Hopf Algebras and Root Systems

Hopf Algebras and Root Systems PDF

Author: István Heckenberger

Publisher: American Mathematical Soc.

Published: 2020-06-19

Total Pages: 582

ISBN-13: 1470452324

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This book is an introduction to Hopf algebras in braided monoidal categories with applications to Hopf algebras in the usual sense. The main goal of the book is to present from scratch and with complete proofs the theory of Nichols algebras (or quantum symmetric algebras) and the surprising relationship between Nichols algebras and generalized root systems. In general, Nichols algebras are not classified by Cartan graphs and their root systems. However, extending partial results in the literature, the authors were able to associate a Cartan graph to a large class of Nichols algebras. This allows them to determine the structure of right coideal subalgebras of Nichols systems which generalize Nichols algebras. As applications of these results, the book contains a classification of right coideal subalgebras of quantum groups and of the small quantum groups, and a proof of the existence of PBW-bases that does not involve case by case considerations. The authors also include short chapter summaries at the beginning of each chapter and historical notes at the end of each chapter. The theory of Cartan graphs, Weyl groupoids, and generalized root systems appears here for the first time in a book form. Hence, the book serves as an introduction to the modern classification theory of pointed Hopf algebras for advanced graduate students and researchers working in categorial aspects and classification theory of Hopf algebras and their generalization.

Classical Hopf Algebras and Their Applications

Classical Hopf Algebras and Their Applications PDF

Author: Pierre Cartier

Publisher: Springer Nature

Published: 2021-09-20

Total Pages: 277

ISBN-13: 3030778452

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This book is dedicated to the structure and combinatorics of classical Hopf algebras. Its main focus is on commutative and cocommutative Hopf algebras, such as algebras of representative functions on groups and enveloping algebras of Lie algebras, as explored in the works of Borel, Cartier, Hopf and others in the 1940s and 50s. The modern and systematic treatment uses the approach of natural operations, illuminating the structure of Hopf algebras by means of their endomorphisms and their combinatorics. Emphasizing notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras and Lie idempotents, the text also covers the important case of enveloping algebras of pre-Lie algebras. A wide range of applications are surveyed, highlighting the main ideas and fundamental results. Suitable as a textbook for masters or doctoral level programs, this book will be of interest to algebraists and anyone working in one of the fields of application of Hopf algebras.

Hopf Algebras and Their Actions on Rings

Hopf Algebras and Their Actions on Rings PDF

Author: Susan Montgomery

Publisher: American Mathematical Soc.

Published: 1993-10-28

Total Pages: 258

ISBN-13: 0821807382

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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.

Hopf Algebra

Hopf Algebra PDF

Author: Sorin Dascalescu

Publisher: CRC Press

Published: 2000-09-15

Total Pages: 420

ISBN-13: 1482270749

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This study covers comodules, rational modules and bicomodules; cosemisimple, semiperfect and co-Frobenius algebras; bialgebras and Hopf algebras; actions and coactions of Hopf algebras on algebras; finite dimensional Hopf algebras, with the Nicholas-Zoeller and Taft-Wilson theorems and character theory; and more.

Hopf Algebras

Hopf Algebras PDF

Author: Eiichi Abe

Publisher: Cambridge University Press

Published: 2004-06-03

Total Pages: 304

ISBN-13: 9780521604895

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An introduction to the basic theory of Hopf algebras for those familiar with basic linear and commutative algebra.

Hopf Algebras

Hopf Algebras PDF

Author: Jeffrey Bergen

Publisher: CRC Press

Published: 2004-01-28

Total Pages: 200

ISBN-13: 1482293153

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This volume publishes key proceedings from the recent International Conference on Hopf Algebras held at DePaul University, Chicago, Illinois. With contributions from leading researchers in the field, this collection deals with current topics ranging from categories of infinitesimal Hopf modules and bimodules to the construction of a Hopf algebraic

Quasi-Hopf Algebras

Quasi-Hopf Algebras PDF

Author: Daniel Bulacu

Publisher: Cambridge University Press

Published: 2019-02-21

Total Pages: 545

ISBN-13: 1108427014

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This self-contained book dedicated to Drinfeld's quasi-Hopf algebras takes the reader from the basics to the state of the art.

Hopf Algebras

Hopf Algebras PDF

Author: David E Radford

Publisher: World Scientific

Published: 2011-12-28

Total Pages: 588

ISBN-13: 9814405108

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The book provides a detailed account of basic coalgebra and Hopf algebra theory with emphasis on Hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups. It is intended to be a graduate text as well as a research monograph. Contents:PreliminariesCoalgebrasRepresentations of CoalgebrasThe Coradical Filtration and Related StructuresBialgebrasThe Convolution AlgebraHopf AlgebrasHopf Modules and Co-Hopf ModulesHopf Algebras as Modules Over Their Hopf SubalgebrasIntegralsActions by Bialgebras and Hopf AlgebrasQuasitriangular Bialgebras and Hopf AlgebrasThe Drinfel'd Double of a Finite-Dimensional Hopf AlgebraCo-Quasitriangular Bialgebras and Hopf AlgebrasPointed Hopf AlgebrasFinite-Dimensional Hopf Algebras in Characteristic 0 Readership: Undergraduates and researchers in algebra and number theory. Keywords:Hopf Algebras;Coalgebras;Quantum GroupsKey Features:Provides a good foundation for those who wish to study Hopf algebras on their ownProvides a firm foundation for those who are more interested in applications to other areasGives many exercises which suggest connections to exploreReviews: "With this monograph, one of the pioneers of the subject provides a comprehensive introduction to the theory of Hopf algebras. As this theory has made great strides in recent years, such a monograph constitutes a very valuable addition to the literature, especially as there are so far comparatively few textbooks on this topic. Radford's book contains at the end of each chapter a very useful set of chapter notes that discuss these references and therefore provide an entry point to the recent research literature, especially to the extensive literature on the classification of finite–dimensional pointed Hopf algebras, a topic not discussed in any of the other books. For all these reasons, Radford's book is a very valuable new textbook on Hopf algebras that will be frequently used both by students and by researchers." Mathematical Reviews "A big number of exercises of different level of difficulty are proposed along the text, which include in particular special features or applications to a variety of concrete examples, further results and categorical aspects of the corresponding material. Interesting and up-to-date historical and bibliographical comments are provided at the end of each of the sixteen chapters." Zentralblatt MATH

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