6160:
6159:
6154:
6152:
6151:
6150:
6137:
6127:
6123:
6122:
6103:
6079:
6077:
6076:
6071:
6069:
6068:
6067:
6054:
6053:
6034:
6032:
6031:
6026:
6024:
6023:
6022:
6005:
6003:
6002:
5997:
5995:
5994:
5973:
5971:
5970:
5965:
5963:
5962:
5946:
5944:
5943:
5938:
5936:
5935:
5930:
5929:
5912:
5910:
5909:
5904:
5902:
5898:
5897:
5826:
5822:
5821:
5771:
5770:
5716:
5712:
5711:
5710:
5672:
5671:
5617:
5615:
5614:
5609:
5591:
5589:
5588:
5583:
5565:
5563:
5562:
5557:
5546:
5545:
5529:
5527:
5526:
5521:
5519:
5518:
5513:
5512:
5490:
5488:
5487:
5482:
5471:
5470:
5450:
5448:
5447:
5442:
5425:
5424:
5408:
5406:
5405:
5400:
5398:
5397:
5392:
5391:
5369:
5367:
5366:
5361:
5350:
5349:
5348:
5342:
5341:
5327:
5325:
5324:
5319:
5307:
5305:
5304:
5299:
5288:
5287:
5286:
5280:
5279:
5265:
5263:
5262:
5257:
5245:
5243:
5242:
5237:
5221:
5220:
5217:
5213:
5188:
5187:
5160:
5159:
5158:
5152:
5151:
5136:
5134:
5133:
5128:
5107:
5105:
5104:
5099:
5094:
5093:
5084:
5083:
5067:
5065:
5064:
5059:
5057:
5056:
5043:
5041:
5040:
5035:
5021:
5018:
5014:
4989:
4988:
4961:
4960:
4959:
4953:
4952:
4937:
4935:
4934:
4929:
4916:
4914:
4913:
4908:
4906:
4905:
4879:
4877:
4876:
4871:
4837:
4835:
4834:
4829:
4827:
4826:
4798:
4796:
4795:
4790:
4779:
4776:
4774:
4769:
4768:
4755:
4753:
4752:
4747:
4745:
4744:
4731:
4729:
4728:
4723:
4711:
4709:
4708:
4703:
4701:
4700:
4691:
4690:
4667:
4665:
4664:
4659:
4657:
4656:
4650:
4645:
4644:
4635:
4634:
4621:
4619:
4618:
4613:
4611:
4610:
4604:
4599:
4598:
4582:
4580:
4579:
4574:
4572:
4571:
4558:
4556:
4555:
4550:
4548:
4547:
4516:
4514:
4513:
4508:
4503:
4502:
4482:
4481:
4478:
4474:
4409:
4408:
4399:
4398:
4368:normal subgroups
4362:
4360:
4359:
4354:
4349:
4348:
4336:
4335:
4326:
4325:
4306:
4304:
4303:
4298:
4296:
4295:
4286:
4285:
4269:
4267:
4266:
4261:
4259:
4258:
4249:
4248:
4227:
4225:
4224:
4219:
4183:
4181:
4180:
4175:
4117:
4115:
4114:
4109:
4098:
4080:
4078:
4077:
4072:
4055:
4054:
4035:. For any field
4030:
4028:
4027:
4022:
4017:
4003:
4002:
3986:
3984:
3983:
3978:
3973:
3959:
3944:
3942:
3941:
3936:
3931:
3917:
3916:
3885:
3883:
3882:
3877:
3872:
3858:
3840:
3838:
3837:
3832:
3827:
3813:
3812:
3783:
3781:
3780:
3775:
3733:
3731:
3730:
3725:
3714:
3713:
3708:
3707:
3690:
3688:
3687:
3682:
3665:
3664:
3622:
3620:
3619:
3614:
3603:
3602:
3586:
3584:
3583:
3578:
3566:
3564:
3563:
3558:
3539:
3537:
3536:
3531:
3519:
3517:
3516:
3511:
3499:
3497:
3496:
3491:
3449:
3447:
3446:
3441:
3426:
3424:
3423:
3418:
3406:
3404:
3403:
3398:
3373:
3371:
3370:
3365:
3336:
3334:
3333:
3328:
3326:
3325:
3313:. Equivalently,
3312:
3310:
3309:
3304:
3302:
3301:
3284:
3282:
3281:
3276:
3274:
3273:
3260:
3258:
3257:
3252:
3250:
3249:
3192:
3190:
3189:
3184:
3173:
3172:
3148:
3146:
3145:
3140:
3138:
3137:
3099:
3097:
3096:
3091:
3039:
3037:
3036:
3031:
3029:
3028:
3015:
3013:
3012:
3007:
2985:
2983:
2982:
2977:
2973:
2913:
2911:
2910:
2905:
2903:
2902:
2889:
2887:
2886:
2881:
2863:
2861:
2860:
2855:
2807:
2805:
2804:
2801:{\displaystyle }
2799:
2756:
2754:
2753:
2748:
2746:
2745:
2732:
2730:
2729:
2724:
2696:
2694:
2693:
2688:
2684:
2568:
2566:
2565:
2560:
2558:
2557:
2544:
2542:
2541:
2536:
2520:
2518:
2517:
2512:
2508:
2464:
2462:
2461:
2456:
2454:
2453:
2440:
2438:
2437:
2432:
2408:
2406:
2405:
2400:
2388:
2386:
2385:
2380:
2363:for all scalars
2358:
2356:
2355:
2350:
2271:
2269:
2268:
2263:
2169:
2167:
2166:
2161:
2159:
2158:
2149:
2148:
2139:
2138:
2104:binary operation
2102:together with a
2101:
2099:
2098:
2093:
2078:
2076:
2075:
2070:
2068:
2067:
2013:
2011:
2010:
2005:
1963:
1961:
1960:
1955:
1943:
1941:
1940:
1935:
1923:
1921:
1920:
1915:
1913:
1912:
1907:
1881:
1879:
1878:
1873:
1866:
1860:
1836:
1809:
1774:
1772:
1771:
1766:
1733:
1731:
1730:
1725:
1683:
1681:
1680:
1675:
1673:
1672:
1667:
1658:
1657:
1625:
1623:
1622:
1617:
1615:
1614:
1593:
1591:
1590:
1585:
1583:
1582:
1525:smooth manifolds
1511:
1509:
1508:
1503:
1449:
1447:
1446:
1443:{\displaystyle }
1441:
1417:
1415:
1414:
1409:
1397:
1395:
1394:
1389:
1369:
1367:
1366:
1361:
1359:
1358:
1349:
1348:
1339:
1338:
1318:
1316:
1315:
1310:
1308:
1307:
1289:
1284:
1283:
1280:
1279:
1276:
1250:
1243:
1236:
1221:Operator algebra
1207:Clifford algebra
1143:
1141:
1140:
1135:
1130:
1129:
1117:
1096:
1094:
1093:
1088:
1086:
1085:
1080:
1059:
1057:
1056:
1051:
1049:
1048:
1043:
1021:Ring of integers
1015:
1012:Integers modulo
963:Euclidean domain
851:
849:
848:
843:
841:
833:
828:
805:
803:
802:
797:
795:
699:Product of rings
685:Fractional ideal
644:
636:
635:
626:
619:
612:
569:Claude Chevalley
426:Complexification
269:Other Lie groups
155:
154:
63:Classical groups
55:
37:
36:
21:
23370:
23369:
23365:
23364:
23363:
23361:
23360:
23359:
23340:
23339:
23312:
23291:
23273:
23239:
23209:
23172:10.2307/1970725
23154:Quillen, Daniel
23142:Robertson, E.F.
23126:Robertson, E.F.
23107:
23078:
23045:
23006:
22976:
22934:
22891:
22861:Fulton, William
22845:
22815:
22794:
22789:
22781:
22777:
22769:
22765:
22757:
22753:
22745:
22741:
22733:
22729:
22721:
22714:
22706:
22702:
22694:
22690:
22682:
22678:
22670:
22666:
22662:, section IV.6.
22658:
22654:
22650:, Ch. III, § 9.
22646:
22642:
22634:
22630:
22622:
22618:
22614:, Theorem 12.1.
22610:
22606:
22602:, Ch. III, § 5.
22598:
22594:
22586:
22582:
22574:
22570:
22566:, section II.3.
22562:
22558:
22554:, section 17.3.
22550:
22546:
22542:, Theorem 10.9.
22538:
22531:
22523:
22519:
22511:
22507:
22499:
22495:
22487:
22480:
22476:, Example 3.27.
22472:
22468:
22460:
22456:
22448:
22444:
22436:
22432:
22424:
22420:
22412:
22408:
22400:
22396:
22388:
22384:
22376:
22372:
22364:
22360:
22352:
22348:
22343:
22339:
22331:
22327:
22319:
22315:
22307:
22303:
22295:
22291:
22283:
22279:
22275:
22270:
22246:
22242:
22238:
22233:
22229:Serre relations
22199:Pre-Lie algebra
22194:Poisson algebra
22144:Leibniz algebra
22108:
22077:
22073:
22056:
22039:
21998:
21995:
21994:
21951:
21928:
21925:
21924:
21872:
21869:
21868:
21849:
21846:
21845:
21784:
21781:
21780:
21761:
21758:
21757:
21738:
21736:
21733:
21732:
21704:
21675:
21664:
21661:
21660:
21605:
21602:
21601:
21549:
21546:
21545:
21502:
21499:
21498:
21491:
21490:are needed. If
21473:category theory
21469:
21447:
21439:
21434:
21432:
21429:
21428:
21386:
21382:
21349:
21345:
21327:
21323:
21315:
21312:
21311:
21280:
21276:
21271:
21259:
21255:
21240:
21236:
21231:
21225:
21221:
21206:
21202:
21197:
21191:
21187:
21185:
21182:
21181:
21158:
21154:
21149:
21143:
21139:
21137:
21134:
21133:
21106:
21102:
21097:
21091:
21087:
21075:
21063:
21060:
21059:
21027:
21024:
21023:
21000:
20996:
20994:
20991:
20990:
20970:
20966:
20957:
20953:
20948:
20945:
20944:
20928:
20925:
20924:
20897:
20893:
20878:
20874:
20865:
20861:
20852:
20848:
20840:
20837:
20836:
20810:
20806:
20797:
20793:
20773:
20770:
20769:
20747:
20744:
20743:
20739:
20719:p-adic integers
20701:
20696:
20695:
20693:
20690:
20689:
20668:
20663:
20662:
20660:
20657:
20656:
20599:
20597:
20594:
20593:
20576:
20575:
20566:
20565:
20556:
20555:
20535:
20532:
20531:
20498:
20497:
20495:
20492:
20491:
20481:
20456:
20454:
20451:
20450:
20432:homotopy groups
20409:
20375:
20374:
20372:
20369:
20368:
20336:
20335:
20333:
20330:
20329:
20310:
20292:
20291:
20289:
20286:
20285:
20266:
20248:
20247:
20245:
20242:
20241:
20224:
20223:
20221:
20218:
20217:
20201:
20194:
20193:
20189:
20183:
20177:
20176:
20175:
20173:
20170:
20169:
20152:
20151:
20149:
20146:
20145:
20122:
20116:
20115:
20114:
20112:
20109:
20108:
20091:
20090:
20088:
20085:
20084:
20078:
20053:
20050:
20049:
20033:
20025:
20022:
20021:
19972:
19970:
19967:
19966:
19940:universal cover
19910:
19909:
19907:
19904:
19903:
19887:
19884:
19883:
19866:
19865:
19863:
19860:
19859:
19843:
19841:
19838:
19837:
19810:
19807:
19806:
19790:
19787:
19786:
19775:
19769:
19734:
19731:
19730:
19711:
19704:
19703:
19699:
19693:
19692:
19690:
19687:
19686:
19670:
19668:
19665:
19664:
19647:
19646:
19644:
19641:
19640:
19620:
19618:
19615:
19614:
19569:
19568:
19566:
19563:
19562:
19560:
19518:
19517:
19515:
19512:
19511:
19509:
19461:
19460:
19458:
19455:
19454:
19452:
19407:
19406:
19404:
19401:
19400:
19398:
19392:
19388:
19384:
19380:
19376:
19372:
19366:
19360:
19354:
19328:
19295:
19294:
19285:
19284:
19272:
19271:
19263:
19260:
19259:
19242:
19241:
19239:
19236:
19235:
19214:
19213:
19211:
19208:
19207:
19125:
19122:
19121:
19101:
19100:
19098:
19095:
19094:
19080:
19040:
19039:
19037:
19034:
19033:
18998:
18997:
18995:
18992:
18991:
18952:
18951:
18949:
18946:
18945:
18910:
18909:
18907:
18904:
18903:
18868:
18867:
18865:
18862:
18861:
18816:
18815:
18782:
18781:
18779:
18776:
18775:
18750:
18747:
18746:
18707:
18706:
18704:
18701:
18700:
18650:
18649:
18628:
18627:
18606:
18605:
18603:
18600:
18599:
18570:
18569:
18548:
18547:
18545:
18542:
18541:
18519:
18516:
18515:
18493:
18490:
18489:
18473:
18471:
18468:
18467:
18438:
18437:
18435:
18432:
18431:
18409:
18406:
18405:
18389:
18387:
18384:
18383:
18354:
18353:
18351:
18348:
18347:
18317:
18314:
18313:
18278:
18277:
18275:
18272:
18271:
18251:
18245:
18244:
18243:
18228:
18222:
18221:
18220:
18211:
18210:
18208:
18205:
18204:
18187:
18186:
18184:
18181:
18180:
18163:
18162:
18160:
18157:
18156:
18133:
18132:
18130:
18127:
18126:
18109:
18108:
18106:
18103:
18102:
18085:
18084:
18082:
18079:
18078:
18061:
18060:
18058:
18055:
18054:
18037:
18036:
18034:
18031:
18030:
18007:
18006:
18004:
18001:
18000:
17997:
17991:
17964:
17961:
17960:
17931:
17930:
17928:
17925:
17924:
17907:
17901:
17900:
17899:
17897:
17894:
17893:
17871:
17868:
17867:
17838:
17837:
17835:
17832:
17831:
17814:
17808:
17807:
17806:
17804:
17801:
17800:
17765:
17764:
17762:
17759:
17758:
17731:
17729:
17726:
17725:
17687:
17681:
17680:
17679:
17674:
17668:
17662:
17661:
17660:
17658:
17655:
17654:
17637:
17631:
17630:
17629:
17627:
17624:
17623:
17622:is an ideal in
17600:
17594:
17593:
17592:
17590:
17587:
17586:
17563:
17562:
17553:
17547:
17546:
17545:
17530:
17524:
17523:
17522:
17513:
17507:
17506:
17505:
17497:
17494:
17493:
17473:
17472:
17470:
17467:
17466:
17431:
17430:
17421:
17420:
17405:
17404:
17395:
17394:
17373:
17372:
17363:
17362:
17347:
17346:
17337:
17336:
17312:
17311:
17302:
17301:
17286:
17285:
17276:
17275:
17257:
17256:
17247:
17246:
17234:
17233:
17231:
17228:
17227:
17197:
17196:
17194:
17191:
17190:
17123:
17122:
17113:
17112:
17092:
17089:
17088:
17068:
17067:
17065:
17062:
17061:
17054:Engel's theorem
17026:
17020:
17019:
17018:
17013:
17007:
17006:
17004:
17001:
17000:
16977:
16971:
16970:
16969:
16964:
16958:
16952:
16951:
16950:
16948:
16945:
16944:
16921:
16920:
16911:
16905:
16904:
16903:
16888:
16882:
16881:
16880:
16871:
16865:
16864:
16863:
16855:
16852:
16851:
16831:
16830:
16828:
16825:
16824:
16807:
16806:
16804:
16801:
16800:
16771:
16770:
16758:
16757:
16745:
16744:
16735:
16734:
16713:
16712:
16700:
16699:
16690:
16689:
16671:
16670:
16661:
16660:
16648:
16647:
16645:
16642:
16641:
16612:
16611:
16609:
16606:
16605:
16581:
16580:
16578:
16575:
16574:
16557:
16556:
16542:
16539:
16538:
16510:
16507:
16506:
16486:
16485:
16476:
16475:
16470:
16467:
16466:
16449:
16448:
16446:
16443:
16442:
16405:
16402:
16401:
16384:
16380:
16378:
16375:
16374:
16358:
16355:
16354:
16337:
16332:
16331:
16329:
16326:
16325:
16305:
16300:
16299:
16297:
16294:
16293:
16276:
16275:
16273:
16270:
16269:
16240:
16239:
16237:
16234:
16233:
16227:solvable groups
16217:Analogously to
16215:
16207:
16173:
16172:
16170:
16167:
16166:
16134:
16132:
16129:
16128:
16099:
16098:
16096:
16093:
16092:
16084:
16059:
16058:
16050:
16047:
16046:
16026:
16025:
16010:
16009:
16007:
16004:
16003:
15982:
15981:
15979:
15976:
15975:
15948:
15947:
15932:
15931:
15929:
15926:
15925:
15908:
15904:
15889:
15885:
15883:
15880:
15879:
15860:
15856:
15855:
15850:
15835:
15831:
15830:
15825:
15819:
15816:
15815:
15795:
15794:
15786:
15783:
15782:
15761:
15760:
15758:
15755:
15754:
15753:is a basis for
15737:
15733:
15718:
15714:
15712:
15709:
15708:
15685:
15676:
15675:
15657:
15656:
15648:
15645:
15644:
15627:
15626:
15612:
15609:
15608:
15562:
15559:
15558:
15538:
15537:
15529:
15526:
15525:
15493:
15490:
15489:
15472:
15471:
15469:
15466:
15465:
15448:
15447:
15445:
15442:
15441:
15408:
15407:
15398:
15397:
15388:
15387:
15372:
15371:
15362:
15361:
15349:
15348:
15330:
15329:
15321:
15318:
15317:
15293:
15292:
15290:
15287:
15286:
15262:
15261:
15246:
15245:
15243:
15240:
15239:
15190:
15187:
15186:
15171:
15165:
15130:
15129:
15127:
15124:
15123:
15106:
15105:
15103:
15100:
15099:
15095:
15073:
15072:
15070:
15067:
15066:
15011:
15008:
15007:
14984:
14983:
14971:
14970:
14961:
14960:
14952:
14949:
14948:
14924:
14923:
14921:
14918:
14917:
14914:
14870:
14869:
14867:
14864:
14863:
14827:
14826:
14824:
14821:
14820:
14799:
14798:
14796:
14793:
14792:
14776:
14773:
14772:
14737:
14736:
14727:
14726:
14718:
14715:
14714:
14690:
14689:
14687:
14684:
14683:
14633:
14630:
14629:
14596:
14595:
14593:
14590:
14589:
14582:
14577:
14571:
14569:Representations
14536:
14533:
14532:
14487:
14484:
14483:
14461:as subalgebras.
14432:
14414:
14413:
14411:
14408:
14407:
14391:
14389:
14386:
14385:
14362:
14360:
14357:
14356:
14352:
14327:
14309:
14308:
14306:
14303:
14302:
14274:
14272:
14269:
14268:
14249:
14231:
14230:
14228:
14225:
14224:
14208:
14201:
14200:
14196:
14178:
14177:
14175:
14172:
14171:
14139:
14138:
14136:
14133:
14132:
14109:
14091:
14090:
14088:
14085:
14084:
14051:
14048:
14047:
14011:
14008:
14007:
13968:
13950:
13949:
13947:
13944:
13943:
13912:
13894:
13893:
13891:
13888:
13887:
13844:
13841:
13840:
13796:
13793:
13792:
13751:
13748:
13747:
13741:
13740:
13739:
13738:
13704:
13701:
13700:
13660:
13657:
13656:
13619:
13616:
13615:
13598:
13590:
13589:
13587:
13584:
13583:
13561:
13543:
13542:
13540:
13537:
13536:
13532:
13531:
13530:
13527:
13519:
13518:
13515:
13507:
13506:
13503:
13471:
13470:
13465:
13459:
13458:
13453:
13446:
13442:
13423:
13422:
13417:
13411:
13410:
13405:
13398:
13394:
13375:
13374:
13366:
13360:
13359:
13354:
13347:
13343:
13335:
13332:
13331:
13308:
13290:
13289:
13287:
13284:
13283:
13282:, is the space
13267:
13265:
13262:
13261:
13227:
13226:
13224:
13221:
13220:
13187:
13186:
13184:
13181:
13180:
13138:
13137:
13135:
13132:
13131:
13102:
13101:
13099:
13096:
13095:
13078:
13073:
13072:
13070:
13067:
13066:
13040:
13036:
13024:
13020:
13011:
13007:
13002:
12999:
12998:
12977:
12973:
12961:
12957:
12948:
12944:
12939:
12936:
12935:
12914:
12910:
12898:
12894:
12885:
12881:
12876:
12873:
12872:
12841:
12840:
12835:
12830:
12824:
12823:
12818:
12813:
12807:
12806:
12801:
12793:
12786:
12782:
12773:
12769:
12758:
12757:
12752:
12747:
12738:
12737:
12732:
12727:
12721:
12720:
12715:
12710:
12703:
12699:
12690:
12686:
12675:
12674:
12669:
12664:
12658:
12657:
12649:
12644:
12638:
12637:
12632:
12627:
12620:
12616:
12607:
12603:
12601:
12598:
12597:
12575:
12573:
12570:
12569:
12536:
12535:
12533:
12530:
12529:
12498:
12492:
12491:
12490:
12488:
12485:
12484:
12467:
12463:
12461:
12458:
12457:
12426:
12411:
12405:
12404:
12403:
12401:
12398:
12397:
12375:
12372:
12371:
12345:
12339:
12338:
12337:
12335:
12332:
12331:
12300:
12299:
12294:
12289:
12283:
12282:
12277:
12272:
12266:
12265:
12260:
12255:
12248:
12244:
12242:
12239:
12238:
12217:
12208:
12203:
12202:
12200:
12197:
12196:
12174:
12165:
12159:
12158:
12157:
12155:
12152:
12151:
12120:
12119:
12114:
12109:
12103:
12102:
12097:
12092:
12086:
12085:
12080:
12075:
12068:
12064:
12047:
12046:
12041:
12036:
12030:
12029:
12024:
12019:
12013:
12012:
12007:
12002:
11995:
11991:
11974:
11973:
11968:
11963:
11957:
11956:
11951:
11946:
11940:
11939:
11934:
11929:
11922:
11918:
11910:
11907:
11906:
11811:
11808:
11807:
11773:
11770:
11769:
11739:
11733:
11732:
11731:
11729:
11726:
11725:
11718:
11695:
11694:
11692:
11689:
11688:
11657:
11651:
11650:
11648:
11645:
11644:
11622:
11619:
11618:
11572:
11569:
11568:
11551:
11550:
11548:
11545:
11544:
11522:
11519:
11518:
11502:
11499:
11498:
11471:
11470:
11465:
11459:
11458:
11453:
11446:
11442:
11425:
11424:
11419:
11413:
11412:
11407:
11400:
11396:
11388:
11385:
11384:
11365:
11364:
11362:
11359:
11358:
11333:
11332:
11327:
11321:
11320:
11315:
11308:
11304:
11302:
11299:
11298:
11277:
11259:
11258:
11256:
11253:
11252:
11235:
11234:
11232:
11229:
11228:
11206:
11203:
11202:
11186:
11172:
11169:
11168:
11146:
11145:
11140:
11134:
11133:
11128:
11121:
11117:
11115:
11112:
11111:
11069:
11066:
11065:
11042:
11022:
11014:
11011:
11010:
10993:
10992:
10990:
10987:
10986:
10952:
10949:
10948:
10914:
10911:
10910:
10878:
10874:
10872:
10869:
10868:
10846:
10843:
10842:
10825:
10824:
10822:
10819:
10818:
10797:
10796:
10794:
10791:
10790:
10779:
10747:
10726:
10725:
10723:
10720:
10719:
10696:
10691:
10690:
10688:
10685:
10684:
10661:
10641:
10639:
10636:
10635:
10616:
10596:
10594:
10591:
10590:
10556:
10554:
10551:
10550:
10525:
10523:
10520:
10519:
10491:
10489:
10486:
10485:
10460:
10442:
10441:
10433:
10425:
10420:
10417:
10416:
10387:
10385:
10382:
10381:
10380:. For example,
10362:
10345:
10343:
10340:
10339:
10314:
10312:
10309:
10308:
10288:
10286:
10283:
10282:
10266:
10264:
10261:
10260:
10234:
10230:
10228:
10225:
10224:
10205:
10187:
10186:
10184:
10181:
10180:
10154:
10153:
10151:
10148:
10147:
10126:
10122:
10120:
10117:
10116:
10096:
10092:
10083:
10079:
10077:
10074:
10073:
10044:
10039:
10038:
10036:
10033:
10032:
10013:
9996:
9994:
9991:
9990:
9965:
9963:
9960:
9959:
9934:
9929:
9928:
9926:
9923:
9922:
9899:
9882:
9880:
9877:
9876:
9857:
9843:
9841:
9838:
9837:
9810:
9792:
9791:
9789:
9786:
9785:
9766:
9749:
9747:
9744:
9743:
9724:
9710:
9708:
9705:
9704:
9668:
9667:
9663:
9661:
9658:
9657:
9638:
9620:
9619:
9617:
9614:
9613:
9584:
9583:
9581:
9578:
9577:
9549:
9547:
9544:
9543:
9518:
9517:
9513:
9511:
9508:
9507:
9487:
9483:
9473:
9472:
9468:
9466:
9463:
9462:
9429:
9427:
9424:
9423:
9406:
9401:
9400:
9398:
9395:
9394:
9369:
9367:
9364:
9363:
9338:
9320:
9319:
9317:
9314:
9313:
9294:
9276:
9275:
9273:
9270:
9269:
9255:
9236:
9218:
9217:
9215:
9212:
9211:
9192:
9175:
9173:
9170:
9169:
9146:
9129:
9127:
9124:
9123:
9102:
9097:
9096:
9094:
9091:
9090:
9080:
9061:
9044:
9042:
9039:
9038:
9005:
8996:
8992:
8981:
8972:
8968:
8960:
8957:
8956:
8937:
8920:
8917:
8916:
8897:
8894:
8893:
8870:
8866:
8855:
8849:
8840:
8836:
8825:
8819:
8787:
8784:
8783:
8764:
8755:
8751:
8740:
8731:
8727:
8719:
8716:
8715:
8686:
8685:
8683:
8680:
8679:
8678:of elements of
8656:
8638:
8637:
8628:
8627:
8625:
8622:
8621:
8575:
8572:
8571:
8554:
8553:
8551:
8548:
8547:
8492:
8481:
8470:
8461:
8457:
8437:
8419:
8418:
8416:
8413:
8412:
8393:
8390:
8389:
8387:identity matrix
8363:
8354:
8350:
8348:
8345:
8344:
8323:
8322:
8320:
8317:
8316:
8293:
8276:
8268:
8265:
8264:
8257:
8252:
8218:
8217:
8215:
8212:
8211:
8185:
8180:
8179:
8177:
8174:
8173:
8157:
8154:
8153:
8122:
8121:
8112:
8107:
8106:
8101:
8092:
8091:
8082:
8077:
8076:
8064:
8063:
8054:
8049:
8048:
8046:
8043:
8042:
8018:
8017:
8008:
8003:
8002:
8000:
7997:
7996:
7995:is an ideal in
7976:
7975:
7966:
7961:
7960:
7958:
7955:
7954:
7934:
7933:
7924:
7919:
7918:
7909:
7908:
7900:
7897:
7896:
7852:
7844:
7843:
7841:
7838:
7837:
7820:
7812:
7811:
7809:
7806:
7805:
7788:
7787:
7779:
7776:
7775:
7676:
7673:
7672:
7652:
7651:
7642:
7641:
7633:
7630:
7629:
7612:
7611:
7609:
7606:
7605:
7585:
7584:
7575:
7570:
7569:
7567:
7564:
7563:
7542:
7541:
7539:
7536:
7535:
7507:
7498:
7497:
7495:
7492:
7491:
7482:of a Lie group
7446:
7444:
7441:
7440:
7411:
7409:
7406:
7405:
7373:
7371:
7368:
7367:
7329:
7325:
7323:
7320:
7319:
7277:
7273:
7264:
7172:
7169:
7168:
7104:
7099:
7098:
7096:
7093:
7092:
7075:
7071:
7065:
7061:
7052:
7048:
7042:
7038:
7026:
7022:
7013:
7009:
7004:
7001:
7000:
6983:
6979:
6977:
6974:
6973:
6956:
6952:
6950:
6947:
6946:
6914:
6911:
6910:
6843:
6840:
6839:
6805:
6802:
6801:
6773:
6747:
6746:
6737:
6736:
6731:
6725:
6724:
6715:
6714:
6712:
6709:
6708:
6691:
6690:
6685:
6679:
6678:
6676:
6673:
6672:
6655:
6654:
6652:
6649:
6648:
6627:
6626:
6624:
6621:
6620:
6603:
6602:
6593:
6592:
6587:
6581:
6580:
6578:
6575:
6574:
6557:
6556:
6551:
6545:
6544:
6535:
6534:
6532:
6529:
6528:
6511:
6510:
6508:
6505:
6504:
6487:
6486:
6484:
6481:
6480:
6463:
6462:
6460:
6457:
6456:
6423:
6391:
6388:
6387:
6366:
6365:
6364:
6355:
6354:
6352:
6349:
6348:
6327:
6326:
6325:
6323:
6320:
6319:
6302:
6301:
6299:
6296:
6295:
6260:
6249:
6208:
6185:
6171:
6168:
6167:
6143:
6142:
6141:
6130:
6118:
6117:
6096:
6085:
6082:
6081:
6060:
6059:
6058:
6049:
6048:
6046:
6043:
6042:
6015:
6014:
6013:
6011:
6008:
6007:
5990:
5989:
5987:
5984:
5983:
5980:
5958:
5957:
5955:
5952:
5951:
5931:
5925:
5924:
5923:
5921:
5918:
5917:
5900:
5899:
5892:
5891:
5886:
5865:
5864:
5844:
5834:
5833:
5824:
5823:
5816:
5815:
5807:
5798:
5797:
5789:
5776:
5775:
5765:
5764:
5756:
5747:
5746:
5738:
5725:
5724:
5717:
5705:
5704:
5699:
5693:
5692:
5687:
5677:
5676:
5666:
5665:
5660:
5654:
5653:
5648:
5638:
5637:
5636:
5632:
5628:
5626:
5623:
5622:
5597:
5594:
5593:
5571:
5568:
5567:
5538:
5537:
5535:
5532:
5531:
5514:
5508:
5507:
5506:
5504:
5501:
5500:
5463:
5462:
5460:
5457:
5456:
5417:
5416:
5414:
5411:
5410:
5393:
5387:
5386:
5385:
5383:
5380:
5379:
5376:
5344:
5343:
5337:
5336:
5335:
5333:
5330:
5329:
5328:is an ideal of
5313:
5310:
5309:
5282:
5281:
5275:
5274:
5273:
5271:
5268:
5267:
5251:
5248:
5247:
5216:
5183:
5182:
5154:
5153:
5147:
5146:
5145:
5143:
5140:
5139:
5122:
5119:
5118:
5089:
5088:
5079:
5078:
5076:
5073:
5072:
5052:
5051:
5049:
5046:
5045:
5017:
4984:
4983:
4955:
4954:
4948:
4947:
4946:
4944:
4941:
4940:
4923:
4920:
4919:
4901:
4900:
4892:
4889:
4888:
4847:
4844:
4843:
4822:
4821:
4807:
4804:
4803:
4775:
4770:
4764:
4763:
4761:
4758:
4757:
4740:
4739:
4737:
4734:
4733:
4717:
4714:
4713:
4696:
4695:
4686:
4685:
4677:
4674:
4673:
4652:
4651:
4646:
4640:
4639:
4630:
4629:
4627:
4624:
4623:
4606:
4605:
4600:
4594:
4593:
4591:
4588:
4587:
4567:
4566:
4564:
4561:
4560:
4543:
4542:
4540:
4537:
4536:
4498:
4497:
4477:
4404:
4403:
4394:
4393:
4385:
4382:
4381:
4344:
4343:
4331:
4330:
4321:
4320:
4315:
4312:
4311:
4291:
4290:
4281:
4280:
4278:
4275:
4274:
4254:
4253:
4244:
4243:
4241:
4238:
4237:
4189:
4186:
4185:
4145:
4142:
4141:
4140:, meaning that
4134:
4129:
4091:
4089:
4086:
4085:
4083:algebraic group
4047:
4046:
4044:
4041:
4040:
4013:
3995:
3994:
3992:
3989:
3988:
3969:
3952:
3950:
3947:
3946:
3927:
3909:
3908:
3906:
3903:
3902:
3886:, the group of
3868:
3851:
3849:
3846:
3845:
3823:
3805:
3804:
3802:
3799:
3798:
3739:
3736:
3735:
3709:
3700:
3699:
3698:
3696:
3693:
3692:
3657:
3656:
3654:
3651:
3650:
3634:, the space of
3595:
3594:
3592:
3589:
3588:
3572:
3569:
3568:
3552:
3549:
3548:
3525:
3522:
3521:
3505:
3502:
3501:
3455:
3452:
3451:
3432:
3429:
3428:
3412:
3409:
3408:
3392:
3389:
3388:
3384:
3359:
3356:
3355:
3352:
3347:
3321:
3320:
3318:
3315:
3314:
3297:
3296:
3294:
3291:
3290:
3289:must be all of
3269:
3268:
3266:
3263:
3262:
3245:
3244:
3242:
3239:
3238:
3199:
3165:
3164:
3162:
3159:
3158:
3115:
3114:
3112:
3109:
3108:
3107:letter such as
3049:
3046:
3045:
3024:
3023:
3021:
3018:
3017:
2995:
2992:
2991:
2932:
2929:
2928:
2898:
2897:
2895:
2892:
2891:
2869:
2866:
2865:
2813:
2810:
2809:
2769:
2766:
2765:
2741:
2740:
2738:
2735:
2734:
2706:
2703:
2702:
2589:
2586:
2585:
2576:Jacobi identity
2553:
2552:
2550:
2547:
2546:
2530:
2527:
2526:
2485:
2482:
2481:
2449:
2448:
2446:
2443:
2442:
2414:
2411:
2410:
2394:
2391:
2390:
2368:
2365:
2364:
2278:
2275:
2274:
2188:
2185:
2184:
2154:
2153:
2144:
2143:
2134:
2133:
2110:
2107:
2106:
2087:
2084:
2083:
2063:
2062:
2059:
2056:
2055:
2052:
2032:Wilhelm Killing
2020:
1969:
1966:
1965:
1949:
1946:
1945:
1929:
1926:
1925:
1908:
1903:
1902:
1894:
1891:
1890:
1783:
1780:
1779:
1739:
1736:
1735:
1692:
1689:
1688:
1668:
1663:
1662:
1653:
1652:
1650:
1647:
1646:
1610:
1609:
1607:
1604:
1603:
1578:
1577:
1575:
1572:
1571:
1549:covering spaces
1541:complex numbers
1467:
1464:
1463:
1423:
1420:
1419:
1403:
1400:
1399:
1383:
1380:
1379:
1372:Jacobi identity
1354:
1353:
1344:
1343:
1334:
1333:
1331:
1328:
1327:
1303:
1302:
1300:
1297:
1296:
1287:
1273:
1269:
1254:
1225:
1224:
1157:
1147:
1146:
1125:
1121:
1113:
1111:
1108:
1107:
1081:
1076:
1075:
1073:
1070:
1069:
1044:
1039:
1038:
1036:
1033:
1032:
1013:
983:Polynomial ring
933:Integral domain
922:
912:
911:
837:
829:
824:
816:
813:
812:
791:
789:
786:
785:
771:Involutive ring
656:
645:
639:
630:
585:
584:
583:
554:Wilhelm Killing
538:
530:
529:
528:
503:
492:
491:
490:
455:
445:
444:
443:
430:
414:
392:Dynkin diagrams
386:
376:
375:
374:
356:
334:Exponential map
323:
313:
312:
311:
292:Conformal group
271:
261:
260:
252:
244:
236:
228:
220:
201:
191:
181:
171:
152:
142:
141:
140:
121:Special unitary
65:
35:
28:
23:
22:
15:
12:
11:
5:
23368:
23358:
23357:
23352:
23338:
23337:
23328:
23310:
23308:on 2010-04-20.
23298:; et al.
23296:Kac, Victor G.
23290:
23289:External links
23287:
23286:
23285:
23272:978-0127505503
23271:
23263:Academic Press
23255:Wigner, Eugene
23251:
23237:
23221:
23207:
23191:
23166:(2): 205–295,
23150:
23134:
23118:
23105:
23089:
23076:
23056:
23043:
23018:
23004:
22988:
22974:
22954:
22933:978-3319134666
22932:
22911:
22889:
22857:
22843:
22831:Erdmann, Karin
22827:
22813:
22793:
22790:
22788:
22787:
22775:
22771:Humphreys 1978
22763:
22751:
22739:
22727:
22712:
22700:
22688:
22686:, section 2.6.
22676:
22664:
22652:
22640:
22638:, section 3.9.
22628:
22616:
22604:
22592:
22580:
22578:, section I.7.
22568:
22556:
22552:Humphreys 1978
22544:
22529:
22517:
22505:
22493:
22478:
22466:
22454:
22452:, Theorem 3.1.
22442:
22430:
22418:
22406:
22394:
22382:
22378:Humphreys 1978
22370:
22358:
22346:
22337:
22325:
22313:
22309:Humphreys 1978
22301:
22289:
22276:
22274:
22271:
22269:
22268:
22266:, Section 2)).
22264:Bourbaki (1989
22239:
22237:
22234:
22232:
22231:
22226:
22221:
22216:
22211:
22206:
22204:Quantum groups
22201:
22196:
22191:
22186:
22181:
22176:
22171:
22166:
22161:
22156:
22151:
22146:
22141:
22136:
22131:
22126:
22120:
22115:
22109:
22107:
22104:
22103:
22102:
22091:
22088:
22085:
22080:
22076:
22072:
22069:
22066:
22062:
22059:
22055:
22052:
22049:
22045:
22042:
22038:
22035:
22032:
22029:
22026:
22023:
22020:
22017:
22014:
22011:
22008:
22005:
22002:
21988:
21987:
21976:
21973:
21970:
21967:
21964:
21961:
21957:
21954:
21950:
21947:
21944:
21941:
21938:
21935:
21932:
21918:
21917:
21906:
21903:
21900:
21897:
21894:
21891:
21888:
21885:
21882:
21879:
21876:
21853:
21842:
21841:
21830:
21827:
21824:
21821:
21818:
21815:
21812:
21809:
21806:
21803:
21800:
21797:
21794:
21791:
21788:
21776:is defined by
21765:
21744:
21741:
21729:
21728:
21717:
21714:
21710:
21707:
21703:
21700:
21697:
21694:
21691:
21688:
21685:
21681:
21678:
21674:
21671:
21668:
21656:is defined as
21645:
21642:
21639:
21636:
21633:
21630:
21627:
21624:
21621:
21618:
21615:
21612:
21609:
21595:
21594:
21583:
21580:
21577:
21574:
21571:
21568:
21565:
21562:
21559:
21556:
21553:
21541:is defined by
21530:
21527:
21524:
21521:
21518:
21515:
21512:
21509:
21506:
21468:
21465:
21464:
21463:
21450:
21446:
21442:
21437:
21425:dihedral group
21420:
21419:
21418:
21417:
21406:
21401:
21398:
21395:
21392:
21389:
21385:
21381:
21378:
21375:
21372:
21369:
21366:
21363:
21358:
21355:
21352:
21348:
21344:
21341:
21336:
21333:
21330:
21326:
21322:
21319:
21295:
21292:
21289:
21286:
21283:
21279:
21274:
21268:
21265:
21262:
21258:
21254:
21249:
21246:
21243:
21239:
21234:
21228:
21224:
21220:
21215:
21212:
21209:
21205:
21200:
21194:
21190:
21167:
21164:
21161:
21157:
21152:
21146:
21142:
21130:
21129:
21128:
21115:
21112:
21109:
21105:
21100:
21094:
21090:
21084:
21081:
21078:
21074:
21070:
21067:
21054:
21053:
21037:
21034:
21031:
21009:
21006:
21003:
20999:
20978:
20973:
20969:
20965:
20960:
20956:
20952:
20932:
20908:
20905:
20900:
20896:
20892:
20889:
20886:
20881:
20877:
20873:
20868:
20864:
20860:
20855:
20851:
20847:
20844:
20824:
20821:
20816:
20813:
20809:
20803:
20800:
20796:
20792:
20789:
20786:
20783:
20780:
20777:
20757:
20754:
20751:
20738:
20735:
20704:
20699:
20686:p-adic numbers
20671:
20666:
20633:) through the
20602:
20579:
20574:
20569:
20564:
20559:
20554:
20551:
20545:
20539:
20530:-bilinear map
20501:
20480:
20477:
20459:
20444:Daniel Quillen
20408:
20405:
20378:
20353:
20350:
20347:
20342:
20339:
20317:
20313:
20309:
20306:
20303:
20298:
20295:
20273:
20269:
20265:
20262:
20259:
20254:
20251:
20227:
20204:
20197:
20192:
20186:
20180:
20155:
20125:
20119:
20094:
20077:
20074:
20057:
20036:
20032:
20029:
19975:
19913:
19891:
19869:
19846:
19814:
19794:
19771:Main article:
19768:
19765:
19744:
19741:
19738:
19714:
19707:
19702:
19696:
19673:
19650:
19623:
19595:
19592:
19589:
19586:
19583:
19580:
19575:
19572:
19556:
19544:
19541:
19538:
19535:
19532:
19529:
19524:
19521:
19505:
19493:
19490:
19487:
19484:
19481:
19478:
19475:
19472:
19467:
19464:
19448:
19436:
19433:
19430:
19427:
19424:
19421:
19418:
19413:
19410:
19394:
19390:
19386:
19382:
19378:
19374:
19368:
19362:
19356:
19350:
19327:
19326:Classification
19324:
19312:
19309:
19306:
19303:
19298:
19293:
19288:
19283:
19280:
19275:
19270:
19267:
19245:
19217:
19204:
19203:
19192:
19189:
19186:
19183:
19180:
19177:
19174:
19171:
19168:
19165:
19162:
19159:
19156:
19153:
19150:
19147:
19144:
19141:
19138:
19135:
19132:
19129:
19104:
19079:
19076:
19063:
19060:
19057:
19054:
19051:
19046:
19043:
19021:
19018:
19015:
19012:
19009:
19004:
19001:
18975:
18972:
18969:
18966:
18963:
18958:
18955:
18933:
18930:
18927:
18924:
18921:
18916:
18913:
18891:
18888:
18885:
18882:
18879:
18874:
18871:
18854:
18853:
18842:
18839:
18836:
18833:
18830:
18827:
18822:
18819:
18814:
18811:
18808:
18805:
18802:
18799:
18796:
18793:
18788:
18785:
18760:
18757:
18754:
18730:
18727:
18724:
18721:
18718:
18713:
18710:
18667:
18664:
18661:
18656:
18653:
18648:
18645:
18642:
18639:
18634:
18631:
18626:
18623:
18620:
18617:
18612:
18609:
18587:
18584:
18581:
18576:
18573:
18568:
18565:
18562:
18559:
18554:
18551:
18529:
18526:
18523:
18503:
18500:
18497:
18476:
18455:
18452:
18449:
18444:
18441:
18419:
18416:
18413:
18392:
18371:
18368:
18365:
18360:
18357:
18327:
18324:
18321:
18301:
18298:
18295:
18292:
18289:
18284:
18281:
18254:
18248:
18242:
18239:
18236:
18231:
18225:
18219:
18214:
18190:
18166:
18136:
18112:
18088:
18064:
18040:
18010:
17999:A Lie algebra
17993:Main article:
17990:
17987:
17974:
17971:
17968:
17948:
17945:
17942:
17937:
17934:
17910:
17904:
17881:
17878:
17875:
17855:
17852:
17849:
17844:
17841:
17817:
17811:
17788:
17785:
17782:
17779:
17776:
17771:
17768:
17734:
17696:
17693:
17690:
17684:
17677:
17671:
17665:
17640:
17634:
17609:
17606:
17603:
17597:
17583:
17582:
17571:
17566:
17561:
17556:
17550:
17544:
17541:
17538:
17533:
17527:
17521:
17516:
17510:
17504:
17501:
17476:
17463:
17462:
17451:
17448:
17445:
17442:
17439:
17434:
17429:
17424:
17419:
17416:
17413:
17408:
17403:
17398:
17393:
17390:
17387:
17384:
17381:
17376:
17371:
17366:
17361:
17358:
17355:
17350:
17345:
17340:
17335:
17332:
17329:
17326:
17323:
17320:
17315:
17310:
17305:
17300:
17297:
17294:
17289:
17284:
17279:
17274:
17271:
17268:
17265:
17260:
17255:
17250:
17245:
17242:
17237:
17221:derived series
17213:is said to be
17200:
17180:
17179:
17168:
17165:
17162:
17159:
17156:
17153:
17150:
17147:
17144:
17141:
17138:
17135:
17131:
17126:
17121:
17116:
17111:
17108:
17105:
17102:
17099:
17096:
17071:
17035:
17032:
17029:
17023:
17016:
17010:
16999:is central in
16986:
16983:
16980:
16974:
16967:
16961:
16955:
16941:
16940:
16929:
16924:
16919:
16914:
16908:
16902:
16899:
16896:
16891:
16885:
16879:
16874:
16868:
16862:
16859:
16834:
16810:
16797:
16796:
16785:
16782:
16779:
16774:
16769:
16766:
16761:
16756:
16753:
16748:
16743:
16738:
16733:
16730:
16727:
16724:
16721:
16716:
16711:
16708:
16703:
16698:
16693:
16688:
16685:
16682:
16679:
16674:
16669:
16664:
16659:
16656:
16651:
16615:
16604:A Lie algebra
16584:
16560:
16555:
16552:
16549:
16546:
16526:
16523:
16520:
16517:
16514:
16494:
16489:
16484:
16479:
16474:
16452:
16415:
16412:
16409:
16387:
16383:
16362:
16340:
16335:
16308:
16303:
16279:
16243:
16232:A Lie algebra
16214:
16211:
16206:
16203:
16190:
16187:
16184:
16179:
16176:
16150:
16147:
16144:
16140:
16137:
16116:
16113:
16110:
16105:
16102:
16083:
16080:
16067:
16062:
16057:
16054:
16034:
16029:
16024:
16021:
16018:
16013:
15985:
15956:
15951:
15946:
15943:
15940:
15935:
15911:
15907:
15903:
15900:
15897:
15892:
15888:
15863:
15859:
15853:
15849:
15845:
15838:
15834:
15828:
15824:
15803:
15798:
15793:
15790:
15764:
15740:
15736:
15732:
15729:
15726:
15721:
15717:
15692:
15688:
15684:
15679:
15674:
15671:
15668:
15665:
15660:
15655:
15652:
15630:
15625:
15622:
15619:
15616:
15596:
15593:
15590:
15587:
15584:
15581:
15578:
15575:
15572:
15569:
15566:
15546:
15541:
15536:
15533:
15510:tensor product
15497:
15475:
15451:
15438:tensor algebra
15434:
15433:
15422:
15419:
15416:
15411:
15406:
15401:
15396:
15391:
15386:
15383:
15380:
15375:
15370:
15365:
15360:
15357:
15352:
15347:
15344:
15341:
15338:
15333:
15328:
15325:
15296:
15270:
15265:
15260:
15257:
15254:
15249:
15224:
15221:
15218:
15215:
15212:
15209:
15206:
15203:
15200:
15197:
15194:
15167:Main article:
15164:
15161:
15148:Weyl's theorem
15133:
15109:
15094:
15091:
15076:
15054:
15051:
15048:
15045:
15042:
15039:
15036:
15033:
15030:
15027:
15024:
15021:
15018:
15015:
15004:
15003:
14992:
14987:
14982:
14977:
14974:
14969:
14964:
14959:
14956:
14927:
14913:
14910:
14893:
14890:
14887:
14884:
14881:
14876:
14873:
14830:
14802:
14780:
14769:
14768:
14757:
14754:
14751:
14748:
14743:
14740:
14735:
14730:
14725:
14722:
14693:
14680:representation
14667:
14664:
14661:
14658:
14655:
14652:
14649:
14646:
14643:
14640:
14637:
14613:
14610:
14607:
14602:
14599:
14581:
14578:
14573:Main article:
14570:
14567:
14566:
14565:
14549:
14546:
14543:
14540:
14506:
14503:
14500:
14497:
14494:
14491:
14473:
14462:
14451:
14439:
14435:
14431:
14428:
14425:
14420:
14417:
14394:
14378:
14365:
14351:
14348:
14347:
14346:
14334:
14330:
14326:
14323:
14320:
14315:
14312:
14290:
14287:
14284:
14280:
14277:
14256:
14252:
14248:
14245:
14242:
14237:
14234:
14211:
14204:
14199:
14195:
14192:
14189:
14184:
14181:
14169:tensor product
14167:, meaning the
14156:
14153:
14150:
14145:
14142:
14116:
14112:
14108:
14105:
14102:
14097:
14094:
14080:
14079:
14067:
14064:
14061:
14058:
14055:
14027:
14024:
14021:
14018:
14015:
13975:
13971:
13967:
13964:
13961:
13956:
13953:
13919:
13915:
13911:
13908:
13905:
13900:
13897:
13883:
13882:
13881:
13880:
13869:
13866:
13863:
13860:
13857:
13854:
13851:
13848:
13838:
13827:
13824:
13821:
13818:
13815:
13812:
13809:
13806:
13803:
13800:
13790:
13779:
13776:
13773:
13770:
13767:
13764:
13761:
13758:
13755:
13726:
13723:
13720:
13717:
13714:
13711:
13708:
13688:
13685:
13682:
13679:
13676:
13673:
13670:
13667:
13664:
13644:
13641:
13638:
13635:
13632:
13629:
13626:
13623:
13601:
13596:
13593:
13581:Riemann sphere
13568:
13564:
13560:
13557:
13554:
13549:
13546:
13535:The action of
13534:
13533:
13528:
13521:
13520:
13516:
13509:
13508:
13504:
13497:
13496:
13495:
13494:
13493:
13492:
13491:
13480:
13476:
13469:
13466:
13464:
13461:
13460:
13457:
13454:
13452:
13449:
13448:
13445:
13441:
13438:
13432:
13428:
13421:
13418:
13416:
13413:
13412:
13409:
13406:
13404:
13401:
13400:
13397:
13393:
13390:
13384:
13380:
13373:
13370:
13367:
13365:
13362:
13361:
13358:
13355:
13353:
13350:
13349:
13346:
13342:
13339:
13328:
13327:
13315:
13311:
13307:
13304:
13301:
13296:
13293:
13270:
13257:
13256:
13244:
13241:
13238:
13233:
13230:
13204:
13201:
13198:
13193:
13190:
13176:
13175:
13168:Spin (physics)
13155:
13152:
13149:
13144:
13141:
13119:
13116:
13113:
13108:
13105:
13081:
13076:
13062:
13061:
13060:
13059:
13048:
13043:
13039:
13035:
13032:
13027:
13023:
13019:
13014:
13010:
13006:
12996:
12985:
12980:
12976:
12972:
12969:
12964:
12960:
12956:
12951:
12947:
12943:
12933:
12922:
12917:
12913:
12909:
12906:
12901:
12897:
12893:
12888:
12884:
12880:
12867:
12866:
12865:
12853:
12846:
12839:
12836:
12834:
12831:
12829:
12826:
12825:
12822:
12819:
12817:
12814:
12812:
12809:
12808:
12805:
12802:
12800:
12797:
12794:
12792:
12789:
12788:
12785:
12781:
12776:
12772:
12767:
12763:
12756:
12753:
12751:
12748:
12746:
12743:
12740:
12739:
12736:
12733:
12731:
12728:
12726:
12723:
12722:
12719:
12716:
12714:
12711:
12709:
12706:
12705:
12702:
12698:
12693:
12689:
12684:
12680:
12673:
12670:
12668:
12665:
12663:
12660:
12659:
12656:
12653:
12650:
12648:
12645:
12643:
12640:
12639:
12636:
12633:
12631:
12628:
12626:
12623:
12622:
12619:
12615:
12610:
12606:
12592:
12591:
12578:
12553:
12550:
12547:
12542:
12539:
12525:
12524:
12512:
12509:
12506:
12501:
12495:
12470:
12466:
12445:
12442:
12439:
12436:
12433:
12429:
12425:
12422:
12419:
12414:
12408:
12385:
12382:
12379:
12359:
12356:
12353:
12348:
12342:
12326:For any field
12323:
12322:
12319:
12318:
12317:
12305:
12298:
12295:
12293:
12290:
12288:
12285:
12284:
12281:
12278:
12276:
12273:
12271:
12268:
12267:
12264:
12261:
12259:
12256:
12254:
12251:
12250:
12247:
12224:
12220:
12216:
12211:
12206:
12181:
12177:
12173:
12168:
12162:
12147:
12146:
12145:
12144:
12132:
12125:
12118:
12115:
12113:
12110:
12108:
12105:
12104:
12101:
12098:
12096:
12093:
12091:
12088:
12087:
12084:
12081:
12079:
12076:
12074:
12071:
12070:
12067:
12063:
12060:
12056:
12052:
12045:
12042:
12040:
12037:
12035:
12032:
12031:
12028:
12025:
12023:
12020:
12018:
12015:
12014:
12011:
12008:
12006:
12003:
12001:
11998:
11997:
11994:
11990:
11987:
11983:
11979:
11972:
11969:
11967:
11964:
11962:
11959:
11958:
11955:
11952:
11950:
11947:
11945:
11942:
11941:
11938:
11935:
11933:
11930:
11928:
11925:
11924:
11921:
11917:
11914:
11897:
11896:
11895:
11883:
11880:
11877:
11874:
11871:
11868:
11865:
11861:
11858:
11855:
11852:
11849:
11846:
11843:
11840:
11836:
11833:
11830:
11827:
11824:
11821:
11818:
11815:
11802:
11801:
11789:
11786:
11783:
11780:
11777:
11753:
11750:
11747:
11742:
11736:
11717:
11714:
11713:
11712:
11698:
11676:
11673:
11670:
11667:
11664:
11660:
11654:
11632:
11629:
11626:
11606:
11603:
11600:
11597:
11594:
11591:
11588:
11585:
11582:
11579:
11576:
11554:
11532:
11529:
11526:
11506:
11497:For any field
11494:
11493:
11492:
11491:
11480:
11476:
11469:
11466:
11464:
11461:
11460:
11457:
11454:
11452:
11449:
11448:
11445:
11441:
11438:
11434:
11430:
11423:
11420:
11418:
11415:
11414:
11411:
11408:
11406:
11403:
11402:
11399:
11395:
11392:
11368:
11355:
11354:
11353:
11342:
11338:
11331:
11328:
11326:
11323:
11322:
11319:
11316:
11314:
11311:
11310:
11307:
11284:
11280:
11276:
11273:
11270:
11265:
11262:
11238:
11216:
11213:
11210:
11189:
11185:
11182:
11179:
11176:
11165:
11164:
11163:
11151:
11144:
11141:
11139:
11136:
11135:
11132:
11129:
11127:
11124:
11123:
11120:
11101:
11100:
11088:
11085:
11082:
11079:
11076:
11073:
11049:
11045:
11041:
11038:
11035:
11031:
11028:
11025:
11021:
11018:
10996:
10974:
10971:
10968:
10965:
10962:
10959:
10956:
10936:
10933:
10930:
10927:
10924:
10921:
10918:
10898:
10895:
10891:
10887:
10884:
10881:
10877:
10856:
10853:
10850:
10828:
10800:
10778:
10777:Two dimensions
10775:
10774:
10773:
10766:
10754:
10750:
10746:
10743:
10740:
10737:
10732:
10729:
10702:
10699:
10694:
10668:
10664:
10660:
10657:
10654:
10651:
10647:
10644:
10623:
10619:
10615:
10612:
10609:
10606:
10602:
10599:
10584:
10572:
10569:
10566:
10562:
10559:
10538:
10535:
10532:
10528:
10507:
10504:
10501:
10497:
10494:
10479:
10467:
10463:
10459:
10456:
10453:
10448:
10445:
10440:
10436:
10432:
10428:
10424:
10400:
10397:
10394:
10390:
10369:
10365:
10361:
10358:
10355:
10351:
10348:
10327:
10324:
10321:
10317:
10291:
10269:
10248:
10245:
10242:
10237:
10233:
10212:
10208:
10204:
10201:
10198:
10193:
10190:
10168:
10165:
10162:
10157:
10129:
10125:
10102:
10099:
10095:
10091:
10086:
10082:
10047:
10042:
10020:
10016:
10012:
10009:
10006:
10002:
9999:
9978:
9975:
9972:
9968:
9952:
9951:
9937:
9932:
9906:
9902:
9898:
9895:
9892:
9888:
9885:
9864:
9860:
9856:
9853:
9850:
9846:
9817:
9813:
9809:
9806:
9803:
9798:
9795:
9773:
9769:
9765:
9762:
9759:
9755:
9752:
9731:
9727:
9723:
9720:
9717:
9713:
9700:
9699:
9683:
9680:
9677:
9671:
9666:
9645:
9641:
9637:
9634:
9631:
9626:
9623:
9601:
9598:
9595:
9590:
9587:
9565:
9562:
9559:
9555:
9552:
9521:
9516:
9493:
9490:
9486:
9482:
9476:
9471:
9442:
9439:
9436:
9432:
9409:
9404:
9382:
9379:
9376:
9372:
9357:
9345:
9341:
9337:
9334:
9331:
9326:
9323:
9301:
9297:
9293:
9290:
9287:
9282:
9279:
9264:matrices with
9243:
9239:
9235:
9232:
9229:
9224:
9221:
9199:
9195:
9191:
9188:
9185:
9181:
9178:
9153:
9149:
9145:
9142:
9139:
9135:
9132:
9105:
9100:
9068:
9064:
9060:
9057:
9054:
9050:
9047:
9012:
9008:
9004:
8999:
8995:
8991:
8988:
8984:
8980:
8975:
8971:
8967:
8964:
8944:
8940:
8936:
8933:
8930:
8927:
8924:
8901:
8881:
8878:
8873:
8869:
8861:
8858:
8854:
8848:
8843:
8839:
8831:
8828:
8824:
8818:
8815:
8812:
8809:
8806:
8803:
8800:
8797:
8794:
8791:
8782:is defined by
8771:
8767:
8763:
8758:
8754:
8750:
8747:
8743:
8739:
8734:
8730:
8726:
8723:
8689:
8663:
8659:
8655:
8652:
8649:
8644:
8641:
8636:
8631:
8609:
8606:
8603:
8600:
8597:
8594:
8591:
8588:
8585:
8582:
8579:
8557:
8544:
8543:
8532:
8529:
8526:
8523:
8520:
8517:
8514:
8511:
8505:
8502:
8499:
8495:
8491:
8488:
8480:
8477:
8473:
8469:
8464:
8460:
8456:
8453:
8450:
8447:
8443:
8440:
8436:
8433:
8430:
8427:
8422:
8397:
8370:
8366:
8362:
8357:
8353:
8326:
8300:
8296:
8292:
8289:
8286:
8282:
8279:
8275:
8272:
8256:
8253:
8251:
8248:
8235:
8232:
8229:
8224:
8221:
8199:
8196:
8193:
8188:
8161:
8130:
8125:
8120:
8115:
8104:
8100:
8095:
8090:
8085:
8075:
8072:
8067:
8062:
8057:
8026:
8021:
8016:
8011:
7984:
7979:
7974:
7969:
7942:
7937:
7932:
7927:
7917:
7912:
7907:
7904:
7884:
7881:
7878:
7875:
7872:
7869:
7866:
7863:
7860:
7855:
7850:
7847:
7823:
7818:
7815:
7791:
7786:
7783:
7768:
7767:
7755:
7752:
7749:
7746:
7743:
7740:
7737:
7734:
7731:
7728:
7725:
7722:
7719:
7716:
7713:
7710:
7707:
7704:
7701:
7698:
7695:
7692:
7689:
7686:
7683:
7680:
7655:
7650:
7645:
7640:
7637:
7615:
7593:
7588:
7583:
7578:
7545:
7520:
7517:
7514:
7506:
7501:
7486:on a manifold
7459:
7456:
7453:
7424:
7421:
7418:
7376:
7343:
7340:
7337:
7332:
7328:
7298:
7297:
7285:
7280:
7276:
7271:
7268:
7263:
7260:
7257:
7254:
7251:
7248:
7245:
7242:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7188:
7185:
7182:
7179:
7176:
7118:
7115:
7112:
7107:
7078:
7074:
7068:
7064:
7060:
7055:
7051:
7045:
7041:
7037:
7034:
7029:
7025:
7021:
7016:
7012:
7008:
6986:
6982:
6959:
6955:
6930:
6927:
6924:
6921:
6918:
6907:
6906:
6895:
6892:
6889:
6886:
6883:
6880:
6877:
6874:
6871:
6868:
6865:
6862:
6859:
6856:
6853:
6850:
6847:
6821:
6818:
6815:
6812:
6809:
6772:
6769:
6750:
6745:
6740:
6734:
6728:
6723:
6718:
6694:
6688:
6682:
6658:
6630:
6606:
6601:
6596:
6590:
6584:
6560:
6554:
6548:
6543:
6538:
6514:
6490:
6466:
6442:
6439:
6436:
6433:
6429:
6426:
6422:
6419:
6416:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6372:
6369:
6363:
6358:
6335:
6330:
6305:
6288:
6287:
6276:
6273:
6270:
6266:
6263:
6259:
6255:
6252:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6214:
6211:
6207:
6204:
6201:
6198:
6195:
6191:
6188:
6184:
6181:
6178:
6175:
6149:
6146:
6140:
6136:
6133:
6126:
6121:
6116:
6113:
6109:
6106:
6102:
6099:
6095:
6092:
6089:
6066:
6063:
6057:
6052:
6021:
6018:
5993:
5979:
5976:
5961:
5934:
5928:
5896:
5890:
5887:
5885:
5882:
5879:
5876:
5873:
5870:
5867:
5866:
5863:
5860:
5857:
5854:
5851:
5848:
5845:
5843:
5840:
5839:
5837:
5832:
5829:
5827:
5825:
5820:
5814:
5811:
5808:
5806:
5803:
5800:
5799:
5796:
5793:
5790:
5788:
5785:
5782:
5781:
5779:
5774:
5769:
5763:
5760:
5757:
5755:
5752:
5749:
5748:
5745:
5742:
5739:
5737:
5734:
5731:
5730:
5728:
5723:
5720:
5718:
5715:
5709:
5703:
5700:
5698:
5695:
5694:
5691:
5688:
5686:
5683:
5682:
5680:
5675:
5670:
5664:
5661:
5659:
5656:
5655:
5652:
5649:
5647:
5644:
5643:
5641:
5635:
5631:
5630:
5607:
5604:
5601:
5581:
5578:
5575:
5555:
5552:
5549:
5544:
5541:
5517:
5511:
5480:
5477:
5474:
5469:
5466:
5440:
5437:
5434:
5431:
5428:
5423:
5420:
5396:
5390:
5375:
5372:
5359:
5356:
5353:
5347:
5340:
5317:
5297:
5294:
5291:
5285:
5278:
5255:
5235:
5232:
5229:
5226:
5212:
5209:
5206:
5203:
5200:
5197:
5194:
5191:
5186:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5157:
5150:
5126:
5116:subalgebra of
5097:
5092:
5087:
5082:
5068:itself is the
5055:
5033:
5030:
5027:
5024:
5013:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4987:
4982:
4979:
4976:
4973:
4970:
4967:
4964:
4958:
4951:
4927:
4904:
4899:
4896:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4825:
4820:
4817:
4814:
4811:
4788:
4785:
4782:
4773:
4767:
4743:
4721:
4699:
4694:
4689:
4684:
4681:
4655:
4649:
4643:
4638:
4633:
4609:
4603:
4597:
4570:
4546:
4528:homomorphism.
4518:
4517:
4506:
4501:
4496:
4493:
4490:
4487:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4412:
4407:
4402:
4397:
4392:
4389:
4373:A Lie algebra
4364:
4363:
4352:
4347:
4342:
4339:
4334:
4329:
4324:
4319:
4294:
4289:
4284:
4257:
4252:
4247:
4234:Lie subalgebra
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4133:
4130:
4128:
4125:
4124:
4123:
4107:
4104:
4101:
4097:
4094:
4070:
4067:
4064:
4061:
4058:
4053:
4050:
4020:
4016:
4012:
4009:
4006:
4001:
3998:
3976:
3972:
3968:
3965:
3962:
3958:
3955:
3934:
3930:
3926:
3923:
3920:
3915:
3912:
3875:
3871:
3867:
3864:
3861:
3857:
3854:
3830:
3826:
3822:
3819:
3816:
3811:
3808:
3790:
3789:
3786:general linear
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3723:
3720:
3717:
3712:
3706:
3703:
3680:
3677:
3674:
3671:
3668:
3663:
3660:
3624:
3612:
3609:
3606:
3601:
3598:
3576:
3567:-vector space
3556:
3541:
3529:
3509:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3439:
3436:
3416:
3396:
3383:
3380:
3363:
3351:
3348:
3346:
3345:Basic examples
3343:
3324:
3300:
3272:
3248:
3198:
3195:
3182:
3179:
3176:
3171:
3168:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3101:
3100:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3042:characteristic
3027:
3005:
3002:
2999:
2988:
2987:
2986:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2923:
2922:
2901:
2879:
2876:
2873:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2759:
2758:
2744:
2722:
2719:
2716:
2713:
2710:
2699:
2698:
2697:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2580:
2579:
2571:
2570:
2556:
2534:
2523:
2522:
2521:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2476:
2475:
2467:
2466:
2452:
2430:
2427:
2424:
2421:
2418:
2398:
2378:
2375:
2372:
2361:
2360:
2359:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2272:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2179:
2178:
2157:
2152:
2147:
2142:
2137:
2132:
2129:
2125:
2122:
2118:
2114:
2091:
2066:
2051:
2048:
2019:
2016:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1953:
1933:
1911:
1906:
1901:
1898:
1883:
1882:
1871:
1865:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1671:
1666:
1661:
1656:
1613:
1581:
1561:classification
1557:correspondence
1523:that are also
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1439:
1436:
1433:
1430:
1427:
1407:
1387:
1357:
1352:
1347:
1342:
1337:
1306:
1256:
1255:
1253:
1252:
1245:
1238:
1230:
1227:
1226:
1218:
1217:
1189:
1188:
1182:
1176:
1170:
1158:
1153:
1152:
1149:
1148:
1145:
1144:
1133:
1128:
1124:
1120:
1116:
1097:
1084:
1079:
1060:
1047:
1042:
1030:-adic integers
1023:
1017:
1008:
994:
993:
992:
991:
985:
979:
978:
977:
965:
959:
953:
947:
941:
923:
918:
917:
914:
913:
910:
909:
908:
907:
895:
894:
893:
887:
875:
874:
873:
855:
854:
853:
852:
840:
836:
832:
827:
823:
820:
806:
794:
773:
767:
761:
755:
741:
740:
734:
728:
714:
713:
707:
701:
695:
694:
693:
687:
675:
669:
657:
655:Basic concepts
654:
653:
650:
649:
632:
631:
629:
628:
621:
614:
606:
603:
602:
601:
600:
595:
587:
586:
582:
581:
576:
574:Harish-Chandra
571:
566:
561:
556:
551:
549:Henri Poincaré
546:
540:
539:
536:
535:
532:
531:
527:
526:
521:
516:
511:
505:
504:
499:Lie groups in
498:
497:
494:
493:
489:
488:
483:
478:
473:
468:
463:
457:
456:
451:
450:
447:
446:
442:
441:
436:
431:
429:
428:
423:
417:
415:
413:
412:
407:
401:
399:
394:
388:
387:
382:
381:
378:
377:
373:
372:
367:
362:
357:
355:
354:
349:
343:
341:
336:
331:
325:
324:
319:
318:
315:
314:
310:
309:
304:
299:
297:Diffeomorphism
294:
289:
284:
279:
273:
272:
267:
266:
263:
262:
257:
256:
255:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
211:
210:
206:
205:
204:
203:
197:
193:
187:
183:
177:
173:
167:
160:
159:
153:
148:
147:
144:
143:
139:
138:
128:
118:
108:
98:
88:
81:Special linear
78:
71:General linear
67:
66:
61:
60:
57:
56:
48:
47:
26:
9:
6:
4:
3:
2:
23367:
23356:
23353:
23351:
23348:
23347:
23345:
23334:
23329:
23325:
23321:
23320:
23315:
23314:"Lie algebra"
23311:
23307:
23303:
23302:
23297:
23293:
23292:
23282:
23278:
23274:
23268:
23264:
23260:
23256:
23252:
23248:
23244:
23240:
23234:
23230:
23226:
23222:
23218:
23214:
23210:
23204:
23200:
23196:
23192:
23189:
23185:
23181:
23177:
23173:
23169:
23165:
23161:
23160:
23155:
23151:
23147:
23143:
23139:
23138:O'Connor, J.J
23135:
23131:
23127:
23123:
23122:O'Connor, J.J
23119:
23116:
23112:
23108:
23102:
23098:
23094:
23090:
23087:
23083:
23079:
23077:0-691-09089-0
23073:
23069:
23065:
23061:
23057:
23054:
23050:
23046:
23044:0-521-59717-X
23040:
23036:
23032:
23028:
23024:
23019:
23015:
23011:
23007:
23001:
22997:
22993:
22989:
22985:
22981:
22977:
22971:
22966:
22965:
22959:
22955:
22951:
22947:
22943:
22939:
22935:
22929:
22925:
22921:
22917:
22912:
22908:
22904:
22900:
22896:
22892:
22886:
22882:
22878:
22874:
22870:
22866:
22862:
22858:
22854:
22850:
22846:
22844:1-84628-040-0
22840:
22836:
22832:
22828:
22824:
22820:
22816:
22810:
22806:
22805:
22800:
22796:
22795:
22784:
22779:
22773:, section 25.
22772:
22767:
22760:
22755:
22748:
22743:
22736:
22731:
22724:
22719:
22717:
22709:
22704:
22697:
22692:
22685:
22680:
22673:
22668:
22661:
22660:Jacobson 1979
22656:
22649:
22648:Jacobson 1979
22644:
22637:
22632:
22625:
22620:
22613:
22608:
22601:
22600:Jacobson 1979
22596:
22590:, p. 24.
22589:
22588:Jacobson 1979
22584:
22577:
22576:Jacobson 1979
22572:
22565:
22564:Jacobson 1979
22560:
22553:
22548:
22541:
22536:
22534:
22526:
22525:Jacobson 1979
22521:
22514:
22509:
22502:
22497:
22490:
22485:
22483:
22475:
22470:
22463:
22458:
22451:
22446:
22439:
22434:
22427:
22422:
22415:
22410:
22403:
22398:
22392:, p. 49.
22391:
22386:
22379:
22374:
22367:
22366:Bourbaki 1989
22362:
22356:, p. 28.
22355:
22354:Jacobson 1979
22350:
22341:
22334:
22333:Bourbaki 1989
22329:
22322:
22321:Bourbaki 1989
22317:
22310:
22305:
22298:
22293:
22286:
22281:
22277:
22265:
22261:
22257:
22253:
22250:
22244:
22240:
22230:
22227:
22225:
22222:
22220:
22217:
22215:
22212:
22210:
22209:Moyal algebra
22207:
22205:
22202:
22200:
22197:
22195:
22192:
22190:
22187:
22185:
22182:
22180:
22177:
22175:
22172:
22170:
22169:Lie coalgebra
22167:
22165:
22164:Lie bialgebra
22162:
22160:
22157:
22155:
22152:
22150:
22147:
22145:
22142:
22140:
22137:
22135:
22132:
22130:
22127:
22124:
22121:
22119:
22116:
22114:
22111:
22110:
22089:
22086:
22078:
22074:
22070:
22067:
22064:
22050:
22036:
22030:
22027:
22024:
22015:
22009:
22006:
22003:
21993:
21992:
21991:
21974:
21971:
21968:
21962:
21959:
21945:
21939:
21936:
21933:
21923:
21922:
21921:
21904:
21898:
21895:
21892:
21889:
21883:
21880:
21877:
21867:
21866:
21865:
21851:
21828:
21825:
21822:
21819:
21816:
21813:
21810:
21804:
21801:
21798:
21795:
21792:
21786:
21779:
21778:
21777:
21763:
21715:
21701:
21698:
21692:
21686:
21683:
21669:
21666:
21659:
21658:
21657:
21643:
21640:
21637:
21634:
21631:
21625:
21622:
21619:
21616:
21613:
21610:
21607:
21600:
21581:
21578:
21575:
21572:
21569:
21563:
21560:
21557:
21551:
21544:
21543:
21542:
21528:
21525:
21522:
21516:
21513:
21510:
21507:
21504:
21497:
21489:
21484:
21482:
21478:
21474:
21444:
21440:
21426:
21422:
21421:
21404:
21399:
21396:
21393:
21390:
21387:
21383:
21376:
21373:
21370:
21364:
21356:
21353:
21350:
21346:
21342:
21339:
21334:
21331:
21328:
21324:
21320:
21310:
21309:
21293:
21290:
21287:
21284:
21281:
21277:
21272:
21266:
21263:
21260:
21256:
21247:
21244:
21241:
21237:
21232:
21226:
21222:
21218:
21213:
21210:
21207:
21203:
21198:
21192:
21188:
21165:
21162:
21159:
21155:
21150:
21144:
21140:
21131:
21113:
21110:
21107:
21103:
21098:
21092:
21088:
21082:
21079:
21076:
21072:
21068:
21065:
21058:
21057:
21056:
21055:
21051:
21035:
21032:
21029:
21007:
21004:
21001:
20997:
20971:
20967:
20963:
20958:
20954:
20930:
20922:
20906:
20903:
20898:
20894:
20890:
20887:
20884:
20879:
20875:
20871:
20866:
20862:
20858:
20853:
20849:
20845:
20842:
20822:
20819:
20814:
20811:
20807:
20801:
20798:
20794:
20790:
20784:
20781:
20778:
20755:
20752:
20749:
20741:
20740:
20734:
20732:
20728:
20724:
20720:
20702:
20687:
20669:
20654:
20650:
20648:
20644:
20640:
20636:
20632:
20628:
20623:
20621:
20617:
20562:
20552:
20543:
20529:
20525:
20521:
20517:
20489:
20486:
20476:
20474:
20449:
20445:
20441:
20437:
20433:
20428:
20426:
20425:chain complex
20422:
20418:
20414:
20404:
20402:
20401:
20396:
20395:
20365:
20348:
20307:
20304:
20263:
20260:
20190:
20184:
20143:
20142:
20123:
20083:
20073:
20071:
20055:
20030:
20027:
20018:
20016:
20015:homeomorphism
20011:
20009:
20005:
20000:
19997:
19993:
19988:
19964:
19960:
19955:
19953:
19949:
19945:
19941:
19937:
19933:
19929:
19889:
19834:
19832:
19812:
19792:
19784:
19779:
19774:
19764:
19762:
19758:
19742:
19739:
19736:
19727:
19700:
19638:
19611:
19609:
19590:
19587:
19584:
19581:
19559:
19539:
19536:
19533:
19530:
19508:
19488:
19485:
19482:
19479:
19476:
19473:
19451:
19431:
19428:
19425:
19422:
19419:
19397:
19371:
19365:
19359:
19353:
19348:
19344:
19341:
19336:
19333:
19323:
19310:
19307:
19291:
19278:
19265:
19233:
19232:nondegenerate
19190:
19181:
19175:
19172:
19166:
19160:
19157:
19151:
19148:
19145:
19139:
19136:
19133:
19127:
19120:
19119:
19118:
19092:
19088:
19084:
19075:
19058:
19055:
19052:
19032:, and all of
19016:
19013:
19010:
18989:
18970:
18967:
18964:
18928:
18925:
18922:
18886:
18883:
18880:
18859:
18840:
18834:
18831:
18828:
18812:
18809:
18806:
18800:
18797:
18794:
18774:
18773:
18772:
18758:
18755:
18752:
18744:
18725:
18722:
18719:
18699:For example,
18697:
18695:
18690:
18688:
18684:
18679:
18662:
18646:
18640:
18624:
18618:
18582:
18566:
18560:
18527:
18524:
18521:
18501:
18498:
18495:
18488:is simple if
18450:
18417:
18414:
18411:
18366:
18345:
18341:
18325:
18322:
18319:
18296:
18293:
18290:
18268:
18252:
18240:
18237:
18234:
18229:
18217:
18154:
18153:
18028:
18027:
17996:
17986:
17972:
17969:
17966:
17943:
17908:
17879:
17876:
17873:
17850:
17815:
17783:
17780:
17777:
17756:
17752:
17747:
17723:
17719:
17714:
17712:
17694:
17691:
17688:
17675:
17669:
17638:
17607:
17604:
17601:
17569:
17559:
17554:
17542:
17539:
17536:
17531:
17519:
17514:
17502:
17499:
17492:
17491:
17490:
17449:
17446:
17427:
17414:
17401:
17385:
17369:
17356:
17343:
17324:
17308:
17295:
17282:
17266:
17253:
17240:
17226:
17225:
17224:
17222:
17218:
17217:
17187:
17185:
17163:
17160:
17157:
17151:
17148:
17142:
17136:
17133:
17129:
17109:
17103:
17097:
17094:
17087:
17086:
17085:
17059:
17055:
17051:
17033:
17030:
17027:
17014:
16984:
16981:
16978:
16965:
16959:
16927:
16917:
16912:
16900:
16897:
16894:
16889:
16877:
16872:
16860:
16857:
16850:
16849:
16848:
16783:
16780:
16767:
16754:
16741:
16722:
16709:
16696:
16680:
16667:
16654:
16640:
16639:
16638:
16637:
16633:
16632:
16602:
16600:
16553:
16550:
16547:
16544:
16521:
16518:
16515:
16482:
16440:
16436:
16431:
16429:
16426:, meaning an
16413:
16410:
16407:
16385:
16381:
16360:
16338:
16324:
16306:
16267:
16263:
16259:
16230:
16228:
16224:
16220:
16210:
16202:
16185:
16164:
16163:hydrogen atom
16145:
16111:
16090:
16079:
16052:
16019:
16001:
15972:
15970:
15941:
15909:
15905:
15901:
15898:
15895:
15890:
15886:
15861:
15857:
15851:
15847:
15843:
15836:
15832:
15826:
15822:
15788:
15780:
15738:
15734:
15730:
15727:
15724:
15719:
15715:
15706:
15690:
15686:
15669:
15666:
15650:
15623:
15620:
15617:
15614:
15591:
15588:
15585:
15579:
15576:
15573:
15570:
15567:
15564:
15531:
15523:
15519:
15515:
15511:
15495:
15439:
15420:
15417:
15404:
15394:
15381:
15368:
15355:
15345:
15342:
15339:
15323:
15316:
15315:
15314:
15312:
15284:
15281:, called the
15255:
15238:
15222:
15219:
15216:
15213:
15210:
15207:
15201:
15198:
15195:
15184:
15180:
15177:over a field
15176:
15170:
15160:
15158:
15154:
15149:
15090:
15049:
15046:
15043:
15037:
15031:
15022:
15016:
15013:
14957:
14954:
14947:
14946:
14945:
14943:
14909:
14907:
14888:
14885:
14882:
14861:
14857:
14853:
14852:Ado's theorem
14849:
14844:
14818:
14778:
14755:
14749:
14723:
14720:
14713:
14712:
14711:
14709:
14681:
14665:
14662:
14659:
14656:
14653:
14650:
14644:
14641:
14638:
14627:
14608:
14587:
14576:
14563:
14544:
14538:
14530:
14526:
14522:
14521:
14517:, called the
14501:
14495:
14489:
14482:
14478:
14474:
14471:
14470:string theory
14467:
14463:
14460:
14456:
14455:Moyal algebra
14452:
14429:
14426:
14383:
14379:
14354:
14353:
14324:
14321:
14285:
14246:
14243:
14197:
14190:
14170:
14151:
14130:
14106:
14103:
14082:
14081:
14062:
14059:
14056:
14045:
14041:
14022:
14019:
14016:
14005:
14001:
13997:
13993:
13989:
13965:
13962:
13941:
13937:
13933:
13909:
13906:
13885:
13884:
13867:
13864:
13861:
13855:
13852:
13849:
13839:
13825:
13822:
13819:
13816:
13813:
13807:
13804:
13801:
13791:
13777:
13774:
13771:
13768:
13762:
13759:
13756:
13746:
13745:
13743:
13742:
13724:
13721:
13715:
13712:
13709:
13686:
13683:
13680:
13677:
13671:
13668:
13665:
13642:
13639:
13636:
13630:
13627:
13624:
13599:
13582:
13558:
13555:
13525:
13513:
13501:
13478:
13474:
13467:
13462:
13455:
13450:
13443:
13439:
13436:
13430:
13426:
13419:
13414:
13407:
13402:
13395:
13391:
13388:
13382:
13378:
13371:
13368:
13363:
13356:
13351:
13344:
13340:
13337:
13330:
13329:
13305:
13302:
13259:
13258:
13239:
13218:
13199:
13178:
13177:
13173:
13169:
13150:
13114:
13079:
13064:
13063:
13046:
13041:
13037:
13033:
13025:
13021:
13017:
13012:
13008:
12997:
12983:
12978:
12974:
12970:
12962:
12958:
12954:
12949:
12945:
12934:
12920:
12915:
12911:
12907:
12899:
12895:
12891:
12886:
12882:
12871:
12870:
12868:
12851:
12844:
12837:
12832:
12827:
12820:
12815:
12810:
12803:
12798:
12795:
12790:
12783:
12779:
12774:
12770:
12765:
12761:
12754:
12749:
12744:
12741:
12734:
12729:
12724:
12717:
12712:
12707:
12700:
12696:
12691:
12687:
12682:
12678:
12671:
12666:
12661:
12654:
12651:
12646:
12641:
12634:
12629:
12624:
12617:
12613:
12608:
12604:
12596:
12595:
12594:
12593:
12567:
12548:
12527:
12526:
12507:
12499:
12468:
12464:
12440:
12437:
12434:
12427:
12420:
12412:
12383:
12380:
12377:
12354:
12346:
12329:
12325:
12324:
12320:
12303:
12296:
12291:
12286:
12279:
12274:
12269:
12262:
12257:
12252:
12245:
12237:
12236:
12209:
12195:
12166:
12149:
12148:
12130:
12123:
12116:
12111:
12106:
12099:
12094:
12089:
12082:
12077:
12072:
12065:
12061:
12058:
12054:
12050:
12043:
12038:
12033:
12026:
12021:
12016:
12009:
12004:
11999:
11992:
11988:
11985:
11981:
11977:
11970:
11965:
11960:
11953:
11948:
11943:
11936:
11931:
11926:
11919:
11915:
11912:
11905:
11904:
11902:
11898:
11881:
11878:
11872:
11869:
11866:
11859:
11856:
11853:
11847:
11844:
11841:
11834:
11831:
11828:
11822:
11819:
11816:
11806:
11805:
11804:
11803:
11787:
11784:
11781:
11778:
11775:
11767:
11764:over a field
11748:
11740:
11724:
11720:
11719:
11671:
11668:
11665:
11658:
11630:
11627:
11624:
11604:
11601:
11598:
11595:
11592:
11589:
11583:
11580:
11577:
11530:
11527:
11524:
11504:
11496:
11495:
11478:
11474:
11467:
11462:
11455:
11450:
11443:
11439:
11436:
11432:
11428:
11421:
11416:
11409:
11404:
11397:
11393:
11390:
11383:
11382:
11356:
11340:
11336:
11329:
11324:
11317:
11312:
11305:
11297:
11296:
11274:
11271:
11214:
11211:
11208:
11183:
11180:
11177:
11174:
11166:
11149:
11142:
11137:
11130:
11125:
11118:
11110:
11109:
11107:
11103:
11102:
11086:
11083:
11080:
11077:
11071:
11063:
11039:
11036:
11019:
11016:
10972:
10969:
10963:
10960:
10957:
10934:
10931:
10925:
10922:
10919:
10896:
10893:
10889:
10885:
10882:
10879:
10875:
10854:
10851:
10848:
10816:
10788:
10787:
10786:
10784:
10771:
10767:
10744:
10741:
10738:
10718:
10700:
10697:
10682:
10658:
10655:
10652:
10613:
10610:
10607:
10589:
10585:
10567:
10533:
10502:
10484:
10480:
10457:
10454:
10438:
10430:
10422:
10414:
10395:
10359:
10356:
10322:
10306:
10246:
10243:
10240:
10235:
10231:
10202:
10199:
10163:
10145:
10127:
10123:
10100:
10097:
10093:
10089:
10084:
10080:
10071:
10068: ×
10067:
10063:
10045:
10010:
10007:
9973:
9958:
9957:unitary group
9954:
9953:
9935:
9920:
9896:
9893:
9854:
9851:
9835:
9831:
9807:
9804:
9763:
9760:
9721:
9718:
9702:
9701:
9697:
9681:
9678:
9675:
9664:
9635:
9632:
9596:
9560:
9542:
9538:
9514:
9491:
9488:
9484:
9480:
9469:
9460:
9456:
9437:
9407:
9377:
9362:
9358:
9335:
9332:
9291:
9288:
9267:
9262:
9259: ×
9258:
9233:
9230:
9189:
9186:
9167:
9143:
9140:
9121:
9103:
9087:
9084: ×
9083:
9058:
9055:
9037:
9033:
9029:
9028:
9027:
9024:
8997:
8993:
8973:
8969:
8965:
8962:
8934:
8931:
8925:
8922:
8913:
8899:
8879:
8876:
8871:
8867:
8859:
8856:
8852:
8846:
8841:
8837:
8829:
8826:
8822:
8816:
8813:
8810:
8807:
8804:
8798:
8792:
8789:
8756:
8752:
8732:
8728:
8724:
8721:
8713:
8709:
8705:
8677:
8653:
8650:
8634:
8607:
8604:
8601:
8598:
8595:
8592:
8586:
8583:
8580:
8530:
8524:
8521:
8515:
8509:
8503:
8500:
8489:
8486:
8478:
8462:
8458:
8454:
8448:
8441:
8438:
8434:
8431:
8425:
8411:
8410:
8409:
8395:
8388:
8384:
8355:
8351:
8342:
8314:
8290:
8287:
8273:
8270:
8262:
8247:
8230:
8194:
8186:
8159:
8150:
8148:
8144:
8113:
8102:
8083:
8073:
8055:
8040:
8009:
7967:
7925:
7905:
7902:
7879:
7876:
7873:
7867:
7861:
7853:
7821:
7784:
7781:
7773:
7747:
7741:
7738:
7735:
7729:
7723:
7720:
7714:
7708:
7702:
7693:
7690:
7687:
7678:
7671:
7670:
7669:
7638:
7635:
7576:
7561:
7558:over a field
7532:
7515:
7489:
7485:
7481:
7477:
7473:
7454:
7438:
7419:
7403:
7399:
7395:
7391:
7365:
7361:
7357:
7338:
7326:
7317:
7313:
7309:
7307:
7303:
7278:
7274:
7269:
7258:
7249:
7246:
7243:
7240:
7234:
7228:
7219:
7216:
7213:
7210:
7204:
7198:
7195:
7186:
7183:
7180:
7177:
7167:
7166:
7165:
7163:
7159:
7155:
7151:
7147:
7143:
7138:
7136:
7132:
7113:
7105:
7076:
7072:
7066:
7062:
7058:
7053:
7049:
7043:
7039:
7035:
7027:
7023:
7019:
7014:
7010:
6984:
6980:
6957:
6953:
6944:
6928:
6925:
6922:
6919:
6916:
6890:
6884:
6881:
6878:
6875:
6869:
6863:
6860:
6854:
6851:
6845:
6838:
6837:
6836:
6835:
6819:
6813:
6810:
6807:
6799:
6795:
6791:
6790:
6785:
6782:over a field
6781:
6778:
6768:
6766:
6743:
6732:
6721:
6686:
6646:
6588:
6552:
6453:
6440:
6437:
6427:
6424:
6420:
6417:
6411:
6405:
6402:
6399:
6370:
6361:
6333:
6293:
6274:
6264:
6261:
6257:
6253:
6250:
6243:
6237:
6234:
6231:
6222:
6212:
6209:
6205:
6202:
6196:
6189:
6186:
6182:
6179:
6166:
6165:
6164:
6147:
6138:
6134:
6131:
6124:
6114:
6111:
6107:
6100:
6097:
6093:
6090:
6064:
6055:
6040:
6039:
6019:
5975:
5948:
5932:
5913:
5894:
5888:
5880:
5877:
5874:
5868:
5858:
5855:
5852:
5846:
5841:
5835:
5830:
5828:
5818:
5812:
5809:
5804:
5801:
5794:
5791:
5786:
5783:
5777:
5772:
5767:
5761:
5758:
5753:
5750:
5743:
5740:
5735:
5732:
5726:
5721:
5719:
5713:
5707:
5701:
5696:
5689:
5684:
5678:
5673:
5668:
5662:
5657:
5650:
5645:
5639:
5633:
5619:
5605:
5602:
5599:
5579:
5576:
5573:
5550:
5515:
5498:
5494:
5493:maximal torus
5475:
5454:
5435:
5432:
5429:
5394:
5378:The subspace
5371:
5354:
5315:
5292:
5253:
5230:
5227:
5224:
5218: for all
5210:
5207:
5201:
5198:
5195:
5189:
5179:
5176:
5170:
5164:
5137:
5124:
5115:
5111:
5071:
5028:
5025:
5022:
5011:
5008:
5002:
4999:
4996:
4990:
4980:
4977:
4971:
4965:
4938:
4925:
4897:
4894:
4886:
4881:
4867:
4864:
4858:
4855:
4852:
4841:
4818:
4815:
4812:
4809:
4800:
4783:
4771:
4719:
4682:
4679:
4671:
4647:
4601:
4586:
4559:and an ideal
4534:
4529:
4527:
4523:
4504:
4494:
4491:
4488:
4485:
4465:
4459:
4456:
4450:
4444:
4438:
4429:
4426:
4423:
4414:
4410:
4390:
4387:
4380:
4379:
4378:
4376:
4371:
4369:
4350:
4340:
4327:
4310:
4309:
4308:
4287:
4273:
4250:
4235:
4231:
4209:
4206:
4203:
4197:
4194:
4168:
4165:
4159:
4156:
4153:
4139:
4121:
4102:
4084:
4065:
4062:
4059:
4038:
4034:
4010:
4007:
3966:
3963:
3924:
3921:
3900:
3896:
3892:
3889:
3865:
3862:
3844:
3820:
3817:
3796:
3792:
3791:
3787:
3771:
3768:
3765:
3762:
3759:
3756:
3750:
3747:
3744:
3718:
3710:
3675:
3672:
3669:
3648:
3644:
3641:
3637:
3633:
3629:
3625:
3607:
3574:
3554:
3546:
3542:
3527:
3507:
3487:
3484:
3481:
3478:
3475:
3472:
3466:
3463:
3460:
3437:
3434:
3414:
3407:over a field
3394:
3386:
3385:
3379:
3377:
3361:
3342:
3340:
3288:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3208:
3204:
3194:
3177:
3156:
3154:
3106:
3087:
3081:
3078:
3075:
3069:
3066:
3060:
3057:
3054:
3043:
3003:
3000:
2997:
2989:
2970:
2964:
2961:
2958:
2952:
2949:
2943:
2940:
2937:
2927:
2926:
2925:
2924:
2920:
2917:
2916:
2915:
2877:
2874:
2871:
2851:
2848:
2842:
2839:
2836:
2830:
2824:
2821:
2818:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2762:
2720:
2717:
2714:
2711:
2708:
2700:
2681:
2678:
2669:
2666:
2663:
2657:
2654:
2648:
2639:
2636:
2633:
2627:
2624:
2618:
2609:
2606:
2603:
2597:
2594:
2584:
2583:
2582:
2581:
2577:
2573:
2572:
2532:
2524:
2505:
2502:
2496:
2493:
2490:
2480:
2479:
2478:
2477:
2473:
2469:
2468:
2428:
2425:
2422:
2419:
2416:
2396:
2376:
2373:
2370:
2362:
2343:
2340:
2337:
2331:
2328:
2322:
2319:
2316:
2310:
2307:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2273:
2259:
2253:
2250:
2247:
2241:
2238:
2232:
2229:
2226:
2220:
2217:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2183:
2182:
2181:
2180:
2176:
2173:
2172:
2171:
2140:
2130:
2123:
2120:
2116:
2105:
2089:
2082:
2047:
2045:
2041:
2038:was given by
2037:
2033:
2029:
2025:
2015:
2001:
1998:
1995:
1992:
1989:
1986:
1980:
1977:
1974:
1951:
1931:
1909:
1899:
1896:
1888:
1869:
1863:
1854:
1851:
1848:
1842:
1839:
1833:
1827:
1824:
1821:
1815:
1812:
1806:
1800:
1797:
1794:
1788:
1785:
1778:
1777:
1776:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1721:
1718:
1715:
1712:
1709:
1703:
1700:
1697:
1687:
1686:cross product
1669:
1659:
1643:
1641:
1635:
1633:
1629:
1601:
1597:
1569:
1564:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1533:commutativity
1530:
1529:tangent space
1526:
1522:
1518:
1513:
1499:
1496:
1493:
1490:
1487:
1484:
1478:
1475:
1472:
1462:Lie bracket,
1461:
1457:
1453:
1434:
1431:
1428:
1405:
1385:
1377:
1373:
1340:
1326:
1322:
1295:
1291:
1290:
1282:
1267:
1263:
1251:
1246:
1244:
1239:
1237:
1232:
1231:
1229:
1228:
1223:
1222:
1216:
1212:
1211:
1210:
1209:
1208:
1203:
1202:
1201:
1196:
1195:
1194:
1187:
1183:
1181:
1177:
1175:
1171:
1169:
1168:Division ring
1165:
1164:
1163:
1162:
1156:
1151:
1150:
1122:
1106:
1104:
1098:
1082:
1068:
1067:-adic numbers
1066:
1061:
1045:
1031:
1029:
1024:
1022:
1018:
1016:
1009:
1007:
1003:
1002:
1001:
1000:
999:
990:
986:
984:
980:
976:
972:
971:
970:
966:
964:
960:
958:
954:
952:
948:
946:
942:
940:
936:
935:
934:
930:
929:
928:
927:
921:
916:
915:
906:
902:
901:
900:
896:
892:
888:
886:
882:
881:
880:
876:
872:
868:
867:
866:
862:
861:
860:
859:
834:
830:
821:
818:
811:
810:Terminal ring
807:
784:
780:
779:
778:
774:
772:
768:
766:
762:
760:
756:
754:
750:
749:
748:
747:
746:
739:
735:
733:
729:
727:
723:
722:
721:
720:
719:
712:
708:
706:
702:
700:
696:
692:
688:
686:
682:
681:
680:
679:Quotient ring
676:
674:
670:
668:
664:
663:
662:
661:
652:
651:
648:
643:→ Ring theory
642:
638:
637:
627:
622:
620:
615:
613:
608:
607:
605:
604:
599:
596:
594:
591:
590:
589:
588:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
550:
547:
545:
542:
541:
534:
533:
525:
522:
520:
517:
515:
512:
510:
507:
506:
502:
496:
495:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
458:
454:
449:
448:
440:
437:
435:
432:
427:
424:
422:
419:
418:
416:
411:
408:
406:
403:
402:
400:
398:
395:
393:
390:
389:
385:
380:
379:
371:
368:
366:
363:
361:
358:
353:
350:
348:
345:
344:
342:
340:
337:
335:
332:
330:
327:
326:
322:
317:
316:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
283:
280:
278:
275:
274:
270:
265:
264:
253:
247:
245:
239:
237:
231:
229:
223:
221:
215:
214:
213:
212:
208:
207:
202:
200:
194:
192:
190:
184:
182:
180:
174:
172:
170:
164:
163:
162:
161:
157:
156:
151:
146:
145:
136:
132:
129:
126:
122:
119:
116:
112:
109:
106:
102:
99:
96:
92:
89:
86:
82:
79:
76:
72:
69:
68:
64:
59:
58:
54:
50:
49:
46:
42:
39:
38:
33:
19:
23355:Lie algebras
23317:
23306:the original
23300:
23258:
23231:. Springer.
23228:
23198:
23163:
23157:
23096:
23093:Milnor, John
23063:
23022:
22996:Lie Algebras
22995:
22963:
22915:
22868:
22837:. Springer.
22834:
22807:. Springer.
22803:
22778:
22766:
22754:
22747:Khukhro 1998
22742:
22735:Quillen 1969
22730:
22703:
22691:
22679:
22667:
22655:
22643:
22631:
22619:
22607:
22595:
22583:
22571:
22559:
22547:
22520:
22508:
22503:, Chapter 8.
22496:
22469:
22457:
22445:
22433:
22421:
22409:
22397:
22385:
22380:, p. 4.
22373:
22361:
22349:
22340:
22328:
22316:
22311:, p. 1.
22304:
22292:
22280:
22259:
22255:
22251:
22243:
22134:Hopf algebra
21989:
21919:
21843:
21730:
21598:
21596:
21495:
21485:
21470:
21049:
20920:
20731:group scheme
20651:
20642:
20638:
20634:
20630:
20624:
20619:
20527:
20519:
20515:
20487:
20482:
20471:to describe
20429:
20410:
20400:compact form
20398:
20392:
20366:
20139:
20079:
20019:
20012:
20001:
19989:
19958:
19956:
19935:
19835:
19828:
19728:
19612:
19557:
19506:
19449:
19395:
19369:
19363:
19357:
19351:
19347:root systems
19342:
19337:
19329:
19205:
19091:Killing form
19081:
18987:
18857:
18855:
18742:
18698:
18691:
18682:
18680:
18343:
18339:
18269:
18150:
18024:
17998:
17754:
17753:and a field
17750:
17748:
17720:. Under the
17715:
17710:
17584:
17464:
17214:
17188:
17181:
17057:
17049:
16942:
16798:
16629:
16603:
16601:of a group.
16438:
16434:
16432:
16427:
16265:
16261:
16257:
16231:
16216:
16208:
16085:
15973:
15778:
15517:
15513:
15508:denotes the
15435:
15310:
15282:
15237:left adjoint
15182:
15178:
15174:
15172:
15096:
15005:
14915:
14905:
14859:
14847:
14845:
14816:
14770:
14707:
14679:
14625:
14585:
14583:
14561:
14528:
14524:
14518:
14481:left adjoint
14476:
14078:-eigenspace.
14043:
14039:
14003:
13999:
13991:
13987:
13935:
13931:
13216:
12327:
11765:
11105:
10841:has a basis
10814:
10780:
10413:circle group
10304:
10142:denotes the
10069:
10065:
9833:
9829:
9694:). See also
9540:
9535:denotes the
9458:
9454:
9260:
9256:
9085:
9081:
9031:
9025:
8914:
8711:
8707:
8545:
8382:
8340:
8312:
8261:matrix group
8258:
8151:
8038:
7953:. The image
7771:
7769:
7559:
7533:
7487:
7483:
7475:
7401:
7397:
7393:
7390:vector field
7363:
7359:
7318:be the ring
7315:
7311:
7310:
7305:
7301:
7299:
7161:
7157:
7153:
7149:
7141:
7139:
7134:
7130:
6908:
6834:Leibniz rule
6797:
6793:
6788:
6783:
6779:
6774:
6503:an ideal of
6454:
6289:
6036:
5981:
5949:
5915:
5621:
5377:
5117:
5109:
5069:
4918:
4882:
4839:
4838:are said to
4801:
4584:
4530:
4521:
4519:
4375:homomorphism
4374:
4372:
4365:
4271:
4233:
4135:
4119:
4036:
3894:
3890:
3794:
3788:Lie algebra.
3785:
3646:
3639:
3635:
3631:
3627:
3626:For a field
3375:
3353:
3338:
3286:
3234:
3230:
3226:
3222:
3218:
3214:
3202:
3200:
3152:
3102:
2763:
2760:
2575:
2471:
2174:
2053:
2043:
2040:Hermann Weyl
2035:
2021:
1884:
1644:
1636:
1631:
1627:
1599:
1595:
1567:
1565:
1519:, which are
1514:
1320:
1294:vector space
1268:(pronounced
1265:
1259:
1219:
1205:
1204:
1200:Free algebra
1198:
1197:
1191:
1190:
1159:
1102:
1064:
1027:
996:
995:
975:Finite field
924:
871:Finite field
857:
856:
783:Initial ring
743:
742:
716:
715:
658:
579:Armand Borel
564:Hermann Weyl
365:Loop algebra
347:Killing form
321:Lie algebras
320:
198:
188:
178:
168:
134:
124:
114:
104:
94:
84:
74:
45:Lie algebras
44:
18:Lie algebras
22865:Harris, Joe
22696:Milnor 2010
22489:Wigner 1959
20923:of a group
19785:at a point
19117:defined by
19087:Élie Cartan
18944:is simple,
16323:torus group
14580:Definitions
14006:) into the
10303:. (Indeed,
10281:, not over
9120:orientation
7836:defined by
6771:Derivations
6763:. See also
4939:: that is,
4885:centralizer
4583:in it, the
4522:isomorphism
4138:associative
4127:Definitions
4033:linear maps
3899:determinant
2472:Alternating
2175:Bilinearity
2036:Lie algebra
1418:is denoted
1321:Lie bracket
1266:Lie algebra
1262:mathematics
1180:Simple ring
891:Jordan ring
765:Graded ring
647:Ring theory
559:Élie Cartan
405:Root system
209:Exceptional
23350:Lie groups
23344:Categories
22783:Serre 2006
22759:Serre 2006
22708:Knapp 2001
22513:Serre 2006
22402:Serre 2006
22273:References
22174:Lie operad
20921:filtration
20721:. Part of
20647:commutator
20394:split form
19930:; see the
19926:. This is
19831:Lie groups
18687:semisimple
18152:semisimple
18149:is called
18053:are 0 and
18023:is called
17585:such that
16943:such that
13996:eigenspace
7668:such that
6789:derivation
5114:normalizer
3888:invertible
3649:, denoted
3235:generators
3219:generators
2046:was used.
2028:Sophus Lie
1517:Lie groups
1460:commutator
1186:Commutator
945:GCD domain
544:Sophus Lie
537:Scientists
410:Weyl group
131:Symplectic
91:Orthogonal
41:Lie groups
23324:EMS Press
23227:(1984) .
23062:(2001) ,
22998:. Dover.
22994:(1979) .
22942:0072-5285
22907:246650103
22540:Hall 2015
22527:, Ch. VI.
22474:Hall 2015
22438:Hall 2015
22075:σ
22068:σ
22051:∘
22037:⊗
22031:⋅
22025:⋅
22016:∘
22010:⋅
22004:⋅
21963:τ
21946:∘
21940:⋅
21934:⋅
21902:→
21896:⊗
21890::
21884:⋅
21878:⋅
21823:⊗
21817:⊗
21802:⊗
21796:⊗
21787:σ
21764:σ
21702:⊗
21699:τ
21693:∘
21687:τ
21684:⊗
21667:σ
21641:⊗
21635:⊗
21629:→
21623:⊗
21617:⊗
21608:σ
21576:⊗
21561:⊗
21552:τ
21526:⊗
21520:→
21514:⊗
21505:τ
21477:morphisms
21253:→
21219:×
21080:≥
21073:⨁
20907:⋯
20904:⊇
20891:⊇
20888:⋯
20885:⊇
20872:⊇
20859:⊇
20812:−
20799:−
20573:→
20563:×
20553::
20191:⊗
20141:real form
19701:⊗
19176:
19161:
19152:
18813:×
18807:≅
18756:≥
18694:reductive
18647:×
18625:≅
18567:≅
18525:≥
18415:≥
18323:≥
18241:×
18238:⋯
18235:×
18218:≅
17970:≥
17877:≥
17692:−
17605:−
17543:⊆
17540:⋯
17537:⊆
17520:⊆
17450:⋯
17447:⊇
17325:⊇
17267:⊇
17241:⊇
17184:nilpotent
17137:
17120:→
17098:
17048:for each
17031:−
16982:−
16901:⊆
16898:⋯
16895:⊆
16878:⊆
16784:⋯
16781:⊇
16723:⊇
16681:⊇
16655:⊇
16631:nilpotent
16554:∈
16411:≥
16400:for some
16223:nilpotent
16017:→
15969:injective
15939:→
15899:…
15844:⋯
15728:…
15624:∈
15580:−
15571:−
15496:⊗
15421:⋯
15418:⊕
15405:⊗
15395:⊗
15382:⊕
15369:⊗
15356:⊕
15346:⊕
15253:↦
15217:−
15017:
15006:given by
14968:→
14958::
14779:π
14771:That is,
14734:→
14724::
14721:π
14660:−
14493:↦
14198:⊗
14060:−
14042:maps the
13990:maps the
13817:−
13681:−
13369:−
12796:−
12742:−
12652:−
12438:⋅
12381:⋅
11800:such that
11669:⋅
11628:⋅
11602:⋅
11596:∈
11528:⋅
11212:≠
11184:∈
11075:↦
10431:⊂
10244:−
10236:∗
10128:∗
10098:−
10085:∗
9679:−
9537:transpose
9489:−
8990:→
8880:⋯
8793:
8749:→
8635:⊂
8602:−
8498:→
8455:∈
8274:⊂
7916:→
7906::
7785:∈
7649:→
7639::
7505:→
7331:∞
7275:ϵ
7247:ϵ
7235:⋅
7217:ϵ
7205:≡
7184:ϵ
7059:−
6926:∈
6817:→
6811::
6744:⋉
6600:→
6542:→
6362:×
6139:∈
6115:∈
6056:×
5878:−
5856:−
5773:−
5577:≥
5228:∈
5208:∈
5180:∈
5026:∈
4981:∈
4898:⊂
4819:∈
4784:ϕ
4720:ϕ
4693:→
4683::
4680:ϕ
4637:→
4526:bijective
4495:∈
4460:ϕ
4445:ϕ
4415:ϕ
4401:→
4391::
4388:ϕ
4341:⊆
4288:⊆
4251:⊆
3766:−
3482:−
3203:dimension
3070:−
2953:−
2474:property,
2151:→
2141:×
2124:⋅
2117:⋅
1993:×
1900:∈
1852:×
1843:×
1825:×
1816:×
1798:×
1789:×
1760:×
1754:−
1745:×
1716:×
1545:connected
1494:−
1351:→
1341:×
1127:∞
905:Semifield
421:Real form
307:Euclidean
158:Classical
23257:(1959).
23197:(2006).
23144:(2005).
23128:(2000).
22960:(1978).
22867:(1991).
22801:(1989).
22749:, Ch. 6.
22725:, §26.1.
22106:See also
21022:for all
20737:Examples
20627:p-groups
20620:Lie ring
20616:integers
20479:Lie ring
20080:Given a
20004:discrete
19994:and the
17216:solvable
15235:) has a
15155:and the
14848:faithful
13130:. Also,
10115:, where
9506:, where
8442:′
8250:Examples
6909:for all
6428:′
6371:′
6334:′
6292:category
6265:′
6254:′
6213:′
6190:′
6148:′
6135:′
6101:′
6065:′
6020:′
5499:.) Here
3643:matrices
2990:for all
2864:for all
2701:for all
2525:for all
1555:). This
899:Semiring
885:Lie ring
667:Subrings
593:Glossary
287:Poincaré
23326:, 2001
23281:0106711
23247:0746308
23217:2179691
23188:0258031
23180:1970725
23115:0830252
23086:1880691
23053:1615819
23014:0559927
22984:0499562
22950:3331229
22899:1153249
22853:2218355
22823:1728312
22792:Sources
22440:, §3.4.
22236:Remarks
21479:in the
21052:.) Then
19555:, and D
17718:radical
17219:if the
16634:if the
16258:abelian
16219:abelian
16000:modules
15488:. Here
15436:be the
13579:on the
12564:of the
10411:is the
9164:is the
8385:at the
6777:algebra
6775:For an
6038:product
5374:Example
4840:commute
4533:kernels
4479:for all
3376:abelian
3105:fraktur
2079:over a
2018:History
1292:) is a
1101:Prüfer
703:•
501:physics
282:Lorentz
111:Unitary
23279:
23269:
23245:
23235:
23215:
23205:
23186:
23178:
23113:
23103:
23084:
23074:
23051:
23041:
23012:
23002:
22982:
22972:
22948:
22940:
22930:
22905:
22897:
22887:
22851:
22841:
22821:
22811:
21731:where
20835:. Let
20547:
20541:
20524:module
20518:is an
20008:center
19783:sphere
19389:, or G
18856:where
18026:simple
16225:, and
15777:as an
15313:, let
14940:, the
14588:, let
13434:
13386:
13217:simple
12849:
12128:
9034:, the
8507:
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6128:
6035:, the
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5070:center
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4483:
4475:
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2974:
2685:
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1861:
1837:
1810:
1521:groups
753:Module
726:Kernel
277:Circle
23176:JSTOR
22254:: an
20919:be a
20514:over
19948:SU(2)
19944:SO(3)
19805:. If
19663:over
18466:over
18382:over
17653:with
17052:. By
16537:with
15878:with
15707:: if
15522:ideal
15309:over
9266:trace
8710:, if
7366:over
7133:over
6796:over
5246:. If
4272:ideal
4230:rings
4118:over
3793:When
3645:over
3211:basis
2081:field
1323:, an
1105:-ring
969:Field
865:Field
673:Ideal
660:Rings
352:Index
23267:ISBN
23233:ISBN
23203:ISBN
23101:ISBN
23072:ISBN
23039:ISBN
23000:ISBN
22970:ISBN
22938:ISSN
22928:ISBN
22903:OCLC
22885:ISBN
22839:ISBN
22809:ISBN
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21597:The
20391:, a
20328:and
19957:For
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19367:, D
19355:, B
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18598:and
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14464:The
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11721:The
11643:and
10947:and
10768:The
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9955:The
9359:The
7770:The
7509:Vect
7448:Vect
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7314:Let
6972:and
6786:, a
6671:and
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6318:and
6006:and
5566:for
4883:The
3543:The
3221:for
3201:The
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2470:The
1537:real
1398:and
1264:, a
302:Loop
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23168:doi
23031:doi
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9921:on
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22715:^
22532:^
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21365::=
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2664:x
2661:[
2658:,
2655:z
2652:[
2649:+
2646:]
2643:]
2640:x
2637:,
2634:z
2631:[
2628:,
2625:y
2622:[
2619:+
2616:]
2613:]
2610:z
2607:,
2604:y
2601:[
2598:,
2595:x
2592:[
2578:,
2569:.
2555:g
2533:x
2506:0
2503:=
2500:]
2497:x
2494:,
2491:x
2488:[
2465:.
2451:g
2429:z
2426:,
2423:y
2420:,
2417:x
2397:F
2377:b
2374:,
2371:a
2347:]
2344:y
2341:,
2338:z
2335:[
2332:b
2329:+
2326:]
2323:x
2320:,
2317:z
2314:[
2311:a
2308:=
2305:]
2302:y
2299:b
2296:+
2293:x
2290:a
2287:,
2284:z
2281:[
2260:,
2257:]
2254:z
2251:,
2248:y
2245:[
2242:b
2239:+
2236:]
2233:z
2230:,
2227:x
2224:[
2221:a
2218:=
2215:]
2212:z
2209:,
2206:y
2203:b
2200:+
2197:x
2194:a
2191:[
2177:,
2156:g
2146:g
2136:g
2131::
2128:]
2121:,
2113:[
2090:F
2065:g
2002:0
1999:=
1996:x
1990:x
1987:=
1984:]
1981:x
1978:,
1975:x
1972:[
1952:v
1932:v
1910:3
1905:R
1897:v
1864:=
1858:)
1855:y
1849:x
1846:(
1840:z
1834:+
1831:)
1828:x
1822:z
1819:(
1813:y
1807:+
1804:)
1801:z
1795:y
1792:(
1786:x
1763:x
1757:y
1751:=
1748:y
1742:x
1722:.
1719:y
1713:x
1710:=
1707:]
1704:y
1701:,
1698:x
1695:[
1670:3
1665:R
1660:=
1655:g
1632:G
1628:G
1612:g
1600:G
1596:G
1580:g
1568:G
1551:(
1500:x
1497:y
1491:y
1488:x
1485:=
1482:]
1479:y
1476:,
1473:x
1470:[
1438:]
1435:y
1432:,
1429:x
1426:[
1406:y
1386:x
1356:g
1346:g
1336:g
1305:g
1281:/
1275:l
1272:/
1249:e
1242:t
1235:v
1132:)
1123:p
1119:(
1115:Z
1103:p
1083:p
1078:Q
1065:p
1046:p
1041:Z
1028:p
1014:n
839:Z
835:1
831:/
826:Z
822:=
819:0
793:Z
625:e
618:t
611:v
251:8
249:E
243:7
241:E
235:6
233:E
227:4
225:F
219:2
217:G
199:n
196:D
189:n
186:C
179:n
176:B
169:n
166:A
137:)
135:n
127:)
125:n
117:)
115:n
107:)
105:n
97:)
95:n
87:)
85:n
77:)
75:n
34:.
20:)
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