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Hopf代数及其在环上的作用
Hopf代数及其在环上的作用

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数理化

  • 电子书积分:10 积分如何计算积分?
  • 作 者:(美)苏珊·蒙哥马利(Susan Montgomery)著
  • 出 版 社:北京:高等教育出版社
  • 出版年份:2018
  • ISBN:9787040502312
  • 页数:238 页
图书介绍:近年来,Hopf 代数出现了许多重大的进展。最著名的是量子群的引进,量子群实际上就是数学物理中的Hopf 代数,现在与许多数学领域都有联系。除此之外,Kaplansky 的许多猜想已得到证明,其中最令人惊讶的是关于Hopf 代数的一类Lagrange定理。关于Hopf 代数作用方面的工作将早先的群作用、Lie代数的作用和分次代数的有关结果统一起来了。本书将这些新近的发展按照Hopf 代数的代数结构和它们的作用及相互作用的观点汇拢在一起。量子群是其中重要的例子,而这并非是它们的终点。书中用两章回顾了基本事实和定义,另外的大部分材料以前并没有以书的形式出现过。本书是关于Hopf 代数的一本优秀的研究生教学参考书,同时也是一本量子群的入门书。
《Hopf代数及其在环上的作用》目录
标签:代数 作用

Chapter 1.Definitions and Examples 1

1.1 Algebras and coalgebras 1

1.2 Duals of algebras and coalgebras 2

1.3 Bialgebras 3

1.4 Convolution and summation notation 6

1.5 Antipodes and Hopf algebras 7

1.6 Modules and comodules 10

1.7 Invariants and coinvariants 13

1.8 Tensor products of H-modules and H-comodules 14

1.9 Hopf modules 14

Chapter 2.Integrals and Semisimplicity 17

2.1 Integrals 17

2.2 Maschke’s Theorem 20

2.3 Commutative semisimple Hopf algebras and restricted enveloping algebras 22

2.4 Cosemisimplicity and integrals on H 25

2.5 Kaplansky’s conjecture and the order of the antipode 27

Chapter 3.Freeness over Subalgebras 28

3.1 The Nichols-Zoeller Theorem 28

3.2 Applications:Hopf algebras of prime dimension and semisimple subHopfalgebras 31

3.3 A normal basis for H over K 32

3.4 The adjoint action,normal subHopfalgebras,and quotients 33

3.5 Freeness and faithful flatness in the infinite-dimensional case 37

Chapter 4.Actions of Finite-Dimensional Hopf Algebras and Smash Products 40

4.1 Module algebras,comodule algebras,and smash products 40

4.2 Integrality and affine invariants:the commutative case 43

4.3 Trace functions and affine invariants:the non-commutative case 45

4.4 Ideals in A#H and A as an AH module 48

4.5 A Morita context relating A#H and AH 52

Chapter 5.Coradicals and Filtrations 56

5.1 Simple subcoalgebras and the coradical 56

5.2 The coradical filtration 60

5.3 Injecti ve coalgebra maps 65

5.4 The coradical filtration of pointed coalgebras 67

5.5 Examples:U(g)and Uq(g) 73

5.6 The structure of pointed cocommutative Hopf algebras 76

5.7 Semisimple cocommutative connected Hopf algebras 83

Chapter 6.Inner Actions 87

6.1 Definitions and examples 87

6.2 A Skolem-Noether theorem for Hopf algebras 89

6.3 Maximal inner subcoalgebras 92

6.4 X-inner actions and extending to quotients 96

Chapter 7.Crossed products 101

7.1 Definitions and examples 101

7.2 Cleft extensions and existence of crossed products 105

7.3 Inner actions and equivalence of crossed products 112

7.4 Generalized Maschke theorems and semiprime crossed products 116

7.5 Twisted H-comodule algebras 121

Chapter 8.Galois Extensions 123

8.1 Definition and examples 123

8.2 The normal basis property and cleft extensions 128

8.3 Galois extensions for finite-dimensional H 131

8.4 Normal bases and Hopf algebra quotients 139

8.5 Relative Hopf modules 144

Chapter 9.Duality 149

9.1 H° 149

9.2 SubHopfalgebras of H°and density 153

9.3 Classical duality 159

9.4 Duality for actions 161

9.5 Duality for graded algebras 172

Chapter 10.New Constructions from Quantum Groups 178

10.1 Quasitriangular and almost cocommutative Hopf algebras 178

10.2 Coquasitriangular and almost commutative Hopf algebras 184

10.3 The Drinfeld double 187

10.4 Braided monoidal categories 197

10.5 Hopf algebras in categories;graded Hopf algebras 203

10.6 Biproducts and Yetter-Drinfeld modules 207

Appendix.Some quantum groups 217

References 223

Index 235

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