41:
1454:
1337:
6563:
is less well understood. In this situation, a representation need not be a direct sum of irreducible representations. And although irreducible representations are indexed by dominant weights, the dimensions and characters of the irreducible representations are known only in some cases. Andersen,
2229:
of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.
2725:(2), as mentioned above. As a result, one can think of linear algebraic groups either as matrix groups or, more abstractly, as smooth affine group schemes over a field. (Some authors use "linear algebraic group" to mean any affine group scheme of finite type over a field.)
1342:
1225:
5610:
5519:
5901:. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer
3754:
1009:
in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include
4534:
2062:
5697:
2663:
6273:
6392:
One reason for the importance of reductive groups comes from representation theory. Every irreducible representation of a unipotent group is trivial. More generally, for any linear algebraic group
3799:
5866:, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism.
3110:
3346:
Over an algebraically closed field, there is a stronger result about algebraic groups as algebraic varieties: every connected linear algebraic group over an algebraically closed field is a
1828:
3341:
3498:
2223:
2192:
2100:
1909:
1768:
5011:
1449:{\displaystyle \left({\begin{array}{cccc}*&*&\dots &*\\0&*&\ddots &\vdots \\\vdots &\ddots &\ddots &*\\0&\dots &0&*\end{array}}\right)}
1332:{\displaystyle \left({\begin{array}{cccc}1&*&\dots &*\\0&1&\ddots &\vdots \\\vdots &\ddots &\ddots &*\\0&\dots &0&1\end{array}}\right)}
6318:
reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. For example, every nondegenerate
1214:
6003:
6441:
3015:
6650:
4929:
4629:
5818:
5290:
5258:
4815:
4783:
4359:
2863:
5524:
2003:
1731:
1694:
1657:
1580:
5436:
5222:
4399:
3823:
3696:
3624:
3383:
2894:
6790:
2141:
3863:) provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.
902:
486:
461:
424:
6050:
1147:
844:
2672:
which satisfy the usual axioms for the multiplication and inverse maps in a group (associativity, identity, inverses). A linear algebraic group is also smooth and of
1855:
1542:
1101:
929:
2161:
1969:
1949:
1929:
1875:
1788:
1620:
1600:
1502:
1478:
1167:
1121:
1066:
953:
6512:. This is the same as what happens in the representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complex
3708:
6108:). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them. Remarkably,
6185:
Every reductive group over a field is the quotient by a finite central subgroup scheme of the product of a torus and some simple groups. For example,
6997:
which are not affine behave very differently. In particular, a smooth connected group scheme which is a projective variety over a field is called an
7039:. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine group scheme over a field
7001:. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian varieties have a rich theory. Even the case of
6462:. This focuses attention on the representation theory of reductive groups. (To be clear, the representations considered here are representations of
2067:
These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their
788:
6659:. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects.
6486:
4440:
3127:
can be identified with an abstract finite group.) Because of this, the study of algebraic groups mostly focuses on connected groups.
2011:
7201:
7144:
2728:
For a full understanding of linear algebraic groups, one has to consider more general (non-smooth) group schemes. For example, let
5638:
2604:
6958:.) By contrast, there are simply connected solvable Lie groups that cannot be viewed as real algebraic groups. For example, the
6520:
of characteristic zero, all these theories are essentially equivalent. In particular, every representation of a reductive group
5049:(for example, an algebraically closed field) has a composition series with all quotient groups isomorphic to the additive group
7170:
6171:
346:
7611:
7514:
6191:
5034:). More generally, every connected solvable linear algebraic group is a semidirect product of a torus with a unipotent group,
7619:
3134:
can be extended to linear algebraic groups. It is straightforward to define what it means for a linear algebraic group to be
4288:
3770:
2810:
th roots of unity. This issue does not arise in characteristic zero. Indeed, every group scheme of finite type over a field
2673:
296:
2798:
7787:
7723:(1948), "Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations",
7186:
3872:
781:
291:
2272:, which allows an elementary definition of a linear algebraic group. First, define a function from the abstract group
7816:
7703:
7660:
7579:
7549:
5780:
is trivial. (Some authors do not require reductive groups to be connected.) A semisimple group is reductive. A group
3061:
7773:
1793:
7867:
3308:
6112:
gave a complete classification of the reductive groups over an algebraically closed field: they are determined by
5390:; conversely, every subgroup containing a Borel subgroup is parabolic. So one can list all parabolic subgroups of
2558:
5312:
3457:
2197:
2166:
2074:
1883:
1742:
7678:
4960:
3041:
707:
7850:
3627:
3412:
1175:
774:
7840:
5964:
6402:
5862:
7122:
2985:
7845:
7769:
7640:
7157:
5605:{\displaystyle \left\{{\begin{bmatrix}*&*&*\\*&*&*\\0&0&*\end{bmatrix}}\right\}.}
391:
205:
6760:
Linear algebraic groups admit variants in several directions. Dropping the existence of the inverse map
6616:
6559:
The representation theory of reductive groups (other than tori) over a field of positive characteristic
5514:{\displaystyle \left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&*&*\end{bmatrix}}\right\}}
4898:
4598:
7779:
7196:
7182:
7067:
can be read from its category of representations. For example, over a field of characteristic zero, Rep
6663:
6656:
5791:
5263:
5231:
4788:
4756:
4332:
3651:
2836:
2239:
2107:
123:
6855:
There are several reasons why a Lie group may not have the structure of a linear algebraic group over
1974:
1702:
1665:
1628:
1551:
6722:
5200:
5120:
4377:
3804:
3677:
3605:
3364:
3146:, by analogy with the definitions in abstract group theory. For example, a linear algebraic group is
2947:
2872:
6938:
showed that every simply connected nilpotent Lie group can be viewed as a unipotent algebraic group
6863:
A Lie group with an infinite group of components G/G cannot be realized as a linear algebraic group.
6763:
2113:
7149:
7102:
7010:
5617:
2831:
2733:
589:
323:
200:
88:
864:
469:
444:
407:
7174:
5159:
5097:
2693:
6029:
5072:
are important for the structure theory of linear algebraic groups. For a linear algebraic group
1351:
1234:
7491:
7191:
7140:
7131:
7056:
6537:
6533:
6513:
6338:
3699:
2554:
1126:
823:
529:
6504:
Chevalley showed that the irreducible representations of a split reductive group over a field
7725:
7166:
6129:
6066:
Reductive groups include the most important linear algebraic groups in practice, such as the
3932:
3390:
2068:
850:
613:
2785:
is not a regular function). In the language of group schemes, there is a clearer reason why
7826:
7797:
7762:
7713:
7670:
7629:
7589:
7559:
7530:
7495:
7106:
7060:
6730:
6675:
6524:
over a field of characteristic zero is a direct sum of irreducible representations, and if
4084:). (These properties are in fact independent of the choice of a faithful representation of
2367:
1833:
1735:
1660:
1515:
1074:
1069:
990:
907:
846:
553:
541:
159:
93:
4250:
that is not contained in any bigger torus. For example, the group of diagonal matrices in
8:
7600:
7178:
7126:
7094:
are the finite simple groups constructed from simple algebraic groups over finite fields.
7074:
6594:
5386:
algebraically closed, the Borel subgroups are exactly the minimal parabolic subgroups of
3846:. In positive characteristic, there can be many different connected subgroups of a group
3545:, left invariance of a derivation is defined as an analogous equality of two linear maps
3049:
2430:
1545:
963:
814:
128:
23:
6748:. In particular, the theory defines open subsets of "stable" and "semistable" points in
7750:
7691:
7030:
6721:-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to
6703:
6699:
6540:
gives a geometric construction of the irreducible representations of a reductive group
6014:
5351:
5084:
means a maximal smooth connected solvable subgroup. For example, one Borel subgroup of
4952:
3998:
3154:
of linear algebraic subgroups such that the quotient groups are commutative. Also, the
3151:
3034:
2972:
2912:
2497:
2226:
2146:
1954:
1934:
1914:
1860:
1773:
1605:
1585:
1487:
1463:
1152:
1106:
1051:
938:
113:
85:
4024:) is said to be semisimple if it becomes diagonalizable over the algebraic closure of
7812:
7783:
7742:
7699:
7656:
7615:
7575:
7545:
7518:
7153:
7097:
7091:
6920:
6899:
6849:
5840:
is semisimple, whereas a nontrivial torus is reductive but not semisimple. Likewise,
4219:
3749:{\displaystyle \operatorname {Ad} \colon G\to \operatorname {Aut} ({\mathfrak {g}}).}
3264:
The assumption of connectedness cannot be omitted in these results. For example, let
3159:
2897:
2489:
2246:
975:
818:
518:
361:
255:
7063:
are constructed using this formalism. Certain properties of a (pro-)algebraic group
6485:
is given by regular functions. It is an important but different problem to classify
684:
7734:
6883:
6852:. Much of the theory of algebraic groups was developed by analogy with Lie groups.
6714:
6545:
6529:
6109:
6098:
6087:
5717:
5323:
3347:
2794:
2410:
1046:
1015:
987:
858:
669:
661:
653:
645:
637:
625:
565:
505:
495:
337:
279:
154:
6124:. It is striking that this classification is independent of the characteristic of
5103:
A basic result of the theory is that any two Borel subgroups of a connected group
2557:. In particular, this defines what it means for two linear algebraic groups to be
7822:
7793:
7758:
7709:
7666:
7652:
7625:
7585:
7571:
7555:
7526:
7118:
6998:
6994:
6959:
6935:
6829:
6509:
6067:
6061:
5889:) if it is semisimple, nontrivial, and every smooth connected normal subgroup of
5772:
5749:
5725:
4196:
4176:
3279:
3139:
3053:
3045:
1034:
1002:
753:
746:
732:
689:
577:
500:
330:
244:
184:
64:
3185:
One may ask to what extent the properties of a connected linear algebraic group
7804:
7636:
7135:
7026:
7002:
6577:
6569:
6319:
6121:
5128:
5068:
3902:
3143:
2866:
2269:
1509:
1505:
971:
760:
696:
386:
366:
303:
268:
189:
179:
164:
149:
103:
80:
6740:
Geometric invariant theory involves further subtleties when a reductive group
1006:
7861:
7746:
7720:
7522:
7048:
7006:
6824:, essentially because real polynomials, which describe the multiplication on
6734:
6725:. In the positive direction, the ring of invariants is finitely generated if
6120:
are classified (up to quotients by finite central subgroup schemes) by their
3242:
3206:
3135:
2681:
2386:
1019:
1011:
679:
601:
435:
308:
174:
7644:
6375:
algebraically closed, and they are understood for some other fields such as
7596:
7537:
6951:
6752:, with the quotient morphism only defined on the set of semistable points.
6376:
4564:
3671:
3131:
2939:
2574:
1027:
534:
233:
222:
169:
144:
139:
98:
69:
32:
7390:
Bröcker & tom Dieck (1985), section III.8; Conrad (2014), section D.3.
4539:
with group structure given by the formula for multiplying complex numbers
2958:. For example, the Hopf algebra corresponding to the multiplicative group
2163:.) By contrast, the only irreducible representation of the additive group
7775:
Algebraic Groups: The Theory of Group
Schemes of Finite Type over a Field
6363:
essentially includes the problem of classifying all quadratic forms over
3359:
3222:
3163:
2309:
967:
802:
6924:, is a Lie group that cannot be viewed as a linear algebraic group over
5382:. That is, Borel subgroups are parabolic subgroups. More precisely, for
4279:). A basic result of the theory is that any two maximal tori in a group
2425:, and these functions must have the property that for every commutative
2071:(as algebraic groups). Every representation of the multiplicative group
7754:
7114:
6717:
showed that the ring of invariants need not be finitely generated as a
3924:
3155:
2103:
1951:, can also be expressed as a matrix group, for example as the subgroup
1030:
constructed much of the theory of algebraic groups as it exists today.
986:-anisotropic and reductive), as can many noncompact groups such as the
854:
701:
429:
6686:
as an algebraic variety. Various complications arise. For example, if
4884:
is nilpotent. As a result, every unipotent group scheme is nilpotent.
6821:
6166:
can be defined in any characteristic (and even as group schemes over
6113:
3916:
3907:
2110:. (Its irreducible representations all have dimension 1, of the form
1481:
959:
932:
522:
7738:
4175:
commute with each other. This reduces the problem of describing the
810:
59:
6116:. In particular, simple groups over an algebraically closed field
4529:{\displaystyle T=\{(x,y)\in A_{\mathbf {R} }^{2}:x^{2}+y^{2}=1\},}
4072:
to be semisimple or unipotent if it is semisimple or unipotent in
7110:
6359:). As a result, the problem of classifying reductive groups over
5170:) is conjugate to a subgroup of the upper-triangular subgroup in
4595:
is a connected linear algebraic group such that every element of
401:
315:
5406:
that contain a fixed Borel subgroup. For example, the subgroups
2355:) is defined by the vanishing of some set of regular functions.
2225:(such as the 2-dimensional representation above) is an iterated
7162:
6793:
6587:
5766:
is trivial. More generally, a connected linear algebraic group
4825:
have the same dimension, although they need not be isomorphic.
2057:{\displaystyle {\begin{pmatrix}1&*\\0&1\end{pmatrix}}.}
40:
6890:(2) is simply connected over any field, whereas the Lie group
7021:
The finite-dimensional representations of an algebraic group
6565:
7504:
th Root of Unity and of
Semisimple Groups in Characteristic
5857:
is a nontrivial smooth connected solvable normal subgroup).
5692:{\displaystyle 0\subset V_{1}\subset V_{2}\subset A_{k}^{3}}
6874:
may be connected as an algebraic group while the Lie group
2658:{\displaystyle m\colon G\times _{k}G\to G,\;i\colon G\to G}
6379:, but for arbitrary fields there are many open questions.
6174:
says that most finite simple groups arise as the group of
3759:
Over a field of characteristic zero, a connected subgroup
3201:). A useful result in this direction is that if the field
3182:, then they are linear algebraic groups as defined above.
2292:
if it can be written as a polynomial in the entries of an
2194:
is the trivial representation. So every representation of
4092:
is perfect, then the semisimple and unipotent parts of a
6268:{\displaystyle GL(n)\cong (G_{m}\times SL(n))/\mu _{n}.}
6178:-points of a simple algebraic group over a finite field
5045:
A smooth connected unipotent group over a perfect field
4430:. An example of a non-split torus over the real numbers
3353:
6012:
has characteristic zero, then one has the more precise
5942:
is (in a unique way) an extension of a reductive group
5776:
if every smooth connected unipotent normal subgroup of
4567:, which is not isomorphic even as an abstract group to
4032:
is perfect, then the semisimple and unipotent parts of
2488:; this is the philosophy of describing a scheme by its
1033:
One of the first uses for the theory was to define the
7490:
5848:) is reductive but not semisimple (because its center
5762:
if every smooth connected solvable normal subgroup of
5537:
5449:
3794:{\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}
3569:). The Lie bracket of two derivations is defined by =
3253:
is commutative, nilpotent, or solvable if and only if
2020:
6766:
6619:
6405:
6194:
6055:
6032:
5967:
5794:
5641:
5527:
5439:
5266:
5234:
5203:
4963:
4901:
4791:
4759:
4601:
4443:
4380:
4335:
3807:
3773:
3711:
3680:
3608:
3460:
3367:
3311:
3064:
2988:
2950:(reversing arrows) between affine group schemes over
2875:
2839:
2684:(as a scheme). Conversely, every affine group scheme
2607:
2564:
In the language of schemes, a linear algebraic group
2200:
2169:
2149:
2116:
2077:
2014:
1977:
1957:
1937:
1917:
1886:
1863:
1836:
1796:
1776:
1745:
1705:
1668:
1631:
1608:
1588:
1554:
1518:
1490:
1466:
1345:
1228:
1178:
1155:
1129:
1109:
1077:
1054:
941:
910:
867:
826:
472:
447:
410:
6690:
is an affine variety, then one can try to construct
6532:
of the irreducible representations are given by the
5402:)) by listing all the linear algebraic subgroups of
4555:. It is not split, because its group of real points
3866:
2942:(coming from the multiplication and inverse maps on
2712:. An example is the embedding of the additive group
6298:which remains maximal over an algebraic closure of
5820:is semisimple or reductive. For example, the group
4862:if it is isomorphic to a closed subgroup scheme of
3174:are naturally viewed as closed subgroup schemes of
2781:is not an isomorphism of algebraic groups (because
6981:cannot be viewed as a linear algebraic group over
6932:has no faithful finite-dimensional representation.
6784:
6644:
6435:
6267:
6044:
5997:
5812:
5743:
5691:
5604:
5513:
5284:
5252:
5216:
5005:
4923:
4858:(for example, a linear algebraic group) is called
4809:
4777:
4623:
4528:
4393:
4353:
3817:
3793:
3748:
3690:
3618:
3492:
3389:can be defined in several equivalent ways: as the
3377:
3335:
3104:
3009:
2888:
2857:
2657:
2217:
2186:
2155:
2135:
2094:
2056:
1997:
1963:
1943:
1923:
1903:
1869:
1849:
1822:
1782:
1762:
1725:
1688:
1651:
1614:
1594:
1574:
1536:
1496:
1472:
1448:
1331:
1208:
1161:
1141:
1115:
1095:
1060:
962:can be viewed as linear algebraic groups over the
947:
923:
896:
838:
480:
455:
418:
7005:(abelian varieties of dimension 1) is central to
6544:in characteristic zero, as spaces of sections of
4850:) with diagonal entries equal to 1, over a field
978:can be regarded as a linear algebraic group over
7859:
7565:
6977:, which is not a linear algebraic group, and so
6508:are finite-dimensional, and they are indexed by
4875:. It is straightforward to check that the group
3249:. For example, under the assumptions mentioned,
1931:-points are isomorphic to the additive group of
1602:. Any unipotent subgroup can be conjugated into
16:Subgroup of the group of invertible n×n matrices
7651:, Lecture Notes in Mathematics, vol. 900,
6792:, one obtains the notion of a linear algebraic
6454:reductive, every irreducible representation of
5921:is simple, and its center is the group scheme μ
4730:as the dimension of any maximal split torus in
3105:{\displaystyle 1\to G^{\circ }\to G\to F\to 1,}
2922:) of regular functions, an affine group scheme
2818:. A group scheme of finite type over any field
7246:Borel (1991), Theorem 18.2 and Corollary 18.4.
6314:. Chevalley showed that the classification of
5162:: every smooth connected solvable subgroup of
4662:). For example, both the multiplicative group
2549:. This makes the linear algebraic groups over
2501:of linear algebraic groups. For example, when
1823:{\displaystyle \mathbf {G} _{\mathrm {m} }(k)}
7469:Deligne & Milne (1982), Corollary II.2.7.
6666:, which aims to construct a quotient variety
6182:, or as minor variants of that construction.
5770:over an algebraically closed field is called
5758:over an algebraically closed field is called
4895:is unipotent if and only if every element of
4842:be the group of upper-triangular matrices in
4711:). As a result, it makes sense to define the
3336:{\displaystyle G({\overline {\mathbf {Q} }})}
2505:is algebraically closed, a homomorphism from
782:
7649:Hodge Cycles, Motives, and Shimura Varieties
7635:
7051:of affine group schemes of finite type over
6588:Group actions and geometric invariant theory
6310:) is a split reductive group over any field
5362:). An important property of Borel subgroups
4520:
4450:
3850:with the same Lie algebra (again, the torus
2495:In either language, one has the notion of a
1001:.) The simple Lie groups were classified by
7566:Bröcker, Theodor; tom Dieck, Tammo (1985),
7544:(2nd ed.), New York: Springer-Verlag,
7478:Deligne & Milne (1982), Remark II.2.28.
7009:, with applications including the proof of
6052:of a reductive group by a unipotent group.
5836:matrices with determinant 1 over any field
5100:(all entries below the diagonal are zero).
4048:). Finally, for any linear algebraic group
3493:{\displaystyle D\lambda _{x}=\lambda _{x}D}
2545:) which is defined by regular functions on
2409:is defined by the vanishing of some set of
2218:{\displaystyle \mathbf {G} _{\mathrm {a} }}
2187:{\displaystyle \mathbf {G} _{\mathrm {a} }}
2095:{\displaystyle \mathbf {G} _{\mathrm {m} }}
1904:{\displaystyle \mathbf {G} _{\mathrm {a} }}
1763:{\displaystyle \mathbf {G} _{\mathrm {m} }}
1219:consisting of matrices of the form, resp.,
1149:matrices, is a linear algebraic group over
6584:, there is not even a precise conjecture.
6501:, or similar problems over other fields.)
6290:if it contains a split maximal torus over
5296:may or may not have a Borel subgroup over
4817:. It follows that any two maximal tori in
4683:. However, it is always true that any two
4307:means the dimension of any maximal torus.
3829:, as one sees in the example of the torus
3767:is uniquely determined by its Lie algebra
2769:induces an isomorphism of abstract groups
2639:
789:
775:
7690:
7077:if and only if the identity component of
6886:groups. For example, the algebraic group
6832:. Likewise, for a linear algebraic group
6018:: every connected linear algebraic group
5006:{\displaystyle B_{n}=T_{n}\ltimes U_{n},}
4187:) to the semisimple and unipotent cases.
2472:), but rather the whole family of groups
474:
449:
412:
7803:
7676:
7399:Conrad (2014), after Proposition 5.1.17.
7202:Distribution on a linear algebraic group
6387:
5860:Every compact connected Lie group has a
4703:torus) are conjugate by some element of
4646:, one cannot expect all maximal tori in
4226:copies of the multiplicative group over
3935:for matrices implies that every element
3825:corresponds to an algebraic subgroup of
1548:that any connected solvable subgroup of
7719:
7500:Representations of Quantum Groups at a
7016:
6662:Part of the theory of group actions is
6572:'s conjecture) when the characteristic
6568:) determined these characters (proving
5934:Every connected linear algebraic group
5150:.) The conjugacy of Borel subgroups in
4120:) can be written uniquely as a product
3951:) can be written uniquely as a product
3537:) is induced by left multiplication by
3209:(for example, of characteristic zero),
2529:) is a homomorphism of abstract groups
1209:{\displaystyle U\subset B\subset GL(n)}
1023:
7860:
7811:(2nd ed.), New York: Birkhäuser,
7680:Linear Algebraic Groups (course notes)
7595:
7444:
6576:is sufficiently large compared to the
6172:classification of finite simple groups
5998:{\displaystyle 1\to U\to G\to R\to 1.}
5946:by a smooth connected unipotent group
3419:is algebraically closed, a derivation
3217:is reductive (as defined below), then
2814:of characteristic zero is smooth over
2444:) is a subgroup of the abstract group
347:Classification of finite simple groups
7768:
7568:Representations of Compact Lie Groups
7536:
7363:Milne (2017), Theorems 7.18 and 8.43.
7145:Weil's conjecture on Tamagawa numbers
6882:) is not connected, and likewise for
6436:{\displaystyle 1\to U\to G\to R\to 1}
5788:is called semisimple or reductive if
5197:such that, over an algebraic closure
3411:), or as the space of left-invariant
3354:The Lie algebra of an algebraic group
3193:are determined by the abstract group
2793:is surjective, but it has nontrivial
6989:
6367:or all central simple algebras over
5430:(3), and the intermediate subgroups
5374:is a projective variety, called the
4587:Every point of a torus over a field
4325:means a linear algebraic group over
3278:(1) of cube roots of unity over the
3010:{\displaystyle x\mapsto x\otimes x.}
2732:be an algebraically closed field of
6733:, proved in characteristic zero by
5754:A connected linear algebraic group
5134:over an algebraically closed field
5107:over an algebraically closed field
5076:over an algebraically closed field
4828:
4699:that are not contained in a bigger
4283:over an algebraically closed field
4205:over an algebraically closed field
3810:
3786:
3776:
3735:
3683:
3611:
3370:
2954:and commutative Hopf algebras over
2319:over an algebraically closed field
13:
7417:Springer (1998), 9.6.2 and 10.1.1.
7237:Milne (2017), Proposition 1.26(b).
6755:
6645:{\displaystyle G\times _{k}X\to X}
6481:-vector spaces, and the action of
6056:Classification of reductive groups
5632:of all chains of linear subspaces
5061:
4924:{\displaystyle G({\overline {k}})}
4624:{\displaystyle G({\overline {k}})}
3877:For an algebraically closed field
3801:. But not every Lie subalgebra of
3261:) has the corresponding property.
3119:is a finite algebraic group. (For
2209:
2178:
2086:
1982:
1979:
1895:
1805:
1754:
1710:
1707:
1673:
1670:
1636:
1633:
1559:
1556:
1484:linear algebraic group, the group
14:
7879:
7833:
7408:Conrad (2014), Proposition 5.4.1.
5813:{\displaystyle G_{\overline {k}}}
5422:of upper-triangular matrices are
5285:{\displaystyle G_{\overline {k}}}
5253:{\displaystyle B_{\overline {k}}}
5193:is defined to be a subgroup over
5123:: for a connected solvable group
5111:are conjugate by some element of
4810:{\displaystyle G_{\overline {k}}}
4778:{\displaystyle T_{\overline {k}}}
4354:{\displaystyle T_{\overline {k}}}
3867:Semisimple and unipotent elements
3289:is a linear algebraic group over
2979:, with comultiplication given by
2858:{\displaystyle G_{\overline {k}}}
2366:are defined as a special case of
1857:of nonzero elements of the field
7345:Borel (1991), Theorem 15.4(iii).
7309:Springer (1998), Theorem 15.2.6.
6946:in a unique way. (As a variety,
5732:; and the dual projective space
5618:projective homogeneous varieties
5418:that contain the Borel subgroup
5146:which is fixed by the action of
4943:of upper-triangular matrices in
4477:
3321:
3020:
2203:
2172:
2080:
1998:{\displaystyle \mathrm {GL} (2)}
1889:
1799:
1748:
1726:{\displaystyle \mathrm {GL} (1)}
1696:of matrices with determinant 1.
1689:{\displaystyle \mathrm {SL} (n)}
1652:{\displaystyle \mathrm {GL} (n)}
1575:{\displaystyle \mathrm {GL} (n)}
39:
7472:
7463:
7454:
7438:
7429:
7420:
7411:
7402:
7393:
7384:
7375:
7366:
7357:
7348:
7339:
7330:
7321:
7312:
7303:
7294:
6655:that satisfies the axioms of a
6580:of the group. For small primes
6382:
5744:Semisimple and reductive groups
5217:{\displaystyle {\overline {k}}}
4675:above occur as maximal tori in
4654:to be conjugate by elements of
4551:is a torus of dimension 1 over
4394:{\displaystyle {\overline {k}}}
4234:. For a linear algebraic group
3818:{\displaystyle {\mathfrak {g}}}
3691:{\displaystyle {\mathfrak {g}}}
3619:{\displaystyle {\mathfrak {g}}}
3378:{\displaystyle {\mathfrak {g}}}
2889:{\displaystyle {\overline {k}}}
2464:is not just the abstract group
1123:, consisting of all invertible
7612:Société Mathématique de France
7515:Société Mathématique de France
7381:Milne (2017), Definition 6.46.
7327:Milne (2017), Corollary 14.12.
7285:
7276:
7267:
7258:
7249:
7240:
7231:
7222:
7213:
6785:{\displaystyle i\colon G\to G}
6776:
6636:
6427:
6421:
6415:
6409:
6371:. These problems are easy for
6244:
6241:
6235:
6213:
6207:
6201:
5989:
5983:
5977:
5971:
5358:(or equivalently, proper over
5119:). (A standard proof uses the
4918:
4905:
4854:. A group scheme over a field
4618:
4605:
4591:is semisimple. Conversely, if
4465:
4453:
3873:Jordan–Chevalley decomposition
3740:
3730:
3721:
3403:) at the identity element 1 ∈
3330:
3315:
3301:) = 1 is not Zariski dense in
3093:
3087:
3081:
3068:
2992:
2739:> 0. Then the homomorphism
2649:
2630:
2245:, much of the structure of an
2233:
2136:{\displaystyle x\mapsto x^{n}}
2120:
1992:
1986:
1817:
1811:
1720:
1714:
1683:
1677:
1646:
1640:
1625:Another algebraic subgroup of
1569:
1563:
1531:
1525:
1203:
1197:
1090:
1084:
974:numbers. (For example, every
708:Infinite dimensional Lie group
1:
7608:Autour des schémas en groupes
7513:, Astérisque, vol. 220,
7484:
7372:Borel (1991), Corollary 11.2.
7300:Milne (2017), Corollary 17.25
7291:Borel (1991), Corollary 11.3.
7228:Milne (2017), Corollary 8.39.
7219:Milne (2017), Corollary 4.10.
6804:For a linear algebraic group
6799:
6744:acts on a projective variety
6497:) for a real reductive group
6477:, the representations are on
6348:determines a reductive group
6329:determines a reductive group
5303:For a closed subgroup scheme
4638:For a linear algebraic group
4413:means a group isomorphic to (
4209:means a group isomorphic to (
3044:containing the point 1) is a
3025:For a linear algebraic group
1544:. It is a consequence of the
1169:. It contains the subgroups
857:equations. An example is the
7460:Milne (2017), Theorem 14.37.
6678:of a linear algebraic group
6598:of a linear algebraic group
5804:
5276:
5244:
5209:
4913:
4801:
4769:
4745:in a linear algebraic group
4613:
4386:
4361:to the algebraic closure of
4345:
3763:of a linear algebraic group
3325:
3229:. Therefore, if in addition
3170:of a linear algebraic group
2881:
2849:
2688:of finite type over a field
2456:). (Thus an algebraic group
1830:is the multiplicative group
897:{\displaystyle M^{T}M=I_{n}}
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
7:
7846:Encyclopedia of Mathematics
7435:Milne (2017), Theorem 22.2.
7354:Borel (1991), Theorem 11.1.
7336:Borel (1991), Theorem 10.6.
7282:Milne (2017), Theorem 9.18.
7264:Milne (2017), section 10.e.
7255:Borel (1991), Remark 14.14.
7158:geometric Langlands program
7084:
7047:in the sense that it is an
7029:of representations, form a
6902:isomorphic to the integers
6812:, the group of real points
6294:(that is, a split torus in
4753:, Grothendieck showed that
2362:, algebraic varieties over
2108:irreducible representations
1508:algebraic group called the
1040:
206:List of group theory topics
10:
7884:
7780:Cambridge University Press
7426:Milne (2017), Lemma 19.16.
7273:Borel (1991), section 7.1.
7197:Differential Galois theory
6965:of the semidirect product
6664:geometric invariant theory
6564:Jantzen and Soergel (
6487:continuous representations
6059:
6045:{\displaystyle R\ltimes U}
5747:
4401:, for some natural number
4230:, for some natural number
4194:
3870:
3443:of the coordinate ring of
3178:. If they are smooth over
2938:) with its structure of a
2930:is determined by the ring
2389:closed subgroup scheme of
2343:) for some natural number
2240:algebraically closed field
861:, defined by the relation
7601:"Reductive group schemes"
6973:has center isomorphic to
6602:on a variety (or scheme)
5905:at least 2 and any field
5869:A linear algebraic group
5121:Borel fixed-point theorem
5098:upper-triangular matrices
4887:A linear algebraic group
3642:), the derivative at 1 ∈
3541:. For an arbitrary field
2948:equivalence of categories
1142:{\displaystyle n\times n}
839:{\displaystyle n\times n}
7841:"Linear algebraic group"
7447:Linear Algebraic Monoids
7207:
7150:Langlands classification
7103:Generalized flag variety
6674:, describing the set of
6396:written as an extension
6026:is a semidirect product
5784:over an arbitrary field
5724:of lines (1-dimensional
5628:are (respectively): the
5426:itself, the whole group
2582:, meaning a scheme over
2401:for some natural number
2331:) of the abstract group
324:Elementary abelian group
201:Glossary of group theory
7868:Linear algebraic groups
7809:Linear Algebraic Groups
7696:Linear Algebraic Groups
7542:Linear Algebraic Groups
7175:cohomological invariant
6954:of some dimension over
6548:over the flag manifold
6514:semisimple Lie algebras
6278:For an arbitrary field
5897:is trivial or equal to
5260:is a Borel subgroup of
5181:For an arbitrary field
5025:is the diagonal torus (
4695:(meaning split tori in
4310:For an arbitrary field
4190:
3343:is a group of order 3.
3233:is infinite, the group
2907:Since an affine scheme
2789:is not an isomorphism:
2694:faithful representation
2358:For an arbitrary field
7677:De Medts, Tom (2019),
7645:"Tannakian categories"
7610:, vol. 1, Paris:
7498:; Soergel, W. (1994),
7318:Borel (1991), 18.2(i).
7192:Pseudo-reductive group
7183:Kneser–Tits conjecture
7141:Adelic algebraic group
7132:Real form (Lie theory)
6808:over the real numbers
6786:
6723:Hilbert's 14th problem
6646:
6534:Weyl character formula
6437:
6339:central simple algebra
6269:
6130:exceptional Lie groups
6046:
5999:
5814:
5693:
5606:
5515:
5394:(up to conjugation by
5286:
5254:
5218:
5080:, a Borel subgroup of
5007:
4925:
4811:
4785:is a maximal torus in
4779:
4741:For any maximal torus
4625:
4530:
4395:
4355:
4262:is a maximal torus in
3819:
3795:
3750:
3700:adjoint representation
3692:
3620:
3494:
3385:of an algebraic group
3379:
3337:
3123:algebraically closed,
3106:
3011:
2890:
2859:
2659:
2376:linear algebraic group
2374:. In that language, a
2314:linear algebraic group
2256:is encoded in its set
2219:
2188:
2157:
2137:
2096:
2069:linear representations
2058:
1999:
1965:
1945:
1925:
1905:
1871:
1851:
1824:
1784:
1764:
1727:
1690:
1653:
1616:
1596:
1576:
1538:
1498:
1474:
1450:
1333:
1210:
1163:
1143:
1117:
1097:
1062:
949:
925:
898:
840:
807:linear algebraic group
740:Linear algebraic group
482:
457:
420:
7726:Annals of Mathematics
7187:Serre's conjecture II
7167:nonabelian cohomology
7123:group of adjoint type
7011:Fermat's Last Theorem
6787:
6647:
6467:as an algebraic group
6438:
6388:Representation theory
6270:
6047:
6000:
5938:over a perfect field
5815:
5694:
5607:
5516:
5334:of a connected group
5287:
5255:
5219:
5008:
4926:
4812:
4780:
4671:and the circle group
4642:over a general field
4626:
4531:
4396:
4356:
4100:are automatically in
3933:Jordan canonical form
3820:
3796:
3751:
3693:
3626:is thus a process of
3621:
3495:
3380:
3338:
3166:of a closed subgroup
3132:abstract group theory
3130:Various notions from
3107:
3012:
2911:is determined by its
2891:
2860:
2828:geometrically reduced
2826:if and only if it is
2660:
2220:
2189:
2158:
2138:
2097:
2059:
2000:
1966:
1946:
1926:
1906:
1872:
1852:
1850:{\displaystyle k^{*}}
1825:
1785:
1765:
1739:, usually denoted by
1728:
1691:
1654:
1617:
1597:
1577:
1539:
1537:{\displaystyle GL(n)}
1499:
1475:
1451:
1334:
1211:
1164:
1144:
1118:
1098:
1096:{\displaystyle GL(n)}
1063:
950:
926:
924:{\displaystyle M^{T}}
899:
853:) that is defined by
851:matrix multiplication
841:
483:
458:
421:
7655:, pp. 101–228,
7445:Renner, Lex (2006),
7107:Bruhat decomposition
7061:motivic Galois group
7055:.) For example, the
7025:, together with the
7017:Tannakian categories
6764:
6617:
6469:. Thus, for a group
6403:
6282:, a reductive group
6192:
6030:
5965:
5792:
5639:
5525:
5437:
5330:. A smooth subgroup
5264:
5232:
5201:
4961:
4899:
4789:
4757:
4631:is semisimple, then
4599:
4441:
4378:
4333:
4106:Jordan decomposition
3931:are equal to 1. The
3923:is unipotent if all
3805:
3771:
3709:
3678:
3606:
3458:
3365:
3309:
3062:
2986:
2873:
2837:
2605:
2308:), where det is the
2198:
2167:
2147:
2114:
2075:
2012:
1975:
1955:
1935:
1915:
1884:
1861:
1834:
1794:
1774:
1743:
1736:multiplicative group
1703:
1666:
1661:special linear group
1629:
1606:
1586:
1552:
1516:
1488:
1464:
1343:
1226:
1176:
1153:
1127:
1107:
1075:
1070:general linear group
1052:
939:
908:
865:
824:
470:
445:
408:
7692:Humphreys, James E.
7614:, pp. 93–444,
7179:essential dimension
7127:parabolic induction
7075:semisimple category
6906:. The double cover
6866:An algebraic group
6128:. For example, the
5931:th roots of unity.
5688:
5185:, a Borel subgroup
5160:Lie–Kolchin theorem
4487:
4291:by some element of
3042:connected component
2830:, meaning that the
2572:is in particular a
1582:is conjugated into
1546:Lie-Kolchin theorem
1504:is an example of a
1480:is an example of a
114:Group homomorphisms
24:Algebraic structure
7805:Springer, Tonny A.
7686:, Ghent University
7508:: Independence of
7092:groups of Lie type
7081:is pro-reductive.
7057:Mumford–Tate group
7031:tannakian category
6782:
6704:ring of invariants
6642:
6538:Borel–Weil theorem
6433:
6265:
6042:
6015:Levi decomposition
5995:
5810:
5689:
5674:
5615:The corresponding
5602:
5589:
5511:
5501:
5282:
5250:
5214:
5092:) is the subgroup
5003:
4953:semidirect product
4921:
4807:
4775:
4685:maximal split tori
4621:
4526:
4471:
4391:
4365:is isomorphic to (
4351:
4329:whose base change
4270:), isomorphic to (
4159:is unipotent, and
3982:is unipotent, and
3815:
3791:
3746:
3688:
3616:
3490:
3375:
3333:
3152:composition series
3102:
3035:identity component
3007:
2973:Laurent polynomial
2886:
2855:
2655:
2480:) for commutative
2215:
2184:
2153:
2133:
2092:
2054:
2045:
1995:
1961:
1941:
1921:
1901:
1867:
1847:
1820:
1780:
1760:
1723:
1686:
1649:
1612:
1592:
1572:
1534:
1494:
1470:
1446:
1440:
1329:
1323:
1206:
1159:
1139:
1113:
1093:
1058:
945:
921:
894:
836:
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
7729:, Second Series,
7621:978-2-85629-794-0
7154:Langlands program
6990:Abelian varieties
6950:is isomorphic to
6928:. More strongly,
6921:metaplectic group
6900:fundamental group
6850:complex Lie group
6731:Haboush's theorem
6729:is reductive, by
6099:symplectic groups
6088:orthogonal groups
5952:unipotent radical
5807:
5279:
5247:
5212:
5158:) amounts to the
4916:
4804:
4772:
4616:
4389:
4348:
4246:means a torus in
4177:conjugacy classes
4108:): every element
4001:with each other.
3630:. For an element
3598:The passage from
3328:
2946:). This gives an
2898:algebraic closure
2884:
2852:
2598:) and morphisms
2490:functor of points
2411:regular functions
2405:. In particular,
2247:algebraic variety
2156:{\displaystyle n}
1964:{\displaystyle U}
1944:{\displaystyle k}
1924:{\displaystyle k}
1870:{\displaystyle k}
1783:{\displaystyle k}
1615:{\displaystyle U}
1595:{\displaystyle B}
1497:{\displaystyle B}
1473:{\displaystyle U}
1162:{\displaystyle k}
1116:{\displaystyle k}
1061:{\displaystyle n}
1026:). In the 1950s,
976:compact Lie group
948:{\displaystyle M}
799:
798:
374:
373:
256:Alternating group
213:
212:
7875:
7854:
7829:
7800:
7765:
7716:
7687:
7685:
7673:
7632:
7605:
7592:
7562:
7533:
7479:
7476:
7470:
7467:
7461:
7458:
7452:
7450:
7442:
7436:
7433:
7427:
7424:
7418:
7415:
7409:
7406:
7400:
7397:
7391:
7388:
7382:
7379:
7373:
7370:
7364:
7361:
7355:
7352:
7346:
7343:
7337:
7334:
7328:
7325:
7319:
7316:
7310:
7307:
7301:
7298:
7292:
7289:
7283:
7280:
7274:
7271:
7265:
7262:
7256:
7253:
7247:
7244:
7238:
7235:
7229:
7226:
7220:
7217:
6995:Algebraic groups
6918:), known as the
6884:simply connected
6830:smooth functions
6791:
6789:
6788:
6783:
6715:Masayoshi Nagata
6651:
6649:
6648:
6643:
6632:
6631:
6510:dominant weights
6458:factors through
6442:
6440:
6439:
6434:
6302:). For example,
6274:
6272:
6271:
6266:
6261:
6260:
6251:
6225:
6224:
6110:Claude Chevalley
6068:classical groups
6051:
6049:
6048:
6043:
6004:
6002:
6001:
5996:
5863:complexification
5819:
5817:
5816:
5811:
5809:
5808:
5800:
5726:linear subspaces
5718:projective space
5698:
5696:
5695:
5690:
5687:
5682:
5670:
5669:
5657:
5656:
5611:
5609:
5608:
5603:
5598:
5594:
5593:
5520:
5518:
5517:
5512:
5510:
5506:
5505:
5324:quasi-projective
5291:
5289:
5288:
5283:
5281:
5280:
5272:
5259:
5257:
5256:
5251:
5249:
5248:
5240:
5223:
5221:
5220:
5215:
5213:
5205:
5012:
5010:
5009:
5004:
4999:
4998:
4986:
4985:
4973:
4972:
4930:
4928:
4927:
4922:
4917:
4909:
4829:Unipotent groups
4816:
4814:
4813:
4808:
4806:
4805:
4797:
4784:
4782:
4781:
4776:
4774:
4773:
4765:
4630:
4628:
4627:
4622:
4617:
4609:
4535:
4533:
4532:
4527:
4513:
4512:
4500:
4499:
4486:
4481:
4480:
4400:
4398:
4397:
4392:
4390:
4382:
4360:
4358:
4357:
4352:
4350:
4349:
4341:
4088:.) If the field
3919:. Equivalently,
3824:
3822:
3821:
3816:
3814:
3813:
3800:
3798:
3797:
3792:
3790:
3789:
3780:
3779:
3755:
3753:
3752:
3747:
3739:
3738:
3697:
3695:
3694:
3689:
3687:
3686:
3625:
3623:
3622:
3617:
3615:
3614:
3499:
3497:
3496:
3491:
3486:
3485:
3473:
3472:
3384:
3382:
3381:
3376:
3374:
3373:
3348:rational variety
3342:
3340:
3339:
3334:
3329:
3324:
3319:
3280:rational numbers
3111:
3109:
3108:
3103:
3080:
3079:
3052:. So there is a
3016:
3014:
3013:
3008:
2895:
2893:
2892:
2887:
2885:
2877:
2864:
2862:
2861:
2856:
2854:
2853:
2845:
2664:
2662:
2661:
2656:
2626:
2625:
2586:together with a
2224:
2222:
2221:
2216:
2214:
2213:
2212:
2206:
2193:
2191:
2190:
2185:
2183:
2182:
2181:
2175:
2162:
2160:
2159:
2154:
2142:
2140:
2139:
2134:
2132:
2131:
2101:
2099:
2098:
2093:
2091:
2090:
2089:
2083:
2063:
2061:
2060:
2055:
2050:
2049:
2004:
2002:
2001:
1996:
1985:
1970:
1968:
1967:
1962:
1950:
1948:
1947:
1942:
1930:
1928:
1927:
1922:
1910:
1908:
1907:
1902:
1900:
1899:
1898:
1892:
1876:
1874:
1873:
1868:
1856:
1854:
1853:
1848:
1846:
1845:
1829:
1827:
1826:
1821:
1810:
1809:
1808:
1802:
1789:
1787:
1786:
1781:
1769:
1767:
1766:
1761:
1759:
1758:
1757:
1751:
1732:
1730:
1729:
1724:
1713:
1695:
1693:
1692:
1687:
1676:
1658:
1656:
1655:
1650:
1639:
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1581:
1579:
1578:
1573:
1562:
1543:
1541:
1540:
1535:
1503:
1501:
1500:
1495:
1479:
1477:
1476:
1471:
1455:
1453:
1452:
1447:
1445:
1441:
1338:
1336:
1335:
1330:
1328:
1324:
1215:
1213:
1212:
1207:
1168:
1166:
1165:
1160:
1148:
1146:
1145:
1140:
1122:
1120:
1119:
1114:
1102:
1100:
1099:
1094:
1067:
1065:
1064:
1059:
1047:positive integer
1035:Chevalley groups
988:simple Lie group
954:
952:
951:
946:
930:
928:
927:
922:
920:
919:
903:
901:
900:
895:
893:
892:
877:
876:
859:orthogonal group
845:
843:
842:
837:
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
485:
484:
479:
477:
462:
460:
459:
454:
452:
425:
423:
422:
417:
415:
338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
228:
227:
54:
53:
43:
30:
19:
18:
7883:
7882:
7878:
7877:
7876:
7874:
7873:
7872:
7858:
7857:
7839:
7836:
7819:
7790:
7739:10.2307/1969111
7706:
7683:
7663:
7653:Springer Nature
7637:Deligne, Pierre
7622:
7603:
7582:
7572:Springer Nature
7552:
7492:Andersen, H. H.
7487:
7482:
7477:
7473:
7468:
7464:
7459:
7455:
7443:
7439:
7434:
7430:
7425:
7421:
7416:
7412:
7407:
7403:
7398:
7394:
7389:
7385:
7380:
7376:
7371:
7367:
7362:
7358:
7353:
7349:
7344:
7340:
7335:
7331:
7326:
7322:
7317:
7313:
7308:
7304:
7299:
7295:
7290:
7286:
7281:
7277:
7272:
7268:
7263:
7259:
7254:
7250:
7245:
7241:
7236:
7232:
7227:
7223:
7218:
7214:
7210:
7119:Cartan subgroup
7087:
7072:
7038:
7019:
7003:elliptic curves
6999:abelian variety
6992:
6960:universal cover
6936:Anatoly Maltsev
6802:
6765:
6762:
6761:
6758:
6756:Related notions
6627:
6623:
6618:
6615:
6614:
6590:
6404:
6401:
6400:
6390:
6385:
6354:
6256:
6252:
6247:
6220:
6216:
6193:
6190:
6189:
6165:
6158:
6151:
6144:
6137:
6122:Dynkin diagrams
6064:
6062:Reductive group
6058:
6031:
6028:
6027:
5966:
5963:
5962:
5926:
5856:
5799:
5795:
5793:
5790:
5789:
5752:
5750:Reductive group
5746:
5715:; a point; the
5710:
5683:
5678:
5665:
5661:
5652:
5648:
5640:
5637:
5636:
5588:
5587:
5582:
5577:
5571:
5570:
5565:
5560:
5554:
5553:
5548:
5543:
5533:
5532:
5528:
5526:
5523:
5522:
5500:
5499:
5494:
5489:
5483:
5482:
5477:
5472:
5466:
5465:
5460:
5455:
5445:
5444:
5440:
5438:
5435:
5434:
5271:
5267:
5265:
5262:
5261:
5239:
5235:
5233:
5230:
5229:
5204:
5202:
5199:
5198:
5069:Borel subgroups
5064:
5062:Borel subgroups
5057:
5033:
5024:
4994:
4990:
4981:
4977:
4968:
4964:
4962:
4959:
4958:
4942:
4908:
4900:
4897:
4896:
4883:
4870:
4841:
4831:
4796:
4792:
4790:
4787:
4786:
4764:
4760:
4758:
4755:
4754:
4670:
4608:
4600:
4597:
4596:
4575:
4508:
4504:
4495:
4491:
4482:
4476:
4475:
4442:
4439:
4438:
4421:
4381:
4379:
4376:
4375:
4373:
4340:
4336:
4334:
4331:
4330:
4278:
4217:
4199:
4197:Algebraic torus
4193:
4174:
4167:
4158:
4152:is semisimple,
4151:
4136:
4130:
4104:. That is (the
4060:) over a field
4028:. If the field
3997:
3990:
3981:
3975:is semisimple,
3974:
3967:
3961:
3875:
3869:
3862:
3841:
3809:
3808:
3806:
3803:
3802:
3785:
3784:
3775:
3774:
3772:
3769:
3768:
3734:
3733:
3710:
3707:
3706:
3682:
3681:
3679:
3676:
3675:
3628:differentiation
3610:
3609:
3607:
3604:
3603:
3594:
3588:
3581:
3575:
3520:
3481:
3477:
3468:
3464:
3459:
3456:
3455:
3398:
3369:
3368:
3366:
3363:
3362:
3356:
3320:
3318:
3310:
3307:
3306:
3273:
3075:
3071:
3063:
3060:
3059:
3054:group extension
3046:normal subgroup
3023:
2987:
2984:
2983:
2966:
2876:
2874:
2871:
2870:
2844:
2840:
2838:
2835:
2834:
2822:is smooth over
2804:
2760:
2751:
2720:
2621:
2617:
2606:
2603:
2602:
2270:rational points
2236:
2208:
2207:
2202:
2201:
2199:
2196:
2195:
2177:
2176:
2171:
2170:
2168:
2165:
2164:
2148:
2145:
2144:
2143:for an integer
2127:
2123:
2115:
2112:
2111:
2085:
2084:
2079:
2078:
2076:
2073:
2072:
2044:
2043:
2038:
2032:
2031:
2026:
2016:
2015:
2013:
2010:
2009:
1978:
1976:
1973:
1972:
1956:
1953:
1952:
1936:
1933:
1932:
1916:
1913:
1912:
1894:
1893:
1888:
1887:
1885:
1882:
1881:
1862:
1859:
1858:
1841:
1837:
1835:
1832:
1831:
1804:
1803:
1798:
1797:
1795:
1792:
1791:
1775:
1772:
1771:
1770:. The group of
1753:
1752:
1747:
1746:
1744:
1741:
1740:
1706:
1704:
1701:
1700:
1669:
1667:
1664:
1663:
1632:
1630:
1627:
1626:
1607:
1604:
1603:
1587:
1584:
1583:
1555:
1553:
1550:
1549:
1517:
1514:
1513:
1489:
1486:
1485:
1465:
1462:
1461:
1439:
1438:
1433:
1428:
1423:
1417:
1416:
1411:
1406:
1401:
1395:
1394:
1389:
1384:
1379:
1373:
1372:
1367:
1362:
1357:
1350:
1346:
1344:
1341:
1340:
1322:
1321:
1316:
1311:
1306:
1300:
1299:
1294:
1289:
1284:
1278:
1277:
1272:
1267:
1262:
1256:
1255:
1250:
1245:
1240:
1233:
1229:
1227:
1224:
1223:
1177:
1174:
1173:
1154:
1151:
1150:
1128:
1125:
1124:
1108:
1105:
1104:
1076:
1073:
1072:
1053:
1050:
1049:
1043:
1003:Wilhelm Killing
940:
937:
936:
915:
911:
909:
906:
905:
888:
884:
872:
868:
866:
863:
862:
825:
822:
821:
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
723:
722:
673:
665:
657:
649:
641:
614:Special unitary
525:
511:
510:
492:
491:
473:
471:
468:
467:
448:
446:
443:
442:
411:
409:
406:
405:
397:
396:
387:Discrete groups
376:
375:
331:Frobenius group
276:
263:
252:
245:Symmetric group
241:
225:
215:
214:
65:Normal subgroup
51:
31:
22:
17:
12:
11:
5:
7881:
7871:
7870:
7856:
7855:
7835:
7834:External links
7832:
7831:
7830:
7817:
7801:
7789:978-1107167483
7788:
7766:
7721:Kolchin, E. R.
7717:
7704:
7688:
7674:
7661:
7633:
7620:
7593:
7580:
7563:
7550:
7534:
7496:Jantzen, J. C.
7486:
7483:
7481:
7480:
7471:
7462:
7453:
7437:
7428:
7419:
7410:
7401:
7392:
7383:
7374:
7365:
7356:
7347:
7338:
7329:
7320:
7311:
7302:
7293:
7284:
7275:
7266:
7257:
7248:
7239:
7230:
7221:
7211:
7209:
7206:
7205:
7204:
7199:
7194:
7189:
7160:
7147:
7138:
7136:Satake diagram
7129:
7100:
7098:Lang's theorem
7095:
7086:
7083:
7068:
7034:
7027:tensor product
7018:
7015:
6991:
6988:
6987:
6986:
6933:
6864:
6801:
6798:
6781:
6778:
6775:
6772:
6769:
6757:
6754:
6653:
6652:
6641:
6638:
6635:
6630:
6626:
6622:
6610:is a morphism
6589:
6586:
6578:Coxeter number
6528:is split, the
6450:unipotent and
6444:
6443:
6432:
6429:
6426:
6423:
6420:
6417:
6414:
6411:
6408:
6389:
6386:
6384:
6381:
6352:
6320:quadratic form
6276:
6275:
6264:
6259:
6255:
6250:
6246:
6243:
6240:
6237:
6234:
6231:
6228:
6223:
6219:
6215:
6212:
6209:
6206:
6203:
6200:
6197:
6163:
6156:
6149:
6142:
6135:
6060:Main article:
6057:
6054:
6041:
6038:
6035:
6006:
6005:
5994:
5991:
5988:
5985:
5982:
5979:
5976:
5973:
5970:
5922:
5852:
5806:
5803:
5798:
5748:Main article:
5745:
5742:
5706:
5700:
5699:
5686:
5681:
5677:
5673:
5668:
5664:
5660:
5655:
5651:
5647:
5644:
5613:
5612:
5601:
5597:
5592:
5586:
5583:
5581:
5578:
5576:
5573:
5572:
5569:
5566:
5564:
5561:
5559:
5556:
5555:
5552:
5549:
5547:
5544:
5542:
5539:
5538:
5536:
5531:
5509:
5504:
5498:
5495:
5493:
5490:
5488:
5485:
5484:
5481:
5478:
5476:
5473:
5471:
5468:
5467:
5464:
5461:
5459:
5456:
5454:
5451:
5450:
5448:
5443:
5313:quotient space
5278:
5275:
5270:
5246:
5243:
5238:
5211:
5208:
5129:proper variety
5063:
5060:
5053:
5029:
5020:
5014:
5013:
5002:
4997:
4993:
4989:
4984:
4980:
4976:
4971:
4967:
4938:
4931:is unipotent.
4920:
4915:
4912:
4907:
4904:
4879:
4866:
4837:
4830:
4827:
4803:
4800:
4795:
4771:
4768:
4763:
4666:
4620:
4615:
4612:
4607:
4604:
4571:
4537:
4536:
4525:
4522:
4519:
4516:
4511:
4507:
4503:
4498:
4494:
4490:
4485:
4479:
4474:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4417:
4388:
4385:
4369:
4347:
4344:
4339:
4274:
4213:
4195:Main article:
4192:
4189:
4172:
4163:
4156:
4149:
4134:
4128:
4004:For any field
3995:
3986:
3979:
3972:
3965:
3959:
3911:if the matrix
3903:diagonalizable
3871:Main article:
3868:
3865:
3858:
3837:
3812:
3788:
3783:
3778:
3757:
3756:
3745:
3742:
3737:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3685:
3613:
3592:
3586:
3579:
3573:
3516:
3501:
3500:
3489:
3484:
3480:
3476:
3471:
3467:
3463:
3449:left-invariant
3396:
3372:
3355:
3352:
3332:
3327:
3323:
3317:
3314:
3269:
3268:be the group μ
3113:
3112:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3078:
3074:
3070:
3067:
3022:
3019:
3018:
3017:
3006:
3003:
3000:
2997:
2994:
2991:
2962:
2883:
2880:
2851:
2848:
2843:
2800:
2799:group scheme μ
2756:
2747:
2734:characteristic
2716:
2666:
2665:
2654:
2651:
2648:
2645:
2642:
2638:
2635:
2632:
2629:
2624:
2620:
2616:
2613:
2610:
2323:is a subgroup
2235:
2232:
2211:
2205:
2180:
2174:
2152:
2130:
2126:
2122:
2119:
2088:
2082:
2065:
2064:
2053:
2048:
2042:
2039:
2037:
2034:
2033:
2030:
2027:
2025:
2022:
2021:
2019:
1994:
1991:
1988:
1984:
1981:
1960:
1940:
1920:
1897:
1891:
1879:additive group
1866:
1844:
1840:
1819:
1816:
1813:
1807:
1801:
1779:
1756:
1750:
1733:is called the
1722:
1719:
1716:
1712:
1709:
1685:
1682:
1679:
1675:
1672:
1648:
1645:
1642:
1638:
1635:
1611:
1591:
1571:
1568:
1565:
1561:
1558:
1533:
1530:
1527:
1524:
1521:
1510:Borel subgroup
1493:
1469:
1458:
1457:
1444:
1437:
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1418:
1415:
1412:
1410:
1407:
1405:
1402:
1400:
1397:
1396:
1393:
1390:
1388:
1385:
1383:
1380:
1378:
1375:
1374:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1352:
1349:
1327:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1301:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1280:
1279:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1257:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1235:
1232:
1217:
1216:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1158:
1138:
1135:
1132:
1112:
1092:
1089:
1086:
1083:
1080:
1057:
1042:
1039:
944:
918:
914:
891:
887:
883:
880:
875:
871:
835:
832:
829:
797:
796:
794:
793:
786:
779:
771:
768:
767:
764:
763:
761:Elliptic curve
757:
756:
750:
749:
743:
742:
736:
731:
730:
727:
726:
721:
720:
717:
714:
710:
706:
705:
704:
699:
697:Diffeomorphism
693:
692:
687:
682:
676:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
633:
622:
621:
610:
609:
598:
597:
586:
585:
574:
573:
562:
561:
554:Special linear
550:
549:
542:General linear
538:
537:
532:
526:
517:
516:
513:
512:
509:
508:
503:
498:
490:
489:
476:
464:
451:
438:
436:Modular groups
434:
433:
432:
427:
414:
398:
395:
394:
389:
383:
382:
381:
378:
377:
372:
371:
370:
369:
364:
359:
356:
350:
349:
343:
342:
341:
340:
334:
333:
327:
326:
321:
312:
311:
309:Hall's theorem
306:
304:Sylow theorems
300:
299:
294:
286:
285:
284:
283:
277:
272:
269:Dihedral group
265:
264:
259:
253:
248:
242:
237:
226:
221:
220:
217:
216:
211:
210:
209:
208:
203:
195:
194:
193:
192:
187:
182:
177:
172:
167:
162:
160:multiplicative
157:
152:
147:
142:
134:
133:
132:
131:
126:
118:
117:
109:
108:
107:
106:
104:Wreath product
101:
96:
91:
89:direct product
83:
81:Quotient group
75:
74:
73:
72:
67:
62:
52:
49:
48:
45:
44:
36:
35:
15:
9:
6:
4:
3:
2:
7880:
7869:
7866:
7865:
7863:
7852:
7848:
7847:
7842:
7838:
7837:
7828:
7824:
7820:
7818:0-8176-4021-5
7814:
7810:
7806:
7802:
7799:
7795:
7791:
7785:
7781:
7777:
7776:
7771:
7767:
7764:
7760:
7756:
7752:
7748:
7744:
7740:
7736:
7732:
7728:
7727:
7722:
7718:
7715:
7711:
7707:
7705:0-387-90108-6
7701:
7697:
7693:
7689:
7682:
7681:
7675:
7672:
7668:
7664:
7662:3-540-11174-3
7658:
7654:
7650:
7646:
7642:
7638:
7634:
7631:
7627:
7623:
7617:
7613:
7609:
7602:
7598:
7597:Conrad, Brian
7594:
7591:
7587:
7583:
7581:0-387-13678-9
7577:
7573:
7569:
7564:
7561:
7557:
7553:
7551:0-387-97370-2
7547:
7543:
7539:
7538:Borel, Armand
7535:
7532:
7528:
7524:
7520:
7516:
7512:
7509:
7505:
7501:
7497:
7493:
7489:
7488:
7475:
7466:
7457:
7448:
7441:
7432:
7423:
7414:
7405:
7396:
7387:
7378:
7369:
7360:
7351:
7342:
7333:
7324:
7315:
7306:
7297:
7288:
7279:
7270:
7261:
7252:
7243:
7234:
7225:
7216:
7212:
7203:
7200:
7198:
7195:
7193:
7190:
7188:
7184:
7180:
7176:
7172:
7171:special group
7168:
7164:
7161:
7159:
7155:
7151:
7148:
7146:
7142:
7139:
7137:
7133:
7130:
7128:
7124:
7120:
7116:
7112:
7108:
7104:
7101:
7099:
7096:
7093:
7089:
7088:
7082:
7080:
7076:
7071:
7066:
7062:
7058:
7054:
7050:
7049:inverse limit
7046:
7045:pro-algebraic
7042:
7037:
7032:
7028:
7024:
7014:
7012:
7008:
7007:number theory
7004:
7000:
6996:
6984:
6980:
6976:
6972:
6968:
6964:
6961:
6957:
6953:
6949:
6945:
6941:
6937:
6934:
6931:
6927:
6923:
6922:
6917:
6913:
6909:
6905:
6901:
6897:
6893:
6889:
6885:
6881:
6877:
6873:
6869:
6865:
6862:
6861:
6860:
6858:
6853:
6851:
6847:
6843:
6839:
6835:
6831:
6827:
6823:
6819:
6815:
6811:
6807:
6797:
6795:
6779:
6773:
6770:
6767:
6753:
6751:
6747:
6743:
6738:
6736:
6732:
6728:
6724:
6720:
6716:
6712:
6708:
6705:
6701:
6697:
6693:
6689:
6685:
6681:
6677:
6673:
6669:
6665:
6660:
6658:
6639:
6633:
6628:
6624:
6620:
6613:
6612:
6611:
6609:
6606:over a field
6605:
6601:
6597:
6596:
6585:
6583:
6579:
6575:
6571:
6567:
6562:
6557:
6555:
6551:
6547:
6543:
6539:
6535:
6531:
6527:
6523:
6519:
6515:
6511:
6507:
6502:
6500:
6496:
6492:
6489:of the group
6488:
6484:
6480:
6476:
6473:over a field
6472:
6468:
6465:
6461:
6457:
6453:
6449:
6430:
6424:
6418:
6412:
6406:
6399:
6398:
6397:
6395:
6380:
6378:
6377:number fields
6374:
6370:
6366:
6362:
6358:
6351:
6347:
6343:
6340:
6337:), and every
6336:
6332:
6328:
6325:over a field
6324:
6321:
6317:
6313:
6309:
6305:
6301:
6297:
6293:
6289:
6285:
6281:
6262:
6257:
6253:
6248:
6238:
6232:
6229:
6226:
6221:
6217:
6210:
6204:
6198:
6195:
6188:
6187:
6186:
6183:
6181:
6177:
6173:
6169:
6162:
6155:
6148:
6141:
6134:
6131:
6127:
6123:
6119:
6115:
6111:
6107:
6103:
6100:
6096:
6092:
6089:
6085:
6081:
6077:
6073:
6069:
6063:
6053:
6039:
6036:
6033:
6025:
6021:
6017:
6016:
6011:
5992:
5986:
5980:
5974:
5968:
5961:
5960:
5959:
5957:
5953:
5950:, called the
5949:
5945:
5941:
5937:
5932:
5930:
5925:
5920:
5916:
5912:
5908:
5904:
5900:
5896:
5892:
5888:
5884:
5880:
5876:
5873:over a field
5872:
5867:
5865:
5864:
5858:
5855:
5851:
5847:
5843:
5839:
5835:
5831:
5827:
5823:
5801:
5796:
5787:
5783:
5779:
5775:
5774:
5769:
5765:
5761:
5757:
5751:
5741:
5739:
5736:of planes in
5735:
5731:
5727:
5723:
5720:
5719:
5714:
5711:of dimension
5709:
5705:
5684:
5679:
5675:
5671:
5666:
5662:
5658:
5653:
5649:
5645:
5642:
5635:
5634:
5633:
5631:
5630:flag manifold
5627:
5623:
5620:
5619:
5599:
5595:
5590:
5584:
5579:
5574:
5567:
5562:
5557:
5550:
5545:
5540:
5534:
5529:
5507:
5502:
5496:
5491:
5486:
5479:
5474:
5469:
5462:
5457:
5452:
5446:
5441:
5433:
5432:
5431:
5429:
5425:
5421:
5417:
5413:
5409:
5405:
5401:
5397:
5393:
5389:
5385:
5381:
5377:
5373:
5369:
5365:
5361:
5357:
5353:
5349:
5345:
5341:
5337:
5333:
5329:
5325:
5321:
5317:
5314:
5310:
5306:
5301:
5299:
5295:
5273:
5268:
5241:
5236:
5227:
5206:
5196:
5192:
5188:
5184:
5179:
5177:
5173:
5169:
5165:
5161:
5157:
5153:
5149:
5145:
5141:
5138:, there is a
5137:
5133:
5130:
5126:
5122:
5118:
5114:
5110:
5106:
5101:
5099:
5095:
5091:
5087:
5083:
5079:
5075:
5071:
5070:
5059:
5056:
5052:
5048:
5043:
5041:
5037:
5032:
5028:
5023:
5019:
5000:
4995:
4991:
4987:
4982:
4978:
4974:
4969:
4965:
4957:
4956:
4955:
4954:
4950:
4946:
4941:
4937:
4932:
4910:
4902:
4894:
4891:over a field
4890:
4885:
4882:
4878:
4874:
4869:
4865:
4861:
4857:
4853:
4849:
4845:
4840:
4836:
4826:
4824:
4821:over a field
4820:
4798:
4793:
4766:
4761:
4752:
4749:over a field
4748:
4744:
4739:
4737:
4733:
4729:
4725:
4721:
4717:
4715:
4710:
4706:
4702:
4698:
4694:
4690:
4686:
4682:
4678:
4674:
4669:
4665:
4661:
4657:
4653:
4649:
4645:
4641:
4636:
4634:
4610:
4602:
4594:
4590:
4585:
4583:
4579:
4574:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4523:
4517:
4514:
4509:
4505:
4501:
4496:
4492:
4488:
4483:
4472:
4468:
4462:
4459:
4456:
4447:
4444:
4437:
4436:
4435:
4433:
4429:
4425:
4420:
4416:
4412:
4408:
4404:
4383:
4372:
4368:
4364:
4342:
4337:
4328:
4324:
4320:
4317:
4313:
4308:
4306:
4302:
4298:
4294:
4290:
4286:
4282:
4277:
4273:
4269:
4265:
4261:
4257:
4253:
4249:
4245:
4241:
4240:maximal torus
4237:
4233:
4229:
4225:
4221:
4216:
4212:
4208:
4204:
4198:
4188:
4186:
4182:
4178:
4171:
4166:
4162:
4155:
4148:
4144:
4140:
4133:
4127:
4123:
4119:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4083:
4079:
4075:
4071:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4015:
4011:
4008:, an element
4007:
4002:
4000:
3994:
3989:
3985:
3978:
3971:
3964:
3958:
3954:
3950:
3946:
3942:
3938:
3934:
3930:
3926:
3922:
3918:
3914:
3910:
3909:
3904:
3900:
3896:
3892:
3888:
3884:
3880:
3874:
3864:
3861:
3857:
3853:
3849:
3845:
3840:
3836:
3832:
3828:
3781:
3766:
3762:
3743:
3727:
3724:
3718:
3715:
3712:
3705:
3704:
3703:
3701:
3698:, giving the
3673:
3669:
3665:
3661:
3657:
3653:
3649:
3645:
3641:
3637:
3633:
3629:
3601:
3596:
3591:
3585:
3578:
3572:
3568:
3564:
3560:
3556:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3524:
3519:
3514:
3510:
3506:
3487:
3482:
3478:
3474:
3469:
3465:
3461:
3454:
3453:
3452:
3450:
3446:
3442:
3438:
3434:
3430:
3426:
3422:
3418:
3414:
3410:
3406:
3402:
3395:
3392:
3391:tangent space
3388:
3361:
3351:
3349:
3344:
3312:
3304:
3300:
3296:
3292:
3288:
3284:
3281:
3277:
3272:
3267:
3262:
3260:
3256:
3252:
3248:
3244:
3243:Zariski dense
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3208:
3204:
3200:
3196:
3192:
3189:over a field
3188:
3183:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3145:
3141:
3137:
3133:
3128:
3126:
3122:
3118:
3099:
3096:
3090:
3084:
3076:
3072:
3065:
3058:
3057:
3056:
3055:
3051:
3047:
3043:
3039:
3036:
3032:
3029:over a field
3028:
3021:Basic notions
3004:
3001:
2998:
2995:
2989:
2982:
2981:
2980:
2978:
2974:
2970:
2965:
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2933:
2929:
2926:over a field
2925:
2921:
2917:
2914:
2910:
2905:
2903:
2899:
2878:
2868:
2846:
2841:
2833:
2829:
2825:
2821:
2817:
2813:
2809:
2805:
2803:
2797:, namely the
2796:
2792:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2759:
2755:
2750:
2746:
2742:
2738:
2735:
2731:
2726:
2724:
2719:
2715:
2711:
2707:
2703:
2699:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2652:
2646:
2643:
2640:
2636:
2633:
2627:
2622:
2618:
2614:
2611:
2608:
2601:
2600:
2599:
2597:
2593:
2589:
2585:
2581:
2577:
2576:
2571:
2568:over a field
2567:
2562:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2512:
2508:
2504:
2500:
2499:
2493:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2432:
2428:
2424:
2420:
2416:
2412:
2408:
2404:
2400:
2396:
2392:
2388:
2384:
2381:over a field
2380:
2377:
2373:
2369:
2365:
2361:
2356:
2354:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2315:
2311:
2307:
2304:and in 1/det(
2303:
2299:
2295:
2291:
2287:
2283:
2279:
2275:
2271:
2267:
2263:
2259:
2255:
2251:
2248:
2244:
2241:
2231:
2228:
2150:
2128:
2124:
2117:
2109:
2105:
2070:
2051:
2046:
2040:
2035:
2028:
2023:
2017:
2008:
2007:
2006:
1989:
1958:
1938:
1918:
1880:
1864:
1842:
1838:
1814:
1777:
1738:
1737:
1717:
1697:
1680:
1662:
1643:
1623:
1609:
1589:
1566:
1547:
1528:
1522:
1519:
1511:
1507:
1491:
1483:
1467:
1442:
1435:
1430:
1425:
1420:
1413:
1408:
1403:
1398:
1391:
1386:
1381:
1376:
1369:
1364:
1359:
1354:
1347:
1325:
1318:
1313:
1308:
1303:
1296:
1291:
1286:
1281:
1274:
1269:
1264:
1259:
1252:
1247:
1242:
1237:
1230:
1222:
1221:
1220:
1200:
1194:
1191:
1188:
1185:
1182:
1179:
1172:
1171:
1170:
1156:
1136:
1133:
1130:
1110:
1103:over a field
1087:
1081:
1078:
1071:
1055:
1048:
1038:
1036:
1031:
1029:
1025:
1021:
1017:
1013:
1008:
1004:
1000:
998:
994:
989:
985:
982:(necessarily
981:
977:
973:
969:
965:
961:
956:
942:
934:
916:
912:
889:
885:
881:
878:
873:
869:
860:
856:
852:
848:
833:
830:
827:
820:
816:
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
345:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
111:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
21:
20:
7844:
7808:
7774:
7770:Milne, J. S.
7730:
7724:
7698:, Springer,
7695:
7679:
7648:
7641:Milne, J. S.
7607:
7567:
7541:
7511:
7507:
7503:
7499:
7474:
7465:
7456:
7446:
7440:
7431:
7422:
7413:
7404:
7395:
7386:
7377:
7368:
7359:
7350:
7341:
7332:
7323:
7314:
7305:
7296:
7287:
7278:
7269:
7260:
7251:
7242:
7233:
7224:
7215:
7078:
7069:
7064:
7052:
7044:
7040:
7035:
7022:
7020:
6993:
6982:
6978:
6974:
6970:
6966:
6962:
6955:
6952:affine space
6947:
6943:
6939:
6929:
6925:
6919:
6915:
6911:
6907:
6903:
6895:
6891:
6887:
6879:
6875:
6871:
6867:
6856:
6854:
6845:
6841:
6837:
6833:
6825:
6817:
6813:
6809:
6805:
6803:
6759:
6749:
6745:
6741:
6739:
6737:and Nagata.
6726:
6718:
6713:). However,
6710:
6706:
6695:
6691:
6687:
6683:
6679:
6671:
6667:
6661:
6657:group action
6654:
6607:
6603:
6599:
6593:
6591:
6581:
6573:
6560:
6558:
6553:
6549:
6546:line bundles
6541:
6525:
6521:
6517:
6505:
6503:
6498:
6494:
6490:
6482:
6478:
6474:
6470:
6466:
6463:
6459:
6455:
6451:
6447:
6445:
6393:
6391:
6383:Applications
6372:
6368:
6364:
6360:
6356:
6349:
6345:
6341:
6334:
6330:
6326:
6322:
6315:
6311:
6307:
6303:
6299:
6295:
6291:
6287:
6283:
6279:
6277:
6184:
6179:
6175:
6167:
6160:
6153:
6146:
6139:
6132:
6125:
6117:
6105:
6101:
6094:
6090:
6083:
6079:
6075:
6071:
6065:
6023:
6019:
6013:
6009:
6007:
5955:
5951:
5947:
5943:
5939:
5935:
5933:
5928:
5923:
5918:
5914:
5910:
5909:, the group
5906:
5902:
5898:
5894:
5890:
5886:
5882:
5878:
5874:
5870:
5868:
5861:
5859:
5853:
5849:
5845:
5841:
5837:
5833:
5829:
5825:
5821:
5785:
5781:
5777:
5771:
5767:
5763:
5759:
5755:
5753:
5737:
5733:
5729:
5721:
5716:
5712:
5707:
5703:
5701:
5629:
5625:
5621:
5616:
5614:
5427:
5423:
5419:
5415:
5411:
5407:
5403:
5399:
5395:
5391:
5387:
5383:
5379:
5376:flag variety
5375:
5371:
5367:
5363:
5359:
5355:
5347:
5343:
5339:
5335:
5331:
5327:
5326:scheme over
5322:is a smooth
5319:
5315:
5308:
5304:
5302:
5297:
5293:
5225:
5194:
5190:
5186:
5182:
5180:
5175:
5171:
5167:
5163:
5155:
5151:
5147:
5143:
5139:
5135:
5131:
5127:acting on a
5124:
5116:
5112:
5108:
5104:
5102:
5093:
5089:
5085:
5081:
5077:
5073:
5067:
5065:
5054:
5050:
5046:
5044:
5039:
5035:
5030:
5026:
5021:
5017:
5015:
4948:
4944:
4939:
4935:
4933:
4892:
4888:
4886:
4880:
4876:
4872:
4867:
4863:
4859:
4855:
4851:
4847:
4843:
4838:
4834:
4832:
4822:
4818:
4750:
4746:
4742:
4740:
4735:
4731:
4727:
4723:
4719:
4713:
4712:
4708:
4704:
4700:
4696:
4692:
4688:
4684:
4680:
4676:
4672:
4667:
4663:
4659:
4655:
4651:
4647:
4643:
4639:
4637:
4635:is a torus.
4632:
4592:
4588:
4586:
4581:
4577:
4572:
4568:
4565:circle group
4560:
4556:
4552:
4548:
4544:
4540:
4538:
4431:
4427:
4423:
4418:
4414:
4410:
4406:
4402:
4370:
4366:
4362:
4326:
4322:
4318:
4315:
4311:
4309:
4304:
4300:
4296:
4292:
4284:
4280:
4275:
4271:
4267:
4263:
4259:
4255:
4251:
4247:
4243:
4239:
4235:
4231:
4227:
4223:
4214:
4210:
4206:
4202:
4200:
4184:
4180:
4169:
4164:
4160:
4153:
4146:
4145:) such that
4142:
4138:
4131:
4125:
4121:
4117:
4113:
4109:
4105:
4101:
4097:
4093:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4036:also lie in
4033:
4029:
4025:
4021:
4017:
4013:
4009:
4005:
4003:
3992:
3987:
3983:
3976:
3969:
3962:
3956:
3952:
3948:
3944:
3940:
3936:
3928:
3920:
3912:
3906:
3898:
3897:) is called
3894:
3890:
3886:
3882:
3878:
3876:
3859:
3855:
3851:
3847:
3843:
3838:
3834:
3830:
3826:
3764:
3760:
3758:
3672:automorphism
3667:
3663:
3659:
3655:
3647:
3643:
3639:
3635:
3631:
3599:
3597:
3589:
3583:
3576:
3570:
3566:
3562:
3558:
3554:
3550:
3546:
3542:
3538:
3534:
3530:
3526:
3522:
3517:
3512:
3508:
3504:
3502:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3416:
3408:
3404:
3400:
3393:
3386:
3357:
3345:
3302:
3298:
3294:
3290:
3286:
3282:
3275:
3270:
3265:
3263:
3258:
3254:
3250:
3246:
3238:
3234:
3230:
3226:
3218:
3214:
3210:
3202:
3198:
3194:
3190:
3186:
3184:
3179:
3175:
3171:
3167:
3150:if it has a
3147:
3129:
3124:
3120:
3116:
3114:
3037:
3030:
3026:
3024:
2976:
2968:
2963:
2959:
2955:
2951:
2943:
2940:Hopf algebra
2935:
2931:
2927:
2923:
2919:
2915:
2908:
2906:
2901:
2827:
2823:
2819:
2815:
2811:
2807:
2801:
2790:
2786:
2782:
2778:
2774:
2770:
2766:
2762:
2757:
2753:
2748:
2744:
2740:
2736:
2729:
2727:
2722:
2717:
2713:
2709:
2705:
2701:
2697:
2689:
2685:
2680:, and it is
2677:
2669:
2667:
2595:
2591:
2587:
2583:
2579:
2575:group scheme
2573:
2569:
2565:
2563:
2550:
2546:
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:homomorphism
2496:
2494:
2485:
2481:
2477:
2473:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2426:
2422:
2418:
2414:
2406:
2402:
2398:
2394:
2390:
2382:
2378:
2375:
2371:
2363:
2359:
2357:
2352:
2348:
2344:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2313:
2305:
2301:
2297:
2293:
2289:
2285:
2281:
2277:
2273:
2265:
2261:
2257:
2253:
2249:
2242:
2237:
2066:
1878:
1734:
1698:
1624:
1459:
1218:
1044:
1032:
1028:Armand Borel
996:
992:
983:
979:
957:
806:
800:
739:
629:
617:
605:
593:
581:
569:
557:
545:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
7733:(1): 1–42,
4722:of a group
4407:split torus
4064:, define a
3925:eigenvalues
3881:, a matrix
3652:conjugation
3413:derivations
3360:Lie algebra
3223:unirational
3164:centralizer
3136:commutative
2971:(1) is the
2832:base change
2761:defined by
2674:finite type
2590:-point 1 ∈
2310:determinant
2234:Definitions
1007:Élie Cartan
803:mathematics
519:Topological
358:alternating
7485:References
7449:, Springer
7115:Weyl group
6800:Lie groups
6530:characters
6286:is called
6097:) and the
5877:is called
5760:semisimple
5352:projective
5338:is called
5142:-point in
4934:The group
4720:split rank
4096:-point of
4068:-point of
3968:such that
3899:semisimple
3515:), where λ
3503:for every
3305:, because
3293:for which
3162:, and the
3156:normalizer
3048:of finite
2559:isomorphic
2484:-algebras
2347:such that
2104:direct sum
1699:The group
1460:The group
960:Lie groups
855:polynomial
819:invertible
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
7851:EMS Press
7807:(1998) ,
7747:0003-486X
7540:(1991) ,
7523:0303-1179
6822:Lie group
6777:→
6771::
6637:→
6625:×
6428:→
6422:→
6416:→
6410:→
6254:μ
6227:×
6211:≅
6114:root data
6037:⋉
5990:→
5984:→
5978:→
5972:→
5805:¯
5773:reductive
5672:⊂
5659:⊂
5646:⊂
5585:∗
5568:∗
5563:∗
5558:∗
5551:∗
5546:∗
5541:∗
5497:∗
5492:∗
5480:∗
5475:∗
5463:∗
5458:∗
5453:∗
5414:(3) over
5340:parabolic
5277:¯
5245:¯
5210:¯
4988:⋉
4914:¯
4871:for some
4860:unipotent
4802:¯
4770:¯
4679:(2) over
4614:¯
4563:) is the
4469:∈
4426:for some
4387:¯
4346:¯
4289:conjugate
3917:nilpotent
3908:unipotent
3901:if it is
3782:⊂
3728:
3722:→
3716::
3650:) of the
3479:λ
3466:λ
3326:¯
3140:nilpotent
3094:→
3088:→
3082:→
3077:∘
3069:→
2999:⊗
2993:↦
2882:¯
2850:¯
2708:for some
2650:→
2644::
2631:→
2619:×
2612::
2312:. Then a
2227:extension
2121:↦
2029:∗
2005: :
1843:∗
1482:unipotent
1436:∗
1426:…
1414:∗
1409:⋱
1404:⋱
1399:⋮
1392:⋮
1387:⋱
1382:∗
1370:∗
1365:…
1360:∗
1355:∗
1309:…
1297:∗
1292:⋱
1287:⋱
1282:⋮
1275:⋮
1270:⋱
1253:∗
1248:…
1243:∗
1189:⊂
1183:⊂
1134:×
1016:Chevalley
933:transpose
831:×
690:Conformal
578:Euclidean
185:nilpotent
7862:Category
7772:(2017),
7694:(1975),
7643:(1982),
7599:(2014),
7085:See also
7059:and the
5366:is that
3670:, is an
3148:solvable
3144:solvable
2869:, where
2555:category
1911:, whose
1790:-points
1506:solvable
1041:Examples
847:matrices
811:subgroup
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
7853:, 2001
7827:1642713
7798:3729270
7763:0024884
7755:1969111
7714:0396773
7671:0654325
7630:3309122
7590:0781344
7560:1102012
7531:1272539
7111:BN pair
6848:) is a
6820:) is a
6735:Hilbert
6702:of the
6570:Lusztig
6170:). The
6086:), the
5917:) over
5292:. Thus
4951:) is a
4547:. Here
4422:) over
4374:) over
4299:). The
4258:) over
4220:product
4218:), the
3999:commute
3915:− 1 is
3842:) over
3439:) over
3285:. Then
3207:perfect
2867:reduced
2777:*, but
2704:) over
2553:into a
2431:algebra
2421:) over
2397:) over
2368:schemes
2300:matrix
2290:regular
2238:For an
1659:is the
1022: (
1020:Kolchin
972:complex
931:is the
849:(under
813:of the
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
7825:
7815:
7796:
7786:
7761:
7753:
7745:
7712:
7702:
7669:
7659:
7628:
7618:
7588:
7578:
7558:
7548:
7529:
7521:
7163:Torsor
6898:) has
6828:, are
6794:monoid
6676:orbits
6595:action
6536:. The
6516:. For
6159:, and
5887:simple
5879:simple
5311:, the
5016:where
3905:, and
3160:center
3158:, the
3115:where
3033:, the
2896:is an
2795:kernel
2692:has a
2682:affine
2387:smooth
2288:to be
1877:. The
1068:, the
1045:For a
1018:, and
1012:Maurer
904:where
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
7751:JSTOR
7684:(PDF)
7604:(PDF)
7208:Notes
7073:is a
6942:over
6870:over
6836:over
6446:with
6344:over
6316:split
6288:split
6022:over
5893:over
5828:) of
5728:) in
5702:with
5354:over
4734:over
4726:over
4716:-rank
4701:split
4691:over
4650:over
4409:over
4321:over
4316:torus
4203:torus
3415:. If
3241:) is
3225:over
3142:, or
3050:index
3040:(the
2975:ring
2721:into
2696:into
2676:over
2668:over
2578:over
2517:) to
2460:over
2385:is a
2370:over
2284:) to
2264:) of
2252:over
2102:is a
964:field
958:Many
815:group
809:is a
719:Sp(∞)
716:SU(∞)
129:image
7813:ISBN
7784:ISBN
7743:ISSN
7700:ISBN
7657:ISBN
7616:ISBN
7576:ISBN
7546:ISBN
7519:ISSN
7090:The
6700:Spec
6566:1994
5881:(or
5624:(3)/
5521:and
5066:The
4833:Let
4580:) =
4405:. A
4314:, a
4301:rank
4287:are
4238:, a
4191:Tori
4168:and
3991:and
3654:map
3553:) →
3529:) →
3431:) →
3358:The
2913:ring
2773:* →
2537:) →
1339:and
1024:1948
1005:and
968:real
805:, a
713:O(∞)
702:Loop
521:and
7735:doi
7043:is
7033:Rep
6914:(2,
6910:of
6894:(2,
6698:as
6682:on
6592:An
6078:),
6008:If
5954:of
5927:of
5378:of
5350:is
5342:if
5307:of
5224:of
5189:of
5178:).
5096:of
4718:or
4687:in
4584:*.
4434:is
4303:of
4242:in
4222:of
4179:in
4137:in
4112:of
4012:of
3939:of
3927:of
3885:in
3854:= (
3833:= (
3725:Aut
3674:of
3668:xgx
3602:to
3561:) ⊗
3507:in
3451:if
3447:is
3245:in
3221:is
3213:if
3205:is
2900:of
2865:is
2806:of
2492:.)
2413:on
2106:of
1971:in
1512:of
991:SL(
970:or
966:of
935:of
817:of
801:In
628:Sp(
616:SU(
592:SO(
556:SL(
544:GL(
7864::
7849:,
7843:,
7823:MR
7821:,
7794:MR
7792:,
7782:,
7778:,
7759:MR
7757:,
7749:,
7741:,
7731:49
7710:MR
7708:,
7667:MR
7665:,
7647:,
7639:;
7626:MR
7624:,
7606:,
7586:MR
7584:,
7574:,
7570:,
7556:MR
7554:,
7527:MR
7525:,
7517:,
7494:;
7185:,
7181:,
7177:,
7173:,
7169:,
7165:,
7156:,
7152:,
7143:,
7134:,
7125:,
7121:,
7117:,
7113:,
7109:,
7105:,
7013:.
6969:⋉
6912:SL
6892:SL
6888:SL
6859:.
6840:,
6796:.
6556:.
6350:SL
6331:SO
6304:GL
6152:,
6145:,
6138:,
6104:(2
6102:Sp
6091:SO
6080:SL
6072:GL
6070::
5993:1.
5958::
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