2707:, the idea being the following: a standard algebraic group is well described by its standard Hopf algebra of regular functions; we can then think of the deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all). While there does not seem to be a direct way to define or manipulate these non-standard objects, one can still work with their Hopf algebras, and indeed one
154:
5011:
5029:
5020:
5375:
4897:
4779:
3496:
5218:
3656:
2898:
983:
305:
2541:
55:
of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.
6994:
848:
746:
7169:
Haldane, F. D. M.; Ha, Z. N. C.; Talstra, J. C.; Bernard, D.; Pasquier, V. (1992). "Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory".
569:
3351:
472:
7146:
5359:
4888:
4770:
4351:
3947:
3807:
5220:
which is unique in the class of natural transformations of functors composed from the structural transformations (associativity, left and right units, transposition, and their inverses) in the category
83:. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. They have diverse applications ranging from
4407:
6139:
6017:
5293:
644:
1080:
is unique ("a bialgebra admits at most 1 Hopf algebra structure"). Thus, the antipode does not pose any extra structure which we can choose: Being a Hopf algebra is a property of a bialgebra.
4088:
5124:
3502:
7039:
5813:
5079:
6748:
6708:
6668:
6628:
7084:
6288:
6057:
5967:
5851:
5691:
5426:
4822:
4566:
4488:
6248:
5578:
2753:
859:
4602:
2989:
5485:
6551:
6518:
6485:
6452:
5927:
5889:
5523:
4704:
3084:
6406:
6342:
5758:
2576:
6772:
5778:
5002:
4524:
6201:
170:
2578:. This observation was actually a source of the notion of Hopf algebra. Using this structure, Hopf proved a structure theorem for the cohomology algebra of Lie groups.
383:
6089:
5607:
4638:
5711:
5453:
5119:
4982:
4942:
6575:
6165:
5653:
3250:
4962:
3264:
6362:
5627:
5239:
5099:
4922:
4662:
4447:
4427:
2626:
2602:
2414:
2402:
2382:
6797:
2352:
Note that functions on a finite group can be identified with the group ring, though these are more naturally thought of as dual – the group ring consists of
1335:(i.e. an algebra ideal in the kernel of the counit, a coalgebra coideal and stable under the antipode). As a consequence one has a quotient Hopf algebra
6912:
3278:, or quantum groupoids, are generalizations of Hopf algebras. Like Hopf algebras, weak Hopf algebras form a self-dual class of algebras; i.e., if
757:
655:
5010:
8207:
Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity".
483:
1155:. As a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite-dimensional Hopf algebra is automatically semisimple.
3491:{\displaystyle (\Delta (1)\otimes 1)(1\otimes \Delta (1))=(1\otimes \Delta (1))(\Delta (1)\otimes 1)=(\Delta \otimes {\mbox{Id}})\Delta (1)}
1123:. In other words, a Hopf subalgebra A is a Hopf algebra in its own right when the multiplication, comultiplication, counit and antipode of
7356:
6779:
1985:(which are quotients of the tensor algebra) are also Hopf algebras with this definition of the comultiplication, counit and antipode
395:
7089:
5302:
4831:
4713:
4294:
7373:
3847:
3707:
7724:
6802:
2223:} and thus has dimension 4. This is the smallest example of a Hopf algebra that is both non-commutative and non-cocommutative.
4356:
3243:
are generalizations of Hopf algebras, where coassociativity only holds up to a twist. They have been used in the study of the
8162:
8126:
8055:
8026:
7969:
7951:
7913:
7607:
Hazewinkel, Michiel (January 2003). "Symmetric
Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions".
7590:
7543:
7485:
7388:
6094:
5972:
5248:
1065:
is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive.
3244:
1152:
580:
7514:
5213:{\displaystyle (A\otimes B)\otimes (C\otimes D){\stackrel {\theta }{\rightarrowtail }}(A\otimes C)\otimes (B\otimes D)}
3995:
3651:{\displaystyle \epsilon (abc)=\sum \epsilon (ab_{(1)})\epsilon (b_{(2)}c)=\sum \epsilon (ab_{(2)})\epsilon (b_{(1)}c)}
8092:
7817:
7270:; Duhr, Claude; Gardi, Einan (2017-12-01). "Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case".
160:
Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless
6999:
5783:
5039:
7406:
4222:
coincide and are non-central commutative algebras in this particular case (the subalgebra of diagonal matrices).
4103:
Note that if Δ(1) = 1 ⊗ 1, these conditions reduce to the two usual conditions on the antipode of a Hopf algebra.
2697:" of those from example 3 which are neither commutative nor co-commutative. These Hopf algebras are often called
6713:
6673:
6633:
6593:
2356:
sums of elements, and thus pairs with functions on the group by evaluating the function on the summed elements.
8265:
7931:
7086:
are the natural transformations of associativity, and of the left and the right units in the monoidal category
7044:
1445:
6253:
6022:
5932:
5891:
in the sense that the diagrams in the section 3 of the definition always commute. As a corollary, each monoid
5818:
5658:
5391:
4789:
8018:
7582:
7477:
6750:(of measures, distributions, analytic functionals and currents) on groups are Hopf algebras in the category (
4964:
are morphisms of comonoids, and (this is equivalent in this situation) at the same time the comultiplication
4533:
4455:
3222:
2893:{\displaystyle a(m\otimes n):=\Delta (a)(m\otimes n)=(a_{1}\otimes a_{2})(m\otimes n)=(a_{1}m\otimes a_{2}n)}
1887:
978:{\displaystyle \nu _{k}^{\;ij}\tau _{j}^{\;m}\mu _{\;im}^{n}=\nu _{k}^{\;jm}\tau _{j}^{\,\;i}\mu _{\;im}^{n}}
8260:
6832:
6792:
6206:
5536:
4130:
algebra is a weak Hopf algebra. In particular, the groupoid algebra on with one pair of invertible arrows
1990:
1014:
4575:
2947:
2228:
5458:
7535:
6827:
6523:
6490:
6457:
6424:
5894:
5856:
5490:
5028:
4671:
3024:
2084:
1403:
which is closed under the algebra and coalgebra operations: in particular, the comultiplication Δ maps
7652:
Hopf, Heinz (1941). "Über die
Topologie der Gruppen–Mannigfaltigkeiten und ihre Verallgemeinerungen".
6367:
6303:
5716:
6553:(of continuous, smooth, holomorphic, regular functions) on groups are Hopf algebras in the category (
4288:
1788:
5019:
2549:
300:{\displaystyle S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\varepsilon (c)1\qquad {\mbox{ for all }}c\in H.}
153:
7962:
Affine Lie algebras and quantum groups. An introduction with applications in conformal field theory
6837:
6588:
5526:
5374:
4896:
4825:
4707:
4244:
Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where
84:
8144:
6757:
5763:
4987:
4497:
6170:
2704:
1690:
7934:(2007), "A Primer of Hopf Algebras", in Cartier, P.; Moussa, P.; Julia, B.; Vanhove, P. (eds.),
7832:
6419:
8255:
7213:
Plefka, J.; Spill, F.; Torrielli, A. (2006). "Hopf algebra structure of the AdS/CFT S-matrix".
4778:
4276:
3208:
2694:
355:
329:
7992:
42 (1941), 22–52. Reprinted in
Selecta Heinz Hopf, pp. 119–151, Springer, Berlin (1964).
7805:
6062:
5583:
4611:
7989:
6894:
5696:
5438:
5104:
4967:
4927:
122:
96:
52:
8050:, Australian Mathematical Society Lecture Series, vol. 19, Cambridge University Press,
6560:
6144:
5632:
8102:
8065:
7574:
7435:
7289:
7232:
7179:
6817:
4947:
2117:
88:
8174:"Pontryagin duality in the theory of topological vector spaces and in topological algebra"
8136:
8110:
8073:
8036:
8004:
7979:
7923:
2536:{\displaystyle H^{*}(G,K)\rightarrow H^{*}(G\times G,K)\cong H^{*}(G,K)\otimes H^{*}(G,K)}
8:
6822:
6807:
6291:
3290:(with respect to the algebra-coalgebra structure obtained from the natural pairing with
3230:
1508:
1392:
311:
145:
115:
36:
7996:
7293:
7236:
7183:
8234:
8216:
8195:
7964:, Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press,
7786:
7768:
7669:
7634:
7616:
7581:, Mathematical Surveys and Monographs, vol. 107 (2nd ed.), Providence, R.I.:
7423:
7313:
7279:
7248:
7222:
6347:
5612:
5224:
5084:
4907:
4647:
4432:
4412:
3258:
3240:
3204:
3112:
ensure that certain natural homomorphisms of vector spaces are indeed homomorphisms of
2611:
2605:
2587:
2387:
2367:
1385:
386:
60:
7744:
8238:
8199:
8158:
8122:
8088:
8051:
8022:
8017:, Regional Conference Series in Mathematics, vol. 82, Providence, Rhode Island:
7965:
7947:
7909:
7813:
7638:
7586:
7539:
7510:
7481:
7472:
Hazewinkel, Michiel; Gubareni, Nadezhda Mikhaĭlovna; Kirichenko, Vladimir V. (2010).
7305:
7195:
5433:
5429:
5004:
are morphisms of monoids; this means that the following diagrams must be commutative:
3275:
3226:
1978:
1135:
is required to be in A). The
Nichols–Zoeller freeness theorem of Warren Nichols and
161:
7790:
7317:
4225:
Early theoretical contributions to weak Hopf algebras are to be found in as well as
8226:
8185:
8150:
8132:
8106:
8069:
8032:
8010:
8000:
7988:, Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen,
7975:
7939:
7919:
7778:
7739:
7661:
7626:
7415:
7401:
7297:
7252:
7240:
7187:
6775:
6578:
1982:
1796:
1136:
1025:
319:
48:
6877:. This is used in the above formula for the comultiplication. For infinite groups
8098:
8082:
8061:
7993:
7943:
7759:
Böhm, Gabriella; Nill, Florian; Szlachanyi, Kornel (1999). "Weak Hopf
Algebras".
7506:
7431:
1800:
1780:
1378:
7301:
7191:
79:), and in numerous other places, making them probably the most familiar type of
7244:
4234:
4112:
3200:
2699:
1895:
32:
8230:
8190:
8173:
8154:
7836:
7630:
6989:{\displaystyle \alpha _{H,H,H}:(H\otimes H)\otimes H\to H\otimes (H\otimes H)}
3267:
introduced by V. G. Turaev in 2000 are also generalizations of Hopf algebras.
8249:
7908:, Pure and Applied Mathematics, vol. 235 (1st ed.), Marcel Dekker,
7309:
3254:
2637:
1696:
1443:. The group-like elements form a group with inverse given by the antipode. A
1010:
843:{\displaystyle \nu _{k}^{\;ij}\nu _{i}^{\;mn}=\nu _{k}^{\;mi}\nu _{i}^{\;nj}}
741:{\displaystyle \mu _{\;ij}^{k}\mu _{\;kn}^{m}=\mu _{\;jn}^{k}\mu _{\;ik}^{m}}
92:
3322:
satisfying all the axioms of Hopf algebra except possibly Δ(1) ≠ 1 ⊗ 1 or ε(
332:(as reflected in the symmetry of the above diagram), so if one can define a
7782:
7199:
6812:
2937:). Furthermore, we can define the trivial representation as the base field
2608:, graded cocommutative Hopf algebra over a field of characteristic 0. Then
2208:
1905:
1883:
1359:
1033:
72:
68:
4275:
In this philosophy, a group can be thought of as a Hopf algebra over the "
2628:(as an algebra) is a free exterior algebra with generators of odd degree.
1787:
with a finite Haar integral arises in this way, giving one formulation of
1001:-linear inverse, which is automatic in the finite-dimensional case, or if
8043:
7267:
2643:
2405:
1996:
1006:
564:{\displaystyle \Delta e_{i}=\sum _{j,k}\nu _{i}^{\;jk}e_{j}\otimes e_{k}}
20:
3817:(the right-hand side is the interesting projection usually denoted by Π(
1358:, a theory analogous to that of normal subgroups and quotient groups in
385:
for the underlying vector space, one may define the algebra in terms of
7985:
7673:
7427:
7227:
3212:
2642:
Most examples above are either commutative (i.e. the multiplication is
2025:
1500:
333:
76:
28:
1882:
Conversely, every commutative Hopf algebra over a field arises from a
7773:
7621:
3257:
where comultiplication from an algebra (with or without unit) to the
3234:
1083:
The antipode is an analog to the inversion map on a group that sends
1058:
108:
80:
44:
40:
7812:. Vol. 43. Cambridge: M.S.R.I. Publications. pp. 211–262.
7665:
7419:
3207:: they are the natural algebraic structure on the direct sum of all
2703:, a term that is so far only loosely defined. They are important in
8087:, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York,
7371:
7354:
7284:
4127:
8221:
7904:
Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Șerban (2001),
4287:
The definition of Hopf algebra is naturally extended to arbitrary
3116:-modules. For instance, the natural isomorphisms of vector spaces
2693:). Other interesting Hopf algebras are certain "deformations" or "
1289:). The two conditions of normality are equivalent if the antipode
3216:
344:
is finite-dimensional), then it is automatically a Hopf algebra.
64:
7842:
5530:
4282:
2711:
them with their Hopf algebras. Hence the name "quantum group".
467:{\displaystyle e_{i}\nabla e_{j}=\sum _{k}\mu _{\;ij}^{k}e_{k}}
107:
Formally, a Hopf algebra is an (associative and coassociative)
3253:
introduced by Alfons Van Daele in 1994 are generalizations of
2631:
7372:
Dăscălescu, Năstăsescu & Raianu (2001). "Remarks 4.2.3".
7141:{\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}
5354:{\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}
4883:{\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}
4765:{\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}
4346:{\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}
4164:
matrices. The weak Hopf algebra structure on this particular
7501:
Mikhalev, Aleksandr Vasilʹevich; Pilz, Günter, eds. (2002).
7474:
Algebras, Rings, and
Modules: Lie Algebras and Hopf Algebras
7471:
3942:{\displaystyle a_{(1)}S(a_{(2)})=\epsilon (1_{(1)}a)1_{(2)}}
3802:{\displaystyle S(a_{(1)})a_{(2)}=1_{(1)}\epsilon (a1_{(2)})}
7355:
Dăscălescu, Năstăsescu & Raianu (2001). "Prop. 4.2.6".
3302:
finite-dimensional algebra and coalgebra with coproduct Δ:
7903:
7871:
7869:
3294:
and its coalgebra-algebra structure). A weak Hopf algebra
1257:
if it is stable under the left adjoint mapping defined by
5455:
as the tensor product, and an arbitrary singletone, say,
2404:
is a Hopf algebra: the multiplication is provided by the
4402:{\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon ,S)}
4248:
is taken to be a set instead of a module. In this case:
7866:
7168:
6905:
6903:
1308:
satisfies the condition (of equality of subsets of H):
7854:
7476:. Mathematical surveys and monographs. Vol. 168.
6134:{\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon )}
6012:{\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon )}
5288:{\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon )}
4262:
there is a natural comultiplication (the diagonal map)
3467:
3261:
of tensor product algebra of the algebra with itself.
279:
7092:
7047:
7002:
6915:
6760:
6716:
6676:
6636:
6596:
6563:
6526:
6493:
6460:
6427:
6370:
6350:
6306:
6256:
6209:
6173:
6147:
6097:
6065:
6025:
5975:
5935:
5897:
5859:
5821:
5786:
5766:
5719:
5699:
5661:
5635:
5615:
5586:
5539:
5493:
5461:
5441:
5394:
5305:
5251:
5227:
5127:
5107:
5087:
5042:
4990:
4970:
4950:
4930:
4910:
4834:
4792:
4716:
4674:
4650:
4614:
4578:
4536:
4500:
4458:
4435:
4415:
4359:
4297:
4268:
the multiplication is the multiplication in the group
4107:
The axioms are partly chosen so that the category of
3998:
3957:(another interesting projection usually denoted by Π(
3850:
3710:
3505:
3354:
3027:
2950:
2756:
2614:
2590:
2552:
2417:
2390:
2370:
2235:
in terms of complete homogeneous symmetric functions
862:
760:
658:
583:
486:
398:
358:
173:
8149:, Progress in Mathematics, vol. 322, Springer,
7938:, vol. II, Berlin: Springer, pp. 537–615,
7265:
7212:
6900:
6344:
is the category of vector spaces over a given field
5629:(but possibly without the invertibility of elements
639:{\displaystyle Se_{i}=\sum _{j}\tau _{i}^{\;j}e_{j}}
7881:
7833:
Group = Hopf algebra « Secret
Blogging Seminar
43:, with these structures' compatibility making it a
7758:
7140:
7078:
7033:
6988:
6893:. In this case the space of functions with finite
6766:
6742:
6702:
6662:
6622:
6569:
6545:
6512:
6479:
6446:
6400:
6356:
6336:
6282:
6242:
6195:
6159:
6133:
6083:
6051:
6011:
5961:
5921:
5883:
5845:
5807:
5772:
5752:
5705:
5685:
5647:
5621:
5601:
5572:
5517:
5479:
5447:
5420:
5353:
5287:
5233:
5212:
5113:
5093:
5073:
4996:
4976:
4956:
4936:
4916:
4882:
4816:
4764:
4698:
4656:
4632:
4596:
4560:
4518:
4482:
4441:
4421:
4401:
4345:
4082:
3941:
3801:
3650:
3490:
3078:
2983:
2892:
2620:
2596:
2570:
2535:
2396:
2376:
977:
842:
740:
638:
563:
466:
377:
299:
7936:Frontiers in Number Theory, Physics, and Geometry
7803:
7732:Transactions of the American Mathematical Society
7534:. Cambridge Tracts in Mathematics. Vol. 74.
2028:and can therefore be uniquely extended to all of
1057:(and the underlying algebra with involution is a
8247:
8146:Monoidal Categories and Topological Field Theory
8142:
7848:
6059:, and vice versa. The existence of the antipode
4083:{\displaystyle S(a_{(1)})a_{(2)}S(a_{(3)})=S(a)}
1107:is a Hopf subalgebra if it is a subcoalgebra of
3831:) with image a separable subalgebra denoted by
63:, where they originated and are related to the
7808:. In Montgomery, S.; Schneider, H.-J. (eds.).
7399:
4890:, i.e. the following diagrams are commutative:
4772:, i.e. the following diagrams are commutative:
8143:Turaev, Vladimir; Virelizier, Alexis (2017),
6897:can be endowed with a Hopf algebra structure.
5609:behave like usual multiplication and unit in
4265:the unit is the identity element of the group
1162:is said to be right normal in a Hopf algebra
988:
7804:Nikshych, Dmitri; Vainerman, Leonid (2002).
7500:
7034:{\displaystyle \lambda _{H}:I\otimes H\to H}
5808:{\displaystyle \varepsilon (x)=\varnothing }
5474:
5468:
5074:{\displaystyle \lambda _{I}:I\otimes I\to I}
4904:3) the structures of monoid and comonoid on
4283:Hopf algebras in braided monoidal categories
1602:(with pointwise addition and multiplication)
1166:if it satisfies the condition of stability,
372:
359:
31:, is a structure that is simultaneously an (
7404:(1989), "A Hopf algebra freeness theorem",
6780:duality theories for non-commutative groups
5969:can naturally be considered as a bialgebra
5853:is compatible with any structure of monoid
5693:is a comonoid in the categorical sense iff
2632:Quantum groups and non-commutative geometry
2359:
1151:is finite-dimensional: a generalization of
8009:
7710:
7645:
7606:
7558:
7447:
6743:{\displaystyle {\mathcal {P}}^{\star }(G)}
6703:{\displaystyle {\mathcal {O}}^{\star }(G)}
6663:{\displaystyle {\mathcal {E}}^{\star }(G)}
6623:{\displaystyle {\mathcal {C}}^{\star }(G)}
5368:4) the diagram of antipode is commutative:
4259:there is a natural counit (map to 1 point)
961:
950:
930:
903:
892:
873:
831:
812:
790:
771:
724:
705:
683:
664:
620:
529:
440:
8220:
8189:
8116:
8048:Quantum groups: A Path To Current Algebra
7806:"Finite groupoids and their applications"
7772:
7743:
7722:
7698:
7686:
7620:
7459:
7342:
7330:
7283:
7226:
7079:{\displaystyle \rho _{H}:H\otimes I\to H}
5295:with the properties 1),2),3) is called a
3144:-modules. Also, the map of vector spaces
1779:Conversely, every commutative involutive
949:
164:, this property can also be expressed as
8080:
8015:Hopf algebras and their actions on rings
6408:are exactly the classical Hopf algebras
6283:{\displaystyle ({\text{Set}},\times ,1)}
6052:{\displaystyle ({\text{Set}},\times ,1)}
5962:{\displaystyle ({\text{Set}},\times ,1)}
5846:{\displaystyle (H,\Delta ,\varepsilon )}
5815:). And any such a structure of comonoid
5686:{\displaystyle (H,\Delta ,\varepsilon )}
5421:{\displaystyle ({\text{Set}},\times ,1)}
5381:The typical examples are the following.
4817:{\displaystyle (H,\Delta ,\varepsilon )}
2714:
1965:) (and extended to higher tensor powers)
1327:. This normality condition implies that
1297:is said to be a normal Hopf subalgebra.
1036:, which is therefore an automorphism if
47:, and that moreover is equipped with an
8206:
8171:
7930:
7887:
7875:
7860:
7573:
6843:
5121:the natural transformation of functors
4561:{\displaystyle \Delta :H\to H\otimes H}
4483:{\displaystyle \nabla :H\otimes H\to H}
314:, one can replace the underlying field
8248:
8042:
7704:
6803:Representation theory of Hopf algebras
6778:. These Hopf algebras are used in the
6409:
4239:
3002:. Finally, the dual representation of
1418:
1053:, then the Hopf algebra is said to be
347:
7959:
7692:
7680:
7453:
6243:{\displaystyle \nabla (x,y)=x\cdot y}
5573:{\displaystyle \nabla (x,y)=x\cdot y}
3342:. Instead one requires the following:
3286:*, the dual space of linear forms on
3270:
3225:generalize Hopf algebras and carry a
3188:is not necessarily a homomorphism of
2364:The cohomology algebra (over a field
1040:was invertible (as may be required).
751:while co-associativity requires that
7651:
7336:
7324:
4291:. A Hopf algebra in such a category
3237:is a locally compact quantum group.
1323:denotes the kernel of the counit on
1139:(1989) established that the natural
1131:(and additionally the identity 1 of
102:
7579:Representations of algebraic groups
7529:
6203:with respect to the multiplication
5531:monoid in the usual algebraic sense
4924:are compatible: the multiplication
4597:{\displaystyle \varepsilon :H\to I}
3195:
3108:The relationship between Δ, ε, and
2984:{\displaystyle a(m):=\epsilon (a)m}
1094:
853:The connecting axiom requires that
51:satisfying a certain property. The
13:
6720:
6680:
6640:
6600:
6529:
6496:
6463:
6430:
6210:
6119:
6107:
5997:
5985:
5907:
5869:
5831:
5720:
5700:
5671:
5540:
5503:
5480:{\displaystyle 1=\{\varnothing \}}
5273:
5261:
4971:
4931:
4802:
4684:
4537:
4459:
4381:
4369:
4228:
3476:
3460:
3433:
3415:
3388:
3358:
2778:
1193:, where the right adjoint mapping
487:
409:
328:The definition of Hopf algebra is
144:) such that the following diagram
14:
8277:
7745:10.1090/S0002-9947-1994-1220906-5
6546:{\displaystyle {\mathcal {P}}(G)}
6513:{\displaystyle {\mathcal {O}}(G)}
6480:{\displaystyle {\mathcal {E}}(G)}
6447:{\displaystyle {\mathcal {C}}(G)}
6141:means exactly that every element
5922:{\displaystyle (H,\nabla ,\eta )}
5884:{\displaystyle (H,\nabla ,\eta )}
5802:
5518:{\displaystyle (H,\nabla ,\eta )}
5471:
4699:{\displaystyle (H,\nabla ,\eta )}
4152:in is isomorphic to the algebra
4119:-module is the separable algebra
3971:) with image a separable algebra
3079:{\displaystyle (af)(m):=f(S(a)m)}
649:Associativity then requires that
59:Hopf algebras occur naturally in
8209:Journal of Mathematical Sciences
8178:Journal of Mathematical Sciences
8119:An introduction to Hopf algebras
6401:{\displaystyle (C,\otimes ,s,I)}
6337:{\displaystyle (C,\otimes ,s,I)}
6250:. Thus, in the category of sets
5753:{\displaystyle \Delta (x)=(x,x)}
5373:
5027:
5018:
5009:
4895:
4777:
3282:is a (weak) Hopf algebra, so is
3245:Knizhnik–Zamolodchikov equations
3203:Hopf algebras are often used in
1153:Lagrange's theorem for subgroups
997:is sometimes required to have a
152:
7837:Group objects and Hopf algebras
7826:
7810:New directions in Hopf algebras
7797:
7752:
7716:
7599:
7567:
7552:
7523:
7503:The Concise Handbook of Algebra
7494:
7465:
7441:
7407:American Journal of Mathematics
6855:
5527:monoid in the categorical sense
5487:, as the unit object) a triple
1249:. Similarly, a Hopf subalgebra
277:
39:and a (counital coassociative)
7906:Hopf Algebras. An introduction
7393:
7382:
7365:
7348:
7272:Journal of High Energy Physics
7259:
7206:
7162:
7135:
7093:
7070:
7025:
6983:
6971:
6962:
6953:
6941:
6737:
6731:
6697:
6691:
6657:
6651:
6617:
6611:
6540:
6534:
6507:
6501:
6474:
6468:
6441:
6435:
6395:
6371:
6331:
6307:
6277:
6257:
6225:
6213:
6128:
6098:
6075:
6046:
6026:
6006:
5976:
5956:
5936:
5916:
5898:
5878:
5860:
5840:
5822:
5796:
5790:
5747:
5735:
5729:
5723:
5680:
5662:
5655:). At the same time, a triple
5596:
5590:
5555:
5543:
5512:
5494:
5415:
5395:
5348:
5306:
5282:
5252:
5207:
5195:
5189:
5177:
5165:
5158:
5146:
5140:
5128:
5065:
4877:
4835:
4811:
4793:
4759:
4717:
4693:
4675:
4624:
4588:
4546:
4510:
4474:
4396:
4360:
4340:
4298:
4256:is replaced by the 1-point set
4077:
4071:
4062:
4057:
4051:
4043:
4035:
4029:
4021:
4016:
4010:
4002:
3934:
3928:
3920:
3912:
3906:
3898:
3889:
3884:
3878:
3870:
3862:
3856:
3796:
3791:
3785:
3774:
3766:
3760:
3747:
3741:
3733:
3728:
3722:
3714:
3645:
3637:
3631:
3623:
3617:
3612:
3606:
3595:
3583:
3575:
3569:
3561:
3555:
3550:
3544:
3533:
3521:
3509:
3485:
3479:
3473:
3457:
3451:
3442:
3436:
3430:
3427:
3424:
3418:
3406:
3400:
3397:
3391:
3379:
3376:
3367:
3361:
3355:
3223:Locally compact quantum groups
3073:
3067:
3061:
3055:
3046:
3040:
3037:
3028:
2975:
2969:
2960:
2954:
2887:
2855:
2849:
2837:
2834:
2808:
2802:
2790:
2787:
2781:
2772:
2760:
2646:) or co-commutative (i.e. Δ =
2571:{\displaystyle G\times G\to G}
2562:
2530:
2518:
2502:
2490:
2474:
2456:
2443:
2440:
2428:
2024:(this rule is compatible with
1365:
271:
265:
256:
251:
245:
237:
229:
223:
210:
204:
196:
191:
185:
177:
1:
8117:Underwood, Robert G. (2011),
8019:American Mathematical Society
7897:
7609:Acta Applicandae Mathematicae
7583:American Mathematical Society
7478:American Mathematical Society
7375:Hopf Algebra: An Introduction
7358:Hopf Algebra: An Introduction
6416:Functional algebras on groups
6294:in the usual algebraic sense.
5780:is defined uniquely as well:
5081:is the left unit morphism in
340:(which is always possible if
7944:10.1007/978-3-540-30308-4_12
7849:Turaev & Virelizier 2017
7563:, Holden-Day, pp. 14–32
7389:Quantum groups lecture notes
7155:
6833:Hopf algebra of permutations
6793:Quasitriangular Hopf algebra
6767:{\displaystyle \circledast }
5773:{\displaystyle \varepsilon }
4997:{\displaystyle \varepsilon }
4519:{\displaystyle \eta :I\to H}
4214:. The separable subalgebras
3172:) is also a homomorphism of
2546:by the group multiplication
2408:, and the comultiplication
1991:Universal enveloping algebra
1293:is bijective, in which case
7:
8121:, Berlin: Springer-Verlag,
7839:, video of Simon Willerton.
7192:10.1103/physrevlett.69.2021
6873:is naturally isomorphic to
6786:
6300:. In the special case when
6196:{\displaystyle x^{-1}\in H}
5388:. In the monoidal category
4289:braided monoidal categories
4271:the antipode is the inverse
3176:-modules. However, the map
2723:be a Hopf algebra, and let
2229:ring of symmetric functions
1467:
10:
8282:
8081:Sweedler, Moss E. (1969),
7725:"Multiplier Hopf algebras"
7723:Van Daele, Alfons (1994).
7536:Cambridge University Press
7245:10.1103/PhysRevD.74.066008
6290:Hopf algebras are exactly
5713:is the diagonal operation
2635:
1147:is free of finite rank if
989:Properties of the antipode
8231:10.1007/s10958-009-9646-1
8155:10.1007/978-3-319-49834-8
6589:stereotype group algebras
5533:, i.e. if the operations
4828:in the monoidal category
4710:in the monoidal category
3298:is usually taken to be a
3140:are also isomorphisms of
2604:be a finite-dimensional,
1300:A normal Hopf subalgebra
378:{\displaystyle \{e_{k}\}}
325:in the above definition.
6848:
6084:{\displaystyle S:H\to H}
5602:{\displaystyle \eta (1)}
4633:{\displaystyle S:H\to H}
4168:is given by coproduct Δ(
3688:has a weakened antipode
3251:Multiplier Hopf algebras
3018:is its dual space, then
2360:Cohomology of Lie groups
1691:Representative functions
85:condensed matter physics
8191:10.1023/A:1020929201133
7631:10.1023/A:1022323609001
7561:Structure of Lie groups
7302:10.1007/jhep12(2017)090
7172:Physical Review Letters
6828:Sweedler's Hopf algebra
6364:, the Hopf algebras in
6298:Classical Hopf algebras
6167:has an inverse element
5706:{\displaystyle \Delta }
5529:if and only if it is a
5448:{\displaystyle \times }
5114:{\displaystyle \theta }
4977:{\displaystyle \Delta }
4937:{\displaystyle \nabla }
4113:rigid monoidal category
3265:Hopf group-(co)algebras
2705:noncommutative geometry
2085:Sweedler's Hopf algebra
1886:in this way, giving an
1594:from a finite group to
477:for co-multiplication:
16:Construction in algebra
8172:Akbarov, S.S. (2003).
7960:Fuchs, Jürgen (1992),
7783:10.1006/jabr.1999.7984
7559:Hochschild, G (1965),
7142:
7080:
7035:
6990:
6889:is a proper subset of
6768:
6744:
6704:
6664:
6624:
6571:
6570:{\displaystyle \odot }
6547:
6514:
6481:
6448:
6402:
6358:
6338:
6284:
6244:
6197:
6161:
6160:{\displaystyle x\in H}
6135:
6085:
6053:
6013:
5963:
5923:
5885:
5847:
5809:
5774:
5754:
5707:
5687:
5649:
5648:{\displaystyle x\in H}
5623:
5603:
5574:
5519:
5481:
5449:
5422:
5355:
5289:
5235:
5214:
5115:
5095:
5075:
4998:
4978:
4958:
4938:
4918:
4884:
4818:
4766:
4700:
4658:
4634:
4598:
4562:
4520:
4484:
4443:
4423:
4403:
4347:
4277:field with one element
4126:For example, a finite
4084:
3943:
3803:
3700:satisfying the axioms:
3652:
3492:
3080:
2985:
2894:
2622:
2598:
2572:
2537:
2398:
2378:
979:
844:
742:
640:
565:
468:
379:
301:
75:(via the concept of a
8266:Representation theory
7990:Annals of Mathematics
7575:Jantzen, Jens Carsten
7143:
7081:
7036:
6991:
6769:
6745:
6705:
6665:
6625:
6572:
6548:
6515:
6482:
6449:
6403:
6359:
6339:
6285:
6245:
6198:
6162:
6136:
6091:for such a bialgebra
6086:
6054:
6014:
5964:
5924:
5886:
5848:
5810:
5775:
5755:
5708:
5688:
5650:
5624:
5604:
5575:
5520:
5482:
5450:
5423:
5356:
5290:
5236:
5215:
5116:
5096:
5076:
4999:
4979:
4959:
4957:{\displaystyle \eta }
4939:
4919:
4885:
4819:
4767:
4701:
4659:
4635:
4599:
4563:
4521:
4485:
4444:
4424:
4404:
4348:
4085:
3982:, anti-isomorphic to
3944:
3804:
3653:
3493:
3229:. The algebra of all
3081:
2986:
2895:
2715:Representation theory
2623:
2599:
2573:
2538:
2399:
2379:
1789:Tannaka–Krein duality
1427:is a nonzero element
980:
845:
743:
641:
566:
469:
380:
302:
53:representation theory
7530:Abe, Eiichi (2004).
7400:Nichols, Warren D.;
7090:
7045:
7000:
6913:
6844:Notes and references
6838:Milnor–Moore theorem
6758:
6714:
6674:
6634:
6594:
6561:
6524:
6491:
6458:
6425:
6368:
6348:
6304:
6254:
6207:
6171:
6145:
6095:
6063:
6023:
5973:
5933:
5895:
5857:
5819:
5784:
5764:
5717:
5697:
5659:
5633:
5613:
5584:
5537:
5491:
5459:
5439:
5392:
5303:
5249:
5225:
5125:
5105:
5085:
5040:
4988:
4968:
4948:
4928:
4908:
4832:
4790:
4714:
4672:
4648:
4612:
4576:
4534:
4498:
4456:
4433:
4413:
4357:
4295:
3996:
3848:
3708:
3503:
3352:
3231:continuous functions
3025:
2948:
2754:
2612:
2588:
2550:
2415:
2388:
2368:
2211:is generated by {1,
2050:(again, extended to
860:
758:
656:
581:
484:
396:
389:for multiplication:
356:
171:
89:quantum field theory
8261:Monoidal categories
7402:Zoeller, M. Bettina
7294:2017JHEP...12..090A
7237:2006PhRvD..74f6008P
7184:1992PhRvL..69.2021H
6823:Anyonic Lie algebra
6808:Ribbon Hopf algebra
6798:Algebra/set analogy
6420:functional algebras
5760:(and the operation
4644:— are morphisms in
4568:(comultiplication),
4240:Analogy with groups
4196:) = 1 and antipode
3241:Quasi-Hopf algebras
3006:can be defined: if
1968:If and only if dim(
1419:Group-like elements
1331:is a Hopf ideal of
1072:admits an antipode
1013:(or more generally
974:
955:
938:
916:
897:
881:
839:
820:
798:
779:
737:
718:
696:
677:
625:
537:
453:
387:structure constants
348:Structure constants
281: for all
7138:
7076:
7031:
6986:
6861:The finiteness of
6764:
6740:
6700:
6660:
6620:
6567:
6543:
6510:
6477:
6444:
6398:
6354:
6334:
6280:
6240:
6193:
6157:
6131:
6081:
6049:
6009:
5959:
5919:
5881:
5843:
5805:
5770:
5750:
5703:
5683:
5645:
5619:
5599:
5570:
5515:
5477:
5445:
5418:
5351:
5285:
5231:
5210:
5111:
5091:
5071:
4994:
4974:
4954:
4934:
4914:
4880:
4814:
4762:
4696:
4654:
4630:
4594:
4558:
4516:
4480:
4439:
4419:
4399:
4343:
4123:mentioned above.
4080:
3939:
3799:
3648:
3488:
3471:
3276:Weak Hopf algebras
3271:Weak Hopf algebras
3259:multiplier algebra
3205:algebraic topology
3076:
2981:
2890:
2618:
2606:graded commutative
2594:
2568:
2533:
2394:
2374:
1783:Hopf algebra over
1693:on a compact group
1425:group-like element
1395:in a Hopf algebra
1386:field of fractions
1253:is left normal in
1158:A Hopf subalgebra
1127:are restricted to
1103:of a Hopf algebra
975:
956:
939:
920:
898:
882:
863:
840:
821:
802:
780:
761:
738:
719:
700:
678:
659:
636:
610:
609:
574:and the antipode:
561:
519:
518:
464:
435:
434:
375:
297:
283:
61:algebraic topology
8164:978-3-319-49833-1
8128:978-0-387-72765-3
8057:978-0-521-69524-4
8028:978-0-8218-0738-5
8011:Montgomery, Susan
7971:978-0-521-48412-1
7953:978-3-540-30307-7
7915:978-0-8247-0481-0
7656:. 2 (in German).
7592:978-0-8218-3527-2
7545:978-0-521-60489-5
7487:978-0-8218-7549-0
7215:Physical Review D
7178:(14): 2021–2025.
6776:stereotype spaces
6579:stereotype spaces
6357:{\displaystyle K}
6263:
6032:
5942:
5622:{\displaystyle H}
5434:cartesian product
5401:
5234:{\displaystyle C}
5174:
5094:{\displaystyle C}
4917:{\displaystyle H}
4657:{\displaystyle C}
4490:(multiplication),
4442:{\displaystyle C}
4422:{\displaystyle H}
3470:
2621:{\displaystyle A}
2597:{\displaystyle A}
2397:{\displaystyle G}
2384:) of a Lie group
2377:{\displaystyle K}
2350:
2349:
2151:⊗ 1, Δ(1) = 1 ⊗ 1
1979:symmetric algebra
1797:Regular functions
1446:primitive element
1111:and the antipode
600:
503:
425:
282:
162:Sweedler notation
103:Formal definition
97:LHC phenomenology
8273:
8242:
8224:
8203:
8193:
8167:
8139:
8113:
8076:
8039:
7982:
7956:
7926:
7891:
7885:
7879:
7873:
7864:
7858:
7852:
7846:
7840:
7830:
7824:
7823:
7801:
7795:
7794:
7776:
7756:
7750:
7749:
7747:
7729:
7720:
7714:
7708:
7702:
7696:
7690:
7684:
7678:
7677:
7649:
7643:
7642:
7624:
7603:
7597:
7595:
7571:
7565:
7564:
7556:
7550:
7549:
7527:
7521:
7520:
7509:. p. 307, C.42.
7498:
7492:
7491:
7469:
7463:
7457:
7451:
7445:
7439:
7438:
7397:
7391:
7386:
7380:
7379:
7369:
7363:
7362:
7352:
7346:
7340:
7334:
7328:
7322:
7321:
7287:
7263:
7257:
7256:
7230:
7210:
7204:
7203:
7166:
7149:
7147:
7145:
7144:
7139:
7085:
7083:
7082:
7077:
7057:
7056:
7040:
7038:
7037:
7032:
7012:
7011:
6995:
6993:
6992:
6987:
6937:
6936:
6907:
6898:
6859:
6773:
6771:
6770:
6765:
6749:
6747:
6746:
6741:
6730:
6729:
6724:
6723:
6709:
6707:
6706:
6701:
6690:
6689:
6684:
6683:
6669:
6667:
6666:
6661:
6650:
6649:
6644:
6643:
6629:
6627:
6626:
6621:
6610:
6609:
6604:
6603:
6576:
6574:
6573:
6568:
6552:
6550:
6549:
6544:
6533:
6532:
6519:
6517:
6516:
6511:
6500:
6499:
6486:
6484:
6483:
6478:
6467:
6466:
6453:
6451:
6450:
6445:
6434:
6433:
6407:
6405:
6404:
6399:
6363:
6361:
6360:
6355:
6343:
6341:
6340:
6335:
6289:
6287:
6286:
6281:
6264:
6261:
6249:
6247:
6246:
6241:
6202:
6200:
6199:
6194:
6186:
6185:
6166:
6164:
6163:
6158:
6140:
6138:
6137:
6132:
6090:
6088:
6087:
6082:
6058:
6056:
6055:
6050:
6033:
6030:
6018:
6016:
6015:
6010:
5968:
5966:
5965:
5960:
5943:
5940:
5928:
5926:
5925:
5920:
5890:
5888:
5887:
5882:
5852:
5850:
5849:
5844:
5814:
5812:
5811:
5806:
5779:
5777:
5776:
5771:
5759:
5757:
5756:
5751:
5712:
5710:
5709:
5704:
5692:
5690:
5689:
5684:
5654:
5652:
5651:
5646:
5628:
5626:
5625:
5620:
5608:
5606:
5605:
5600:
5579:
5577:
5576:
5571:
5524:
5522:
5521:
5516:
5486:
5484:
5483:
5478:
5454:
5452:
5451:
5446:
5427:
5425:
5424:
5419:
5402:
5399:
5377:
5360:
5358:
5357:
5352:
5299:in the category
5294:
5292:
5291:
5286:
5240:
5238:
5237:
5232:
5219:
5217:
5216:
5211:
5176:
5175:
5173:
5168:
5163:
5120:
5118:
5117:
5112:
5100:
5098:
5097:
5092:
5080:
5078:
5077:
5072:
5052:
5051:
5031:
5022:
5013:
5003:
5001:
5000:
4995:
4983:
4981:
4980:
4975:
4963:
4961:
4960:
4955:
4943:
4941:
4940:
4935:
4923:
4921:
4920:
4915:
4899:
4889:
4887:
4886:
4881:
4823:
4821:
4820:
4815:
4781:
4771:
4769:
4768:
4763:
4705:
4703:
4702:
4697:
4663:
4661:
4660:
4655:
4639:
4637:
4636:
4631:
4603:
4601:
4600:
4595:
4567:
4565:
4564:
4559:
4525:
4523:
4522:
4517:
4489:
4487:
4486:
4481:
4448:
4446:
4445:
4440:
4429:is an object in
4428:
4426:
4425:
4420:
4408:
4406:
4405:
4400:
4352:
4350:
4349:
4344:
4089:
4087:
4086:
4081:
4061:
4060:
4039:
4038:
4020:
4019:
3948:
3946:
3945:
3940:
3938:
3937:
3916:
3915:
3888:
3887:
3866:
3865:
3808:
3806:
3805:
3800:
3795:
3794:
3770:
3769:
3751:
3750:
3732:
3731:
3657:
3655:
3654:
3649:
3641:
3640:
3616:
3615:
3579:
3578:
3554:
3553:
3497:
3495:
3494:
3489:
3472:
3468:
3196:Related concepts
3085:
3083:
3082:
3077:
2990:
2988:
2987:
2982:
2899:
2897:
2896:
2891:
2883:
2882:
2867:
2866:
2833:
2832:
2820:
2819:
2735:-modules. Then,
2627:
2625:
2624:
2619:
2603:
2601:
2600:
2595:
2577:
2575:
2574:
2569:
2542:
2540:
2539:
2534:
2517:
2516:
2489:
2488:
2455:
2454:
2427:
2426:
2403:
2401:
2400:
2395:
2383:
2381:
2380:
2375:
2120:different from 2
2116:is a field with
1983:exterior algebra
1480:Comultiplication
1472:
1471:
1343:and epimorphism
1095:Hopf subalgebras
1026:antihomomorphism
984:
982:
981:
976:
973:
968:
954:
947:
937:
928:
915:
910:
896:
890:
880:
871:
849:
847:
846:
841:
838:
829:
819:
810:
797:
788:
778:
769:
747:
745:
744:
739:
736:
731:
717:
712:
695:
690:
676:
671:
645:
643:
642:
637:
635:
634:
624:
618:
608:
596:
595:
570:
568:
567:
562:
560:
559:
547:
546:
536:
527:
517:
499:
498:
473:
471:
470:
465:
463:
462:
452:
447:
433:
421:
420:
408:
407:
384:
382:
381:
376:
371:
370:
320:commutative ring
306:
304:
303:
298:
284:
280:
255:
254:
233:
232:
214:
213:
195:
194:
156:
121:together with a
49:antihomomorphism
8281:
8280:
8276:
8275:
8274:
8272:
8271:
8270:
8246:
8245:
8165:
8129:
8095:
8058:
8029:
7972:
7954:
7932:Cartier, Pierre
7916:
7900:
7895:
7894:
7886:
7882:
7874:
7867:
7859:
7855:
7847:
7843:
7831:
7827:
7820:
7802:
7798:
7757:
7753:
7727:
7721:
7717:
7711:Montgomery 1993
7709:
7705:
7697:
7693:
7685:
7681:
7666:10.2307/1968985
7650:
7646:
7604:
7600:
7593:
7572:
7568:
7557:
7553:
7546:
7528:
7524:
7517:
7507:Springer-Verlag
7499:
7495:
7488:
7480:. p. 149.
7470:
7466:
7458:
7454:
7448:Montgomery 1993
7446:
7442:
7420:10.2307/2374514
7398:
7394:
7387:
7383:
7370:
7366:
7353:
7349:
7341:
7337:
7329:
7325:
7266:Abreu, Samuel;
7264:
7260:
7211:
7207:
7167:
7163:
7158:
7153:
7152:
7091:
7088:
7087:
7052:
7048:
7046:
7043:
7042:
7007:
7003:
7001:
6998:
6997:
6920:
6916:
6914:
6911:
6910:
6908:
6901:
6860:
6856:
6851:
6846:
6789:
6759:
6756:
6755:
6725:
6719:
6718:
6717:
6715:
6712:
6711:
6685:
6679:
6678:
6677:
6675:
6672:
6671:
6645:
6639:
6638:
6637:
6635:
6632:
6631:
6605:
6599:
6598:
6597:
6595:
6592:
6591:
6562:
6559:
6558:
6528:
6527:
6525:
6522:
6521:
6495:
6494:
6492:
6489:
6488:
6462:
6461:
6459:
6456:
6455:
6429:
6428:
6426:
6423:
6422:
6418:. The standard
6410:described above
6369:
6366:
6365:
6349:
6346:
6345:
6305:
6302:
6301:
6260:
6255:
6252:
6251:
6208:
6205:
6204:
6178:
6174:
6172:
6169:
6168:
6146:
6143:
6142:
6096:
6093:
6092:
6064:
6061:
6060:
6029:
6024:
6021:
6020:
5974:
5971:
5970:
5939:
5934:
5931:
5930:
5896:
5893:
5892:
5858:
5855:
5854:
5820:
5817:
5816:
5785:
5782:
5781:
5765:
5762:
5761:
5718:
5715:
5714:
5698:
5695:
5694:
5660:
5657:
5656:
5634:
5631:
5630:
5614:
5611:
5610:
5585:
5582:
5581:
5538:
5535:
5534:
5492:
5489:
5488:
5460:
5457:
5456:
5440:
5437:
5436:
5398:
5393:
5390:
5389:
5379:
5364:
5304:
5301:
5300:
5250:
5247:
5246:
5226:
5223:
5222:
5169:
5164:
5162:
5161:
5126:
5123:
5122:
5106:
5103:
5102:
5086:
5083:
5082:
5047:
5043:
5041:
5038:
5037:
5033:
5024:
5015:
4989:
4986:
4985:
4984:and the counit
4969:
4966:
4965:
4949:
4946:
4945:
4929:
4926:
4925:
4909:
4906:
4905:
4901:
4833:
4830:
4829:
4791:
4788:
4787:
4783:
4715:
4712:
4711:
4673:
4670:
4669:
4649:
4646:
4645:
4613:
4610:
4609:
4577:
4574:
4573:
4535:
4532:
4531:
4499:
4496:
4495:
4457:
4454:
4453:
4434:
4431:
4430:
4414:
4411:
4410:
4358:
4355:
4354:
4296:
4293:
4292:
4285:
4242:
4231:
4229:Hopf algebroids
4212:
4205:
4194:
4187:
4180:
4173:
4142:
4135:
4050:
4046:
4028:
4024:
4009:
4005:
3997:
3994:
3993:
3980:
3966:
3927:
3923:
3905:
3901:
3877:
3873:
3855:
3851:
3849:
3846:
3845:
3840:
3826:
3784:
3780:
3759:
3755:
3740:
3736:
3721:
3717:
3709:
3706:
3705:
3630:
3626:
3605:
3601:
3568:
3564:
3543:
3539:
3504:
3501:
3500:
3466:
3353:
3350:
3349:
3273:
3198:
3026:
3023:
3022:
2949:
2946:
2945:
2936:
2929:
2878:
2874:
2862:
2858:
2828:
2824:
2815:
2811:
2755:
2752:
2751:
2717:
2640:
2634:
2613:
2610:
2609:
2589:
2586:
2585:
2551:
2548:
2547:
2512:
2508:
2484:
2480:
2450:
2446:
2422:
2418:
2416:
2413:
2412:
2389:
2386:
2385:
2369:
2366:
2365:
2362:
2337:
2330:
2317:
2303:
2297:
2290:
2280:
2270:
2262:
2255:
2243:
2207:The underlying
2070:if and only if
1933:, Δ(1) = 1 ⊗ 1
1890:of categories.
1888:antiequivalence
1875:if and only if
1846:
1801:algebraic group
1772:if and only if
1743:
1679:if and only if
1650:
1576:if and only if
1470:
1421:
1379:integral domain
1368:
1288:
1278:
1262:
1232:
1225:
1205:
1198:
1171:
1137:Bettina Zoeller
1097:
1068:If a bialgebra
1052:
1015:quasitriangular
991:
969:
960:
948:
943:
929:
924:
911:
902:
891:
886:
872:
867:
861:
858:
857:
830:
825:
811:
806:
789:
784:
770:
765:
759:
756:
755:
732:
723:
713:
704:
691:
682:
672:
663:
657:
654:
653:
630:
626:
619:
614:
604:
591:
587:
582:
579:
578:
555:
551:
542:
538:
528:
523:
507:
494:
490:
485:
482:
481:
458:
454:
448:
439:
429:
416:
412:
403:
399:
397:
394:
393:
366:
362:
357:
354:
353:
352:Fixing a basis
350:
278:
244:
240:
222:
218:
203:
199:
184:
180:
172:
169:
168:
158:
105:
17:
12:
11:
5:
8279:
8269:
8268:
8263:
8258:
8244:
8243:
8215:(4): 459–586.
8204:
8184:(2): 179–349.
8169:
8163:
8140:
8127:
8114:
8093:
8078:
8056:
8040:
8027:
8007:
7983:
7970:
7957:
7952:
7928:
7914:
7899:
7896:
7893:
7892:
7880:
7865:
7863:, p. 482.
7853:
7841:
7825:
7818:
7796:
7767:(2): 385–438.
7751:
7738:(2): 917–932.
7715:
7703:
7699:Underwood 2011
7691:
7687:Underwood 2011
7679:
7644:
7615:(1–3): 55–83.
7598:
7591:
7566:
7551:
7544:
7538:. p. 59.
7522:
7516:978-0792370727
7515:
7493:
7486:
7464:
7460:Underwood 2011
7452:
7440:
7414:(2): 381–385,
7392:
7381:
7378:. p. 151.
7364:
7361:. p. 153.
7347:
7343:Underwood 2011
7335:
7331:Underwood 2011
7323:
7258:
7228:hep-th/0608038
7205:
7160:
7159:
7157:
7154:
7151:
7150:
7137:
7134:
7131:
7128:
7125:
7122:
7119:
7116:
7113:
7110:
7107:
7104:
7101:
7098:
7095:
7075:
7072:
7069:
7066:
7063:
7060:
7055:
7051:
7030:
7027:
7024:
7021:
7018:
7015:
7010:
7006:
6985:
6982:
6979:
6976:
6973:
6970:
6967:
6964:
6961:
6958:
6955:
6952:
6949:
6946:
6943:
6940:
6935:
6932:
6929:
6926:
6923:
6919:
6899:
6853:
6852:
6850:
6847:
6845:
6842:
6841:
6840:
6835:
6830:
6825:
6820:
6815:
6810:
6805:
6800:
6795:
6788:
6785:
6784:
6783:
6763:
6739:
6736:
6733:
6728:
6722:
6699:
6696:
6693:
6688:
6682:
6659:
6656:
6653:
6648:
6642:
6619:
6616:
6613:
6608:
6602:
6585:Group algebras
6582:
6566:
6542:
6539:
6536:
6531:
6509:
6506:
6503:
6498:
6476:
6473:
6470:
6465:
6443:
6440:
6437:
6432:
6413:
6397:
6394:
6391:
6388:
6385:
6382:
6379:
6376:
6373:
6353:
6333:
6330:
6327:
6324:
6321:
6318:
6315:
6312:
6309:
6295:
6279:
6276:
6273:
6270:
6267:
6259:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6212:
6192:
6189:
6184:
6181:
6177:
6156:
6153:
6150:
6130:
6127:
6124:
6121:
6118:
6115:
6112:
6109:
6106:
6103:
6100:
6080:
6077:
6074:
6071:
6068:
6048:
6045:
6042:
6039:
6036:
6028:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5958:
5955:
5952:
5949:
5946:
5938:
5918:
5915:
5912:
5909:
5906:
5903:
5900:
5880:
5877:
5874:
5871:
5868:
5865:
5862:
5842:
5839:
5836:
5833:
5830:
5827:
5824:
5804:
5801:
5798:
5795:
5792:
5789:
5769:
5749:
5746:
5743:
5740:
5737:
5734:
5731:
5728:
5725:
5722:
5702:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5644:
5641:
5638:
5618:
5598:
5595:
5592:
5589:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5514:
5511:
5508:
5505:
5502:
5499:
5496:
5476:
5473:
5470:
5467:
5464:
5444:
5417:
5414:
5411:
5408:
5405:
5397:
5371:
5370:
5369:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5314:
5311:
5308:
5284:
5281:
5278:
5275:
5272:
5269:
5266:
5263:
5260:
5257:
5254:
5245:The quintuple
5243:
5242:
5230:
5209:
5206:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5172:
5167:
5160:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5110:
5090:
5070:
5067:
5064:
5061:
5058:
5055:
5050:
5046:
5025:
5016:
5007:
5006:
5005:
4993:
4973:
4953:
4933:
4913:
4893:
4892:
4891:
4879:
4876:
4873:
4870:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4837:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4786:2) the triple
4775:
4774:
4773:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4668:1) the triple
4653:
4642:
4641:
4629:
4626:
4623:
4620:
4617:
4606:
4605:
4593:
4590:
4587:
4584:
4581:
4570:
4569:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4528:
4527:
4515:
4512:
4509:
4506:
4503:
4492:
4491:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4438:
4418:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4353:is a sextuple
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4284:
4281:
4273:
4272:
4269:
4266:
4263:
4260:
4257:
4241:
4238:
4235:Hopf algebroid
4230:
4227:
4210:
4203:
4192:
4185:
4178:
4171:
4140:
4133:
4111:-modules is a
4105:
4104:
4100:
4099:
4079:
4076:
4073:
4070:
4067:
4064:
4059:
4056:
4053:
4049:
4045:
4042:
4037:
4034:
4031:
4027:
4023:
4018:
4015:
4012:
4008:
4004:
4001:
3991:
3978:
3962:
3936:
3933:
3930:
3926:
3922:
3919:
3914:
3911:
3908:
3904:
3900:
3897:
3894:
3891:
3886:
3883:
3880:
3876:
3872:
3869:
3864:
3861:
3858:
3854:
3843:
3838:
3822:
3798:
3793:
3790:
3787:
3783:
3779:
3776:
3773:
3768:
3765:
3762:
3758:
3754:
3749:
3746:
3743:
3739:
3735:
3730:
3727:
3724:
3720:
3716:
3713:
3702:
3701:
3682:
3681:
3661:
3660:
3659:
3658:
3647:
3644:
3639:
3636:
3633:
3629:
3625:
3622:
3619:
3614:
3611:
3608:
3604:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3577:
3574:
3571:
3567:
3563:
3560:
3557:
3552:
3549:
3546:
3542:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3498:
3487:
3484:
3481:
3478:
3475:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3344:
3343:
3314:and counit ε:
3272:
3269:
3197:
3194:
3087:
3086:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
2992:
2991:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2934:
2927:
2901:
2900:
2889:
2886:
2881:
2877:
2873:
2870:
2865:
2861:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2831:
2827:
2823:
2818:
2814:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2747:-module, with
2716:
2713:
2700:quantum groups
2673:is defined by
2650:∘ Δ where the
2636:Main article:
2633:
2630:
2617:
2593:
2582:Theorem (Hopf)
2567:
2564:
2561:
2558:
2555:
2544:
2543:
2532:
2529:
2526:
2523:
2520:
2515:
2511:
2507:
2504:
2501:
2498:
2495:
2492:
2487:
2483:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2453:
2449:
2445:
2442:
2439:
2436:
2433:
2430:
2425:
2421:
2393:
2373:
2361:
2358:
2348:
2347:
2345:
2342:
2339:
2335:
2328:
2320:
2315:
2307:
2301:
2295:
2285:
2275:
2268:
2260:
2253:
2239:
2233:
2231:
2225:
2224:
2205:
2202:
2199:
2170:
2152:
2121:
2118:characteristic
2111:
2081:
2080:
2078:
2075:
2068:
2055:
2042:) = 0 for all
2033:
2002:
1994:
1987:
1986:
1976:
1973:
1966:
1944:
1934:
1911:
1903:
1896:Tensor algebra
1892:
1891:
1880:
1873:
1870:
1848:
1842:
1828:
1805:
1803:
1793:
1792:
1777:
1770:
1767:
1745:
1739:
1725:
1702:
1694:
1687:
1686:
1684:
1677:
1674:
1652:
1646:
1632:
1609:
1603:
1587:
1586:
1584:
1581:
1574:
1553:
1545:) = 1 for all
1536:
1514:
1506:
1497:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1469:
1466:
1420:
1417:
1367:
1364:
1286:
1276:
1260:
1230:
1223:
1203:
1200:is defined by
1196:
1169:
1096:
1093:
1048:
990:
987:
986:
985:
972:
967:
964:
959:
953:
946:
942:
936:
933:
927:
923:
919:
914:
909:
906:
901:
895:
889:
885:
879:
876:
870:
866:
851:
850:
837:
834:
828:
824:
818:
815:
809:
805:
801:
796:
793:
787:
783:
777:
774:
768:
764:
749:
748:
735:
730:
727:
722:
716:
711:
708:
703:
699:
694:
689:
686:
681:
675:
670:
667:
662:
647:
646:
633:
629:
623:
617:
613:
607:
603:
599:
594:
590:
586:
572:
571:
558:
554:
550:
545:
541:
535:
532:
526:
522:
516:
513:
510:
506:
502:
497:
493:
489:
475:
474:
461:
457:
451:
446:
443:
438:
432:
428:
424:
419:
415:
411:
406:
402:
374:
369:
365:
361:
349:
346:
308:
307:
296:
293:
290:
287:
276:
273:
270:
267:
264:
261:
258:
253:
250:
247:
243:
239:
236:
231:
228:
225:
221:
217:
212:
209:
206:
202:
198:
193:
190:
187:
183:
179:
176:
150:
104:
101:
27:, named after
15:
9:
6:
4:
3:
2:
8278:
8267:
8264:
8262:
8259:
8257:
8256:Hopf algebras
8254:
8253:
8251:
8240:
8236:
8232:
8228:
8223:
8218:
8214:
8210:
8205:
8201:
8197:
8192:
8187:
8183:
8179:
8175:
8170:
8166:
8160:
8156:
8152:
8148:
8147:
8141:
8138:
8134:
8130:
8124:
8120:
8115:
8112:
8108:
8104:
8100:
8096:
8094:9780805392548
8090:
8086:
8085:
8084:Hopf algebras
8079:
8075:
8071:
8067:
8063:
8059:
8053:
8049:
8045:
8041:
8038:
8034:
8030:
8024:
8020:
8016:
8012:
8008:
8006:
8002:
7998:
7995:
7991:
7987:
7984:
7981:
7977:
7973:
7967:
7963:
7958:
7955:
7949:
7945:
7941:
7937:
7933:
7929:
7925:
7921:
7917:
7911:
7907:
7902:
7901:
7889:
7884:
7877:
7872:
7870:
7862:
7857:
7850:
7845:
7838:
7834:
7829:
7821:
7819:9780521815123
7815:
7811:
7807:
7800:
7792:
7788:
7784:
7780:
7775:
7770:
7766:
7762:
7755:
7746:
7741:
7737:
7733:
7726:
7719:
7713:, p. 203
7712:
7707:
7700:
7695:
7688:
7683:
7675:
7671:
7667:
7663:
7659:
7655:
7648:
7640:
7636:
7632:
7628:
7623:
7618:
7614:
7610:
7602:
7596:, section 2.3
7594:
7588:
7584:
7580:
7576:
7570:
7562:
7555:
7547:
7541:
7537:
7533:
7532:Hopf Algebras
7526:
7518:
7512:
7508:
7504:
7497:
7489:
7483:
7479:
7475:
7468:
7461:
7456:
7449:
7444:
7437:
7433:
7429:
7425:
7421:
7417:
7413:
7409:
7408:
7403:
7396:
7390:
7385:
7377:
7376:
7368:
7360:
7359:
7351:
7344:
7339:
7332:
7327:
7319:
7315:
7311:
7307:
7303:
7299:
7295:
7291:
7286:
7281:
7277:
7273:
7269:
7262:
7254:
7250:
7246:
7242:
7238:
7234:
7229:
7224:
7221:(6): 066008.
7220:
7216:
7209:
7201:
7197:
7193:
7189:
7185:
7181:
7177:
7173:
7165:
7161:
7132:
7129:
7126:
7123:
7120:
7117:
7114:
7111:
7108:
7105:
7102:
7099:
7096:
7073:
7067:
7064:
7061:
7058:
7053:
7049:
7028:
7022:
7019:
7016:
7013:
7008:
7004:
6980:
6977:
6974:
6968:
6965:
6959:
6956:
6950:
6947:
6944:
6938:
6933:
6930:
6927:
6924:
6921:
6917:
6906:
6904:
6896:
6892:
6888:
6884:
6880:
6876:
6872:
6868:
6865:implies that
6864:
6858:
6854:
6839:
6836:
6834:
6831:
6829:
6826:
6824:
6821:
6819:
6816:
6814:
6811:
6809:
6806:
6804:
6801:
6799:
6796:
6794:
6791:
6790:
6781:
6777:
6761:
6753:
6734:
6726:
6694:
6686:
6654:
6646:
6614:
6606:
6590:
6586:
6583:
6580:
6564:
6556:
6537:
6504:
6471:
6438:
6421:
6417:
6414:
6411:
6392:
6389:
6386:
6383:
6380:
6377:
6374:
6351:
6328:
6325:
6322:
6319:
6316:
6313:
6310:
6299:
6296:
6293:
6274:
6271:
6268:
6265:
6237:
6234:
6231:
6228:
6222:
6219:
6216:
6190:
6187:
6182:
6179:
6175:
6154:
6151:
6148:
6125:
6122:
6116:
6113:
6110:
6104:
6101:
6078:
6072:
6069:
6066:
6043:
6040:
6037:
6034:
6003:
6000:
5994:
5991:
5988:
5982:
5979:
5953:
5950:
5947:
5944:
5913:
5910:
5904:
5901:
5875:
5872:
5866:
5863:
5837:
5834:
5828:
5825:
5799:
5793:
5787:
5767:
5744:
5741:
5738:
5732:
5726:
5677:
5674:
5668:
5665:
5642:
5639:
5636:
5616:
5593:
5587:
5567:
5564:
5561:
5558:
5552:
5549:
5546:
5532:
5528:
5509:
5506:
5500:
5497:
5465:
5462:
5442:
5435:
5431:
5412:
5409:
5406:
5403:
5387:
5384:
5383:
5382:
5378:
5376:
5367:
5366:
5365:
5362:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5298:
5279:
5276:
5270:
5267:
5264:
5258:
5255:
5228:
5204:
5201:
5198:
5192:
5186:
5183:
5180:
5170:
5155:
5152:
5149:
5143:
5137:
5134:
5131:
5108:
5088:
5068:
5062:
5059:
5056:
5053:
5048:
5044:
5035:
5034:
5032:
5030:
5023:
5021:
5014:
5012:
4991:
4951:
4944:and the unit
4911:
4903:
4902:
4900:
4898:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4827:
4808:
4805:
4799:
4796:
4785:
4784:
4782:
4780:
4756:
4753:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4709:
4690:
4687:
4681:
4678:
4667:
4666:
4665:
4651:
4627:
4621:
4618:
4615:
4608:
4607:
4591:
4585:
4582:
4579:
4572:
4571:
4555:
4552:
4549:
4543:
4540:
4530:
4529:
4513:
4507:
4504:
4501:
4494:
4493:
4477:
4471:
4468:
4465:
4462:
4452:
4451:
4450:
4436:
4416:
4393:
4390:
4387:
4384:
4378:
4375:
4372:
4366:
4363:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4290:
4280:
4278:
4270:
4267:
4264:
4261:
4258:
4255:
4251:
4250:
4249:
4247:
4237:
4236:
4226:
4223:
4221:
4217:
4213:
4206:
4199:
4195:
4188:
4181:
4174:
4167:
4163:
4159:
4155:
4151:
4147:
4143:
4136:
4129:
4124:
4122:
4118:
4114:
4110:
4102:
4101:
4097:
4093:
4074:
4068:
4065:
4054:
4047:
4040:
4032:
4025:
4013:
4006:
3999:
3992:
3989:
3985:
3981:
3974:
3970:
3965:
3960:
3956:
3952:
3931:
3924:
3917:
3909:
3902:
3895:
3892:
3881:
3874:
3867:
3859:
3852:
3844:
3841:
3834:
3830:
3825:
3820:
3816:
3812:
3788:
3781:
3777:
3771:
3763:
3756:
3752:
3744:
3737:
3725:
3718:
3711:
3704:
3703:
3699:
3695:
3691:
3687:
3684:
3683:
3679:
3675:
3671:
3667:
3663:
3662:
3642:
3634:
3627:
3620:
3609:
3602:
3598:
3592:
3589:
3586:
3580:
3572:
3565:
3558:
3547:
3540:
3536:
3530:
3527:
3524:
3518:
3515:
3512:
3506:
3499:
3482:
3463:
3454:
3448:
3445:
3439:
3421:
3412:
3409:
3403:
3394:
3385:
3382:
3373:
3370:
3364:
3348:
3347:
3346:
3345:
3341:
3337:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3300:
3299:
3297:
3293:
3289:
3285:
3281:
3277:
3268:
3266:
3262:
3260:
3256:
3255:Hopf algebras
3252:
3248:
3246:
3242:
3238:
3236:
3232:
3228:
3224:
3220:
3218:
3215:groups of an
3214:
3210:
3206:
3202:
3193:
3191:
3187:
3183:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3151:
3147:
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3115:
3111:
3106:
3104:
3100:
3096:
3092:
3070:
3064:
3058:
3052:
3049:
3043:
3034:
3031:
3021:
3020:
3019:
3017:
3013:
3009:
3005:
3001:
2997:
2978:
2972:
2966:
2963:
2957:
2951:
2944:
2943:
2942:
2940:
2933:
2926:
2922:
2918:
2914:
2910:
2906:
2884:
2879:
2875:
2871:
2868:
2863:
2859:
2852:
2846:
2843:
2840:
2829:
2825:
2821:
2816:
2812:
2805:
2799:
2796:
2793:
2784:
2775:
2769:
2766:
2763:
2757:
2750:
2749:
2748:
2746:
2742:
2738:
2734:
2730:
2726:
2722:
2712:
2710:
2706:
2702:
2701:
2696:
2695:quantizations
2692:
2688:
2684:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
2653:
2649:
2645:
2639:
2638:quantum group
2629:
2615:
2607:
2591:
2583:
2579:
2565:
2559:
2556:
2553:
2527:
2524:
2521:
2513:
2509:
2505:
2499:
2496:
2493:
2485:
2481:
2477:
2471:
2468:
2465:
2462:
2459:
2451:
2447:
2437:
2434:
2431:
2423:
2419:
2411:
2410:
2409:
2407:
2391:
2371:
2357:
2355:
2346:
2343:
2340:
2338:
2331:
2324:
2321:
2318:
2311:
2308:
2306:
2304:
2294:
2288:
2284:
2278:
2274:
2267:
2263:
2256:
2247:
2242:
2238:
2234:
2232:
2230:
2227:
2226:
2222:
2218:
2214:
2210:
2206:
2203:
2200:
2198:
2194:
2190:
2186:
2182:
2178:
2174:
2171:
2168:
2164:
2160:
2156:
2153:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2119:
2115:
2112:
2109:
2105:
2101:
2097:
2093:
2089:
2086:
2083:
2082:
2079:
2076:
2073:
2069:
2067:
2063:
2059:
2056:
2053:
2049:
2045:
2041:
2037:
2034:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
2001:
1998:
1995:
1992:
1989:
1988:
1984:
1980:
1977:
1974:
1971:
1967:
1964:
1960:
1956:
1952:
1948:
1945:
1942:
1938:
1935:
1932:
1928:
1924:
1920:
1916:
1912:
1910:
1907:
1904:
1901:
1897:
1894:
1893:
1889:
1885:
1881:
1878:
1874:
1871:
1868:
1864:
1860:
1856:
1852:
1849:
1845:
1840:
1836:
1832:
1829:
1826:
1822:
1818:
1814:
1810:
1806:
1804:
1802:
1798:
1795:
1794:
1790:
1786:
1782:
1778:
1775:
1771:
1768:
1765:
1761:
1757:
1753:
1749:
1746:
1742:
1737:
1733:
1729:
1726:
1723:
1719:
1715:
1711:
1707:
1703:
1701:
1698:
1697:compact group
1695:
1692:
1689:
1688:
1685:
1682:
1678:
1675:
1672:
1668:
1664:
1660:
1656:
1653:
1649:
1644:
1640:
1636:
1633:
1630:
1626:
1622:
1618:
1614:
1610:
1608:
1605:finite group
1604:
1601:
1597:
1593:
1589:
1588:
1585:
1582:
1579:
1575:
1573:
1569:
1565:
1561:
1557:
1554:
1552:
1548:
1544:
1540:
1537:
1535:
1531:
1527:
1523:
1519:
1515:
1513:
1510:
1507:
1505:
1502:
1501:group algebra
1499:
1498:
1494:
1492:Cocommutative
1491:
1488:
1485:
1482:
1479:
1476:
1474:
1473:
1465:
1463:
1459:
1455:
1451:
1448:
1447:
1442:
1438:
1434:
1430:
1426:
1416:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1387:
1383:
1380:
1376:
1373:
1363:
1361:
1357:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1318:
1315:
1311:
1307:
1303:
1298:
1296:
1292:
1285:
1281:
1275:
1271:
1267:
1263:
1256:
1252:
1248:
1244:
1240:
1236:
1229:
1222:
1218:
1214:
1210:
1206:
1199:
1192:
1188:
1184:
1180:
1176:
1172:
1165:
1161:
1156:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1099:A subalgebra
1092:
1090:
1086:
1081:
1079:
1075:
1071:
1066:
1064:
1060:
1056:
1051:
1046:
1041:
1039:
1035:
1031:
1027:
1023:
1018:
1016:
1012:
1011:cocommutative
1008:
1004:
1000:
996:
993:The antipode
970:
965:
962:
957:
951:
944:
940:
934:
931:
925:
921:
917:
912:
907:
904:
899:
893:
887:
883:
877:
874:
868:
864:
856:
855:
854:
835:
832:
826:
822:
816:
813:
807:
803:
799:
794:
791:
785:
781:
775:
772:
766:
762:
754:
753:
752:
733:
728:
725:
720:
714:
709:
706:
701:
697:
692:
687:
684:
679:
673:
668:
665:
660:
652:
651:
650:
631:
627:
621:
615:
611:
605:
601:
597:
592:
588:
584:
577:
576:
575:
556:
552:
548:
543:
539:
533:
530:
524:
520:
514:
511:
508:
504:
500:
495:
491:
480:
479:
478:
459:
455:
449:
444:
441:
436:
430:
426:
422:
417:
413:
404:
400:
392:
391:
390:
388:
367:
363:
345:
343:
339:
335:
331:
326:
324:
321:
317:
313:
294:
291:
288:
285:
274:
268:
262:
259:
248:
241:
234:
226:
219:
215:
207:
200:
188:
181:
174:
167:
166:
165:
163:
157:
155:
149:
147:
143:
139:
135:
131:
127:
125:
120:
117:
113:
110:
100:
98:
94:
93:string theory
90:
86:
82:
78:
74:
70:
66:
62:
57:
54:
50:
46:
42:
38:
35:associative)
34:
30:
26:
22:
8212:
8208:
8181:
8177:
8145:
8118:
8083:
8047:
8044:Street, Ross
8014:
7961:
7935:
7905:
7888:Akbarov 2009
7883:
7876:Akbarov 2003
7861:Akbarov 2009
7856:
7844:
7828:
7809:
7799:
7774:math/9805116
7764:
7760:
7754:
7735:
7731:
7718:
7706:
7701:, p. 36
7694:
7689:, p. 57
7682:
7660:(1): 22–52.
7657:
7654:Ann. of Math
7653:
7647:
7622:math/0410468
7612:
7608:
7601:
7578:
7569:
7560:
7554:
7531:
7525:
7502:
7496:
7473:
7467:
7462:, p. 82
7455:
7450:, p. 36
7443:
7411:
7405:
7395:
7384:
7374:
7367:
7357:
7350:
7345:, p. 62
7338:
7333:, p. 55
7326:
7275:
7271:
7268:Britto, Ruth
7261:
7218:
7214:
7208:
7175:
7171:
7164:
6890:
6886:
6882:
6878:
6874:
6870:
6866:
6862:
6857:
6813:Superalgebra
6751:
6584:
6554:
6415:
6297:
5385:
5380:
5372:
5363:
5296:
5244:
5026:
5017:
5008:
4894:
4776:
4643:
4286:
4274:
4253:
4245:
4243:
4232:
4224:
4219:
4215:
4208:
4201:
4197:
4190:
4183:
4176:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4138:
4131:
4125:
4120:
4116:
4108:
4106:
4095:
4091:
3987:
3983:
3976:
3972:
3968:
3963:
3958:
3954:
3950:
3836:
3832:
3828:
3823:
3818:
3814:
3810:
3697:
3693:
3689:
3685:
3677:
3673:
3669:
3665:
3339:
3335:
3331:
3327:
3323:
3319:
3315:
3311:
3307:
3303:
3295:
3291:
3287:
3283:
3279:
3274:
3263:
3249:
3239:
3221:
3199:
3189:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3145:
3141:
3137:
3133:
3129:
3125:
3121:
3117:
3113:
3109:
3107:
3102:
3098:
3094:
3090:
3088:
3015:
3014:-module and
3011:
3007:
3003:
2999:
2995:
2993:
2938:
2931:
2924:
2920:
2916:
2912:
2908:
2904:
2902:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2718:
2708:
2698:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2658:
2654:
2651:
2647:
2641:
2581:
2580:
2545:
2363:
2353:
2351:
2333:
2326:
2322:
2313:
2309:
2299:
2292:
2286:
2282:
2276:
2272:
2265:
2258:
2251:
2249:
2248:≥ 1):
2245:
2240:
2236:
2220:
2216:
2212:
2209:vector space
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2166:
2162:
2158:
2154:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2113:
2107:
2103:
2099:
2095:
2091:
2087:
2071:
2065:
2061:
2057:
2051:
2047:
2043:
2039:
2035:
2029:
2021:
2017:
2013:
2009:
2005:
1999:
1969:
1962:
1958:
1954:
1950:
1946:
1940:
1936:
1930:
1926:
1922:
1918:
1914:
1908:
1906:vector space
1899:
1884:group scheme
1876:
1866:
1862:
1858:
1854:
1850:
1843:
1838:
1834:
1830:
1824:
1820:
1816:
1812:
1808:
1784:
1773:
1763:
1759:
1755:
1751:
1747:
1740:
1735:
1731:
1727:
1721:
1717:
1713:
1709:
1705:
1699:
1680:
1670:
1666:
1662:
1658:
1654:
1647:
1642:
1638:
1634:
1628:
1624:
1620:
1616:
1612:
1606:
1599:
1595:
1591:
1577:
1571:
1567:
1563:
1559:
1555:
1550:
1546:
1542:
1538:
1533:
1529:
1525:
1521:
1517:
1511:
1503:
1477:Depending on
1461:
1457:
1453:
1452:satisfies Δ(
1449:
1444:
1440:
1436:
1432:
1431:such that Δ(
1428:
1424:
1422:
1412:
1408:
1404:
1400:
1396:
1388:
1381:
1374:
1371:
1369:
1360:group theory
1355:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1313:
1309:
1305:
1301:
1299:
1294:
1290:
1283:
1279:
1273:
1269:
1265:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1227:
1220:
1216:
1212:
1208:
1201:
1194:
1190:
1186:
1182:
1178:
1174:
1167:
1163:
1159:
1157:
1148:
1144:
1140:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1098:
1088:
1084:
1082:
1077:
1073:
1069:
1067:
1062:
1054:
1049:
1044:
1042:
1037:
1034:homomorphism
1029:
1021:
1020:In general,
1019:
1002:
998:
994:
992:
852:
750:
648:
573:
476:
351:
341:
337:
327:
322:
315:
309:
159:
151:
141:
140:(called the
137:
133:
129:
123:
118:
111:
106:
73:group theory
69:group scheme
67:concept, in
58:
25:Hopf algebra
24:
18:
4189:, counit ε(
4115:. The unit
3334:) for some
2743:is also an
2644:commutative
2406:cup product
2026:commutators
1997:Lie algebra
1489:Commutative
1366:Hopf orders
1007:commutative
71:theory, in
21:mathematics
8250:Categories
8137:1234.16022
8111:0194.32901
8074:1117.16031
8037:0793.16029
8005:0025.09303
7986:Heinz Hopf
7980:0925.17031
7924:0962.16026
7898:References
7761:J. Algebra
7285:1704.07931
7278:(12): 90.
6818:Supergroup
5432:(with the
4664:such that
4640:(antipode)
4252:the field
3213:cohomology
3192:-modules.
2709:identifies
2161:) = 1 and
2074:is abelian
2016:for every
2012:⊗ 1 + 1 ⊗
1921:⊗ 1 + 1 ⊗
1879:is abelian
1776:is abelian
1683:is abelian
1590:functions
1580:is abelian
1372:Hopf order
1055:involutive
77:group ring
29:Heinz Hopf
8239:115153766
8222:0806.3205
8200:115297067
7639:189899056
7310:1029-8479
7156:Citations
7133:γ
7127:ρ
7121:λ
7115:α
7103:⊗
7071:→
7065:⊗
7050:ρ
7026:→
7020:⊗
7005:λ
6978:⊗
6969:⊗
6963:→
6957:⊗
6948:⊗
6918:α
6762:⊛
6727:⋆
6687:⋆
6647:⋆
6607:⋆
6565:⊙
6381:⊗
6317:⊗
6269:×
6235:⋅
6211:∇
6188:∈
6180:−
6152:∈
6126:ε
6120:Δ
6114:η
6108:∇
6076:→
6038:×
6004:ε
5998:Δ
5992:η
5986:∇
5948:×
5914:η
5908:∇
5876:η
5870:∇
5838:ε
5832:Δ
5803:∅
5788:ε
5768:ε
5721:Δ
5701:Δ
5678:ε
5672:Δ
5640:∈
5588:η
5565:⋅
5541:∇
5510:η
5504:∇
5472:∅
5443:×
5407:×
5346:γ
5340:ρ
5334:λ
5328:α
5316:⊗
5297:bialgebra
5280:ε
5274:Δ
5268:η
5262:∇
5202:⊗
5193:⊗
5184:⊗
5171:θ
5166:↣
5153:⊗
5144:⊗
5135:⊗
5109:θ
5066:→
5060:⊗
5045:λ
4992:ε
4972:Δ
4952:η
4932:∇
4875:γ
4869:ρ
4863:λ
4857:α
4845:⊗
4809:ε
4803:Δ
4757:γ
4751:ρ
4745:λ
4739:α
4727:⊗
4691:η
4685:∇
4625:→
4604:(counit),
4589:→
4580:ε
4553:⊗
4547:→
4538:Δ
4511:→
4502:η
4475:→
4469:⊗
4460:∇
4388:ε
4382:Δ
4376:η
4370:∇
4338:γ
4332:ρ
4326:λ
4320:α
4308:⊗
3896:ϵ
3772:ϵ
3621:ϵ
3593:ϵ
3590:∑
3559:ϵ
3531:ϵ
3528:∑
3507:ϵ
3477:Δ
3464:⊗
3461:Δ
3446:⊗
3434:Δ
3416:Δ
3413:⊗
3389:Δ
3386:⊗
3371:⊗
3359:Δ
3235:Lie group
2967:ϵ
2872:⊗
2844:⊗
2822:⊗
2797:⊗
2779:Δ
2767:⊗
2652:twist map
2563:→
2557:×
2514:∗
2506:⊗
2486:∗
2478:≅
2463:×
2452:∗
2444:→
2424:∗
2332:) = (−1)
2281:+ ... +
1059:*-algebra
958:μ
941:τ
922:ν
900:μ
884:τ
865:ν
823:ν
804:ν
782:ν
763:ν
721:μ
702:μ
680:μ
661:μ
612:τ
602:∑
549:⊗
521:ν
505:∑
488:Δ
437:μ
427:∑
410:∇
330:self-dual
289:∈
263:ε
109:bialgebra
81:bialgebra
45:bialgebra
41:coalgebra
8046:(2007),
8013:(1993),
7791:14889155
7577:(2003),
7318:54981897
7200:10046379
6787:See also
4826:comonoid
4144:between
4128:groupoid
4090:for all
3949:for all
3809:for all
3664:for all
3227:topology
3209:homology
2257:) = 1 ⊗
2102:= 0 and
1957:for all
1566:for all
1528:for all
1495:Remarks
1486:Antipode
1468:Examples
1377:over an
1233:for all
1185:for all
1143:-module
312:algebras
146:commutes
142:antipode
8103:0252485
8066:2294803
7878:, 10.3.
7674:1968985
7436:0987762
7428:2374514
7290:Bibcode
7253:2370323
7233:Bibcode
7180:Bibcode
6895:support
4526:(unit),
3217:H-space
1781:reduced
1460:⊗1 + 1⊗
1076:, then
318:with a
310:As for
126:-linear
114:over a
65:H-space
37:algebra
8237:
8198:
8161:
8135:
8125:
8109:
8101:
8091:
8072:
8064:
8054:
8035:
8025:
8003:
7978:
7968:
7950:
7922:
7912:
7851:, 6.2.
7816:
7789:
7672:
7637:
7589:
7542:
7513:
7484:
7434:
7426:
7316:
7308:
7251:
7198:
6587:. The
6292:groups
5386:Groups
5101:, and
5036:where
4708:monoid
4449:, and
4409:where
3961:) or ε
3821:) or ε
3672:, and
3326:) ≠ ε(
3201:Graded
3089:where
3010:is an
2919:and Δ(
2354:finite
2319:) = 0
1961:in 'T(
1799:on an
1483:Counit
1391:is an
1319:where
1061:). If
1024:is an
33:unital
8235:S2CID
8217:arXiv
8196:S2CID
7787:S2CID
7769:arXiv
7728:(PDF)
7670:JSTOR
7635:S2CID
7617:arXiv
7424:JSTOR
7314:S2CID
7280:arXiv
7249:S2CID
7223:arXiv
6909:Here
6849:Notes
6774:) of
6577:) of
5525:is a
4824:is a
4706:is a
3233:on a
3156:with
2941:with
2923:) = (
2305:⊗ 1.
2195:) = −
2169:) = 0
2135:, Δ(
2098:= 1,
2064:) = −
1972:)=0,1
1953:) = −
1943:) = 0
1509:group
1399:over
1393:order
1384:with
1119:into
1115:maps
1032:is a
1028:, so
116:field
8159:ISBN
8123:ISBN
8089:ISBN
8052:ISBN
8023:ISBN
7997:4784
7966:ISBN
7948:ISBN
7910:ISBN
7814:ISBN
7605:See
7587:ISBN
7540:ISBN
7511:ISBN
7482:ISBN
7306:ISSN
7276:2017
7196:PMID
5580:and
5430:sets
4233:See
4218:and
4207:) =
4175:) =
4148:and
4137:and
3986:via
3128:and
3097:and
2994:for
2903:for
2727:and
2719:Let
2685:) =
2584:Let
2187:and
2179:) =
2139:) =
2127:) =
2008:) =
1993:U(g)
1981:and
1917:) =
1861:) =
1837:) =
1819:) =
1758:) =
1734:) =
1716:) =
1665:) =
1641:) =
1623:) =
1562:) =
1520:) =
1456:) =
1435:) =
1272:) =
1215:) =
1181:) ⊆
1047:= id
334:dual
128:map
95:and
87:and
23:, a
8227:doi
8213:162
8186:doi
8182:113
8151:doi
8133:Zbl
8107:Zbl
8070:Zbl
8033:Zbl
8001:Zbl
7976:Zbl
7940:doi
7920:Zbl
7779:doi
7765:221
7740:doi
7736:342
7662:doi
7627:doi
7416:doi
7412:111
7298:doi
7241:doi
7188:doi
6752:Ste
6555:Ste
6262:Set
6031:Set
6019:in
5941:Set
5929:in
5428:of
5400:Set
4279:".
4156:of
4094:in
3975:or
3953:in
3835:or
3813:in
3676:in
3338:in
3336:a,b
3330:)ε(
3211:or
2731:be
2344:yes
2341:yes
2106:= −
2077:yes
2046:in
2020:in
1975:yes
1929:in
1872:yes
1769:yes
1676:yes
1583:yes
1570:in
1549:in
1532:in
1407:to
1304:in
1287:(2)
1277:(1)
1245:in
1237:in
1231:(2)
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