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Hopf algebras / David E. Radford.

By: Radford, David E.
Material type: materialTypeLabelBookSeries: Series on knots and everything ; v. 49.Publisher: Singapore ; Hackensack, NJ : World Scientific, c2012Description: xxii, 559 p. : il. ; 24 cm.ISBN: 9789814335997 (hardcover : alk. paper); 9814335991 (hardcover : alk. paper).Subject(s): Hopf algebrasOther classification: 16-01 (16T05 16Txx)
Contents:
1. Preliminaries [1] -- 1.1 Notation and terminology conventions [1] -- 1.2 Rank of a tensor [2] -- 1.3 Topological aspects of vector space duals [7] -- 2. Coalgebras [19] -- 2.1 Algebras and coalgebras, basic definitions [19] -- 2.2 Comatrix identities, the fundamental theorem of coalgebras [35] -- 2.3 The dual algebra [40] -- 2.4 The wedge product [52] -- 2.5 The dual coalgebra [58] -- 2.6 Double duals [66] -- 2.7 The cofree coalgebra on a vector space [70] -- 3. Representations of coalgebras [77] -- 3.1 Rational modules of the dual algebra [77] -- 3.2 Comodules [85] -- 3.3 Mr and Mr [98] -- 3.4 The coradical of a coalgebra [98] -- 3.5 Injective comodules [103] -- 3.6 Coalgebras which are submodules of their dual algebras [114] -- 3.7 Indecomposable coalgebras [118] -- 4. The coradical filtration and related structures [123] -- 4.1 Filtrations of coalgebras [124] -- 4.2 The wedge product and the coradical filtration [129] -- 4.3 Idempotents and the coradical filtration [132] -- 4.4 Graded algebras and coalgebras [136] -- 4.5 The cofree pointed irreducible coalgebra on a vector space [146] -- 4.6 The radical of the dual algebra [151] -- 4.7 Free pointed coalgebras associated to coalgebras [153] -- 4.8 Linked simple subcoalgebras [158] -- 5. Bialgebras [165] -- 5.1 Basic definitions and results [165] -- 5.2 The dual bialgebra [177] -- 5.3 The free bialgebra on a coalgebra and related constructions [180] -- 5.4 The universal enveloping algebra [191] -- 5.5 The cofree bialgebra on an algebra [193] -- 5.6 Filtrations and gradings of bialgebras [197] -- 5.7 Representations of bialgebras [200] -- 6. The convolution algebra [203] -- 6.1 Definition and basic properties [203] -- 6.2 Invertible elements in the convolution algebra [206] -- 7. Hopf algebras [211] -- 7.1 Definition of Hopf algebra, the antipode [211] -- 7.2 Q-binomial symbols [217] -- 7.3 Two families of examples [220] -- 7.4 The dual Hopf algebra [225] -- 7.5 The free Hopf algebra on a coalgebra [227] -- 7.6 When a bialgebra is a Hopf algebra [231] -- 7.7 Two-cocycles, pairings, and skew pairings of bialgebras [237] -- 7.8 Twists of bialgebras [243] -- 7.9 Filtrations and gradings on Hopf algebras [246] -- 7.10 The cofree pointed irreducible Hopf algebra on an algebra [249] -- 7.11 The shuffle algebra [250] -- 8. Hopf modules and co-Hopf modules [259] -- 8.1 Definition of Hopf module and examples [259] -- 8.2 The structure of Hopf modules [263] -- 8.3 Co-Hopf modules [268] -- 8.4 A basic co-Hopf module and its dual [270] -- 9. Hopf algebras as modules over Hopf subalgebras [273] -- 9.1 Filtrations whose base term is a Hopf subalgebra [273] -- 9.2 Relative Hopf modules [276] -- 9.3 When Hopf algebras free over their Hopf subalgebras [279] -- 9.4 An example of a Hopf algebra which is not free over some Hopf subalgebra [282] -- 10. Integrals [289] -- 10.1 Definition of integrals for a bialgebra and its dual algebra [289] -- 10.2 Existence and uniqueness of integrals for a Hopf algebra [293] -- 10.3 Integrals and semisimplicity [298] -- 10.4 Integrals and the trace function [302] -- 10.5 Integrals and the antipode [305] -- 10.6 Generalized integrals and grouplike elements [313] -- 10.7 Integrals, the center, and cocommutative elements of the dual [318] -- 10.8 Integrals and co-semisimplicity [324] -- 10.9 Existence and uniqueness results for integrals of the dual algebra of a Hopf algebra [329] -- 11. Actions by bialgebras and Hopf algebras [343] -- 11.1 Monoidal categories [345] -- 11.2 Module actions and module algebras, coalgebras [349] -- 11.3 Comodule actions and comodule algebras, coalgebras [355] -- 11.4 Duality between the smash product and smash coproduct [360] -- 11.5 Prebraiding, braiding structures on a monoidal category [363] -- 11.6 Yetter-Drinfel’d modules and biproducts [367] -- 11.7 Abstract characterization of biproducts [380] -- 12. Quasitriangular bialgebras and Hopf algebras [387] -- 12.1 The quantum Yang-Baxter and braid equations, Yang-Baxter algebras [387] -- 12.2 Almost cocommutative Hopf algebras, quasitriangular bialgebras and Hopf algebras [391] -- 12.3 Grouplike and ribbon elements [400] -- 12.4 Factorizable Hopf algebras [405] -- 13. The Drinfel’d double of a finite-dimensional Hopf algebra [413] -- 13.1 The double and its category of representations [413] -- 13.2 Basic properties of the double [420] -- 13.3 Characterizations of the double as a quasitriangular Hopf algebra [423] -- 13.4 The dual of the double [427] -- 13.5 The double of a quasitriangular Hopf algebra [435] -- 13.6 The double of a factorizable Hopf algebra [439] -- 13.7 Quasi-ribbon and ribbon elements of the double [440] -- 13.8 Generalized doubles and their duals [443] -- 14. Coquasitriangular bialgebras and Hopf algebras [447] -- 14.1 Coquasitriangular and Yang-Baxter coalgebras [447] -- 14.2 Coquasitriangular bialgebras and Hopf algebras [452] -- 14.3 The square of the antipode of a coquasitriangular Hopf algebra [456] -- 14.4 The free coquasitriangular bialgebra on a coquasitriangular coalgebra [459] -- 15. Pointed Hopf algebras [467] -- 15.1 Crossed products [468] -- 15.2 Pointed Hopf algebras as crossed products [472] -- 15.3 Cocommutative pointed Hopf algebras; the characteristic 0 case [478] -- 15.4 Minimal-pointed Hopf algebras [479] -- 15.5 Pointed Hopf algebras, biproducts, and Nichols algebras [485] -- 15.6 Quantized enveloping algebras and their generalizations [498] -- 15.7 Ore extensions and pointed Hopf algebras [509] -- 16. Finite-dimensional Hopf algebras in characteristic 0 [515] -- 16.1 Characterizations of semisimple Hopf algebras [516] -- 16.2 Isomorphism types of Hopf algebras of the same dimension [520] -- 16.3 Some very basic classification results [525] -- Bibliography [537] --
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1. Preliminaries [1] --
1.1 Notation and terminology conventions [1] --
1.2 Rank of a tensor [2] --
1.3 Topological aspects of vector space duals [7] --
2. Coalgebras [19] --
2.1 Algebras and coalgebras, basic definitions [19] --
2.2 Comatrix identities, the fundamental theorem of coalgebras [35] --
2.3 The dual algebra [40] --
2.4 The wedge product [52] --
2.5 The dual coalgebra [58] --
2.6 Double duals [66] --
2.7 The cofree coalgebra on a vector space [70] --
3. Representations of coalgebras [77] --
3.1 Rational modules of the dual algebra [77] --
3.2 Comodules [85] --
3.3 Mr and Mr [98] --
3.4 The coradical of a coalgebra [98] --
3.5 Injective comodules [103] --
3.6 Coalgebras which are submodules of their dual algebras [114] --
3.7 Indecomposable coalgebras [118] --
4. The coradical filtration and related structures [123] --
4.1 Filtrations of coalgebras [124] --
4.2 The wedge product and the coradical filtration [129] --
4.3 Idempotents and the coradical filtration [132] --
4.4 Graded algebras and coalgebras [136] --
4.5 The cofree pointed irreducible coalgebra on a vector space [146] --
4.6 The radical of the dual algebra [151] --
4.7 Free pointed coalgebras associated to coalgebras [153] --
4.8 Linked simple subcoalgebras [158] --
5. Bialgebras [165] --
5.1 Basic definitions and results [165] --
5.2 The dual bialgebra [177] --
5.3 The free bialgebra on a coalgebra and related constructions [180] --
5.4 The universal enveloping algebra [191] --
5.5 The cofree bialgebra on an algebra [193] --
5.6 Filtrations and gradings of bialgebras [197] --
5.7 Representations of bialgebras [200] --
6. The convolution algebra [203] --
6.1 Definition and basic properties [203] --
6.2 Invertible elements in the convolution algebra [206] --
7. Hopf algebras [211] --
7.1 Definition of Hopf algebra, the antipode [211] --
7.2 Q-binomial symbols [217] --
7.3 Two families of examples [220] --
7.4 The dual Hopf algebra [225] --
7.5 The free Hopf algebra on a coalgebra [227] --
7.6 When a bialgebra is a Hopf algebra [231] --
7.7 Two-cocycles, pairings, and skew pairings of bialgebras [237] --
7.8 Twists of bialgebras [243] --
7.9 Filtrations and gradings on Hopf algebras [246] --
7.10 The cofree pointed irreducible Hopf algebra on an algebra [249] --
7.11 The shuffle algebra [250] --
8. Hopf modules and co-Hopf modules [259] --
8.1 Definition of Hopf module and examples [259] --
8.2 The structure of Hopf modules [263] --
8.3 Co-Hopf modules [268] --
8.4 A basic co-Hopf module and its dual [270] --
9. Hopf algebras as modules over Hopf subalgebras [273] --
9.1 Filtrations whose base term is a Hopf subalgebra [273] --
9.2 Relative Hopf modules [276] --
9.3 When Hopf algebras free over their Hopf subalgebras [279] --
9.4 An example of a Hopf algebra which is not free over some Hopf subalgebra [282] --
10. Integrals [289] --
10.1 Definition of integrals for a bialgebra and its dual algebra [289] --
10.2 Existence and uniqueness of integrals for a Hopf algebra [293] --
10.3 Integrals and semisimplicity [298] --
10.4 Integrals and the trace function [302] --
10.5 Integrals and the antipode [305] --
10.6 Generalized integrals and grouplike elements [313] --
10.7 Integrals, the center, and cocommutative elements of the dual [318] --
10.8 Integrals and co-semisimplicity [324] --
10.9 Existence and uniqueness results for integrals of the dual algebra of a Hopf algebra [329] --
11. Actions by bialgebras and Hopf algebras [343] --
11.1 Monoidal categories [345] --
11.2 Module actions and module algebras, coalgebras [349] --
11.3 Comodule actions and comodule algebras, coalgebras [355] --
11.4 Duality between the smash product and smash coproduct [360] --
11.5 Prebraiding, braiding structures on a monoidal category [363] --
11.6 Yetter-Drinfel’d modules and biproducts [367] --
11.7 Abstract characterization of biproducts [380] --
12. Quasitriangular bialgebras and Hopf algebras [387] --
12.1 The quantum Yang-Baxter and braid equations, Yang-Baxter algebras [387] --
12.2 Almost cocommutative Hopf algebras, quasitriangular bialgebras and Hopf algebras [391] --
12.3 Grouplike and ribbon elements [400] --
12.4 Factorizable Hopf algebras [405] --
13. The Drinfel’d double of a finite-dimensional Hopf algebra [413] --
13.1 The double and its category of representations [413] --
13.2 Basic properties of the double [420] --
13.3 Characterizations of the double as a quasitriangular Hopf algebra [423] --
13.4 The dual of the double [427] --
13.5 The double of a quasitriangular Hopf algebra [435] --
13.6 The double of a factorizable Hopf algebra [439] --
13.7 Quasi-ribbon and ribbon elements of the double [440] --
13.8 Generalized doubles and their duals [443] --
14. Coquasitriangular bialgebras and Hopf algebras [447] --
14.1 Coquasitriangular and Yang-Baxter coalgebras [447] --
14.2 Coquasitriangular bialgebras and Hopf algebras [452] --
14.3 The square of the antipode of a coquasitriangular Hopf algebra [456] --
14.4 The free coquasitriangular bialgebra on a coquasitriangular coalgebra [459] --
15. Pointed Hopf algebras [467] --
15.1 Crossed products [468] --
15.2 Pointed Hopf algebras as crossed products [472] --
15.3 Cocommutative pointed Hopf algebras; the characteristic 0 case [478] --
15.4 Minimal-pointed Hopf algebras [479] --
15.5 Pointed Hopf algebras, biproducts, and Nichols algebras [485] --
15.6 Quantized enveloping algebras and their generalizations [498] --
15.7 Ore extensions and pointed Hopf algebras [509] --
16. Finite-dimensional Hopf algebras in characteristic 0 [515] --
16.1 Characterizations of semisimple Hopf algebras [516] --
16.2 Isomorphism types of Hopf algebras of the same dimension [520] --
16.3 Some very basic classification results [525] --
Bibliography [537] --

MR, MR2894855

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