**CBMS Regional Conference Series in Mathematics**

Volume: 82;
1993;
238 pp;
Softcover

MSC: Primary 16;
Secondary 17; 20; 14

Print ISBN: 978-0-8218-0738-5

Product Code: CBMS/82

List Price: $36.00

Individual Price: $28.80

**Electronic ISBN: 978-1-4704-2442-8
Product Code: CBMS/82.E**

List Price: $36.00

Individual Price: $28.80

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# Hopf Algebras and Their Actions on Rings

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*Susan Montgomery*

A co-publication of the AMS and CBMS

The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions. Quantum groups are treated as an important example, rather than as an end in themselves. The two introductory chapters review definitions and basic facts; otherwise, most of the material has not previously appeared in book form. Providing an accessible introduction to Hopf algebras, this book would make an excellent graduate textbook for a course in Hopf algebras or an introduction to quantum groups.

#### Table of Contents

# Table of Contents

## Hopf Algebras and Their Actions on Rings

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xiii14 free
- Chapter 1. Definitions and Examples 116 free
- 1.1 Algebras and coalgebras 116
- 1.2 Duals of algebras and coalgebras 217
- 1.3 Bialgebras 318
- 1.4 Convolution and summation notation 621
- 1.5 Antipodes and Hopf algebras 722
- 1.6 Modules and comodules 1025
- 1.7 Invariants and coinvariants 1328
- 1.8 Tensor products of H-modules and H-comodules 1429
- 1.9 Hopf modules 1429

- Chapter 2. Integrals and Semisimplicity 1732
- Chapter 3. Freeness over Subalgebras 2843
- 3.1 The Nichols-Zoeller Theorem 2843
- 3.2 Applications: Hopf algebras of prime dimension and semisimple subHopf algebras 3146
- 3.3 A normal basis for H over K 3247
- 3.4 The adjoint action, normal subHopf algebras, and quotients 3348
- 3.5 Freeness and faithful flatness in the infinite-dimensional case 3752

- Chapter 4. Actions of Finite-Dimensional Hopf Algebras and Smash Products 4055
- 4.1 Module algebras, comodule algebras, and smash products 4055
- 4.2 Integrality and affine invariants: the commutative case 4358
- 4.3 Trace functions and affine invariants: the non-commutative case 4560
- 4.4 Ideals in A#H and A as an A[sup(H)]-module 4863
- 4.5 A Morita context relating A#H and A[sup(H)] 5267

- Chapter 5. Coradicals and Filtrations 5671
- 5.1 Simple subcoalgebras and the coradical 5671
- 5.2 The coradical filtration 6075
- 5.3 Injective coalgebra maps 6580
- 5.4 The coradical filtration of pointed coalgebras 6782
- 5.5 Examples: U(g) and U[sub(q)](g) 7388
- 5.6 The structure of pointed cocommutative Hopf algebras 7691
- 5.7 Semisimple cocommutative connected Hopf algebras 8398

- Chapter 6. Inner Actions 87102
- Chapter 7. Crossed products 101116
- Chapter 8. Galois Extensions 123138
- Chapter 9. Duality 149164
- Chapter 10. New Constructions from Quantum Groups 178193
- 10.1 Quasitriangular and almost cocommutative Hopf algebras 178193
- 10.2 Coquasitriangular and almost commutative Hopf algebras 184199
- 10.3 The Drinfeld double 187202
- 10.4 Braided monoidal categories 197212
- 10.5 Hopf algebras in categories; graded Hopf algebras 203218
- 10.6 Biproducts and Yetter-Drinfeld modules 207222

- Appendix. Some quantum groups 217232
- References 223238
- Index 235250 free
- Back Cover Back Cover1255

#### Readership

Research mathematicians and graduate students.

#### Reviews

I found that Montgomery's book is an excellent outline of all the topics mentioned and can serve as a useful guide in structuring [a course on Hopf algebras and quantum groups] … there is an excellent bibliography … the author has performed a highly useful service to the mathematical community.

-- Zentralblatt MATH

A good guidebook to recent developments in Hopf algebra theory with an emphasis on their actions and coactions on algebras … Most of the results have not previously appeared in book form … This book could also be used as a textbook for graduate students … The author, S. Montgomery, has contributed much to group actions on rings and their generalizations to Hopf algebras (especially finite dimensional).

-- Bulletin of the AMS

This book should be very useful as a comprehensive, up to date introduction to the algebraic aspects of Hopf algebras.

-- Monatshefte für Mathematik