Author : Robert G. Underwood
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 45,42 MB
Release : 2011-08-30
Category : Mathematics
ISBN : 0387727655
Only book on Hopf algebras aimed at advanced undergraduates
Author : Sorin Dascalescu
Publisher : CRC Press
Page : 420 pages
File Size : 38,95 MB
Release : 2000-09-15
Category : Mathematics
ISBN : 1482270749
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.
Author : Susan Montgomery
Publisher : American Mathematical Soc.
Page : 258 pages
File Size : 26,64 MB
Release : 1993-10-28
Category : Mathematics
ISBN : 0821807382
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.
Author : David E Radford
Publisher : World Scientific
Page : 584 pages
File Size : 43,55 MB
Release : 2011-12-28
Category : Mathematics
ISBN : 9814405108
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.
Author : István Heckenberger
Publisher : American Mathematical Soc.
Page : 606 pages
File Size : 14,79 MB
Release : 2020-06-19
Category : Education
ISBN : 1470452324
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.
Author : Pierre Cartier
Publisher : Springer Nature
Page : 277 pages
File Size : 16,77 MB
Release : 2021-09-20
Category : Mathematics
ISBN : 3030778452
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.
Author : Daniel Bulacu
Publisher : Cambridge University Press
Page : 545 pages
File Size : 29,32 MB
Release : 2019-02-21
Category : Mathematics
ISBN : 1108427014
This self-contained book dedicated to Drinfeld's quasi-Hopf algebras takes the reader from the basics to the state of the art.
Author : Eiichi Abe
Publisher : Cambridge University Press
Page : 304 pages
File Size : 19,20 MB
Release : 2004-06-03
Category : Mathematics
ISBN : 9780521604895
An introduction to the basic theory of Hopf algebras for those familiar with basic linear and commutative algebra.
Author : Jeffrey Bergen
Publisher : CRC Press
Page : 282 pages
File Size : 15,99 MB
Release : 2004-01-28
Category : Mathematics
ISBN : 9780824755669
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 Morita invariant. It uses the newly introduced theory of bi-Frobenius algebras to investigate a notion of group-like algebras and summarizes results on the classification of Hopf algebras of dimension pq. It also explores pre-Lie, dendriform, and Nichols algebras and discusses support cones for infinitesimal group schemes.
Author : W.C. Waterhouse
Publisher : Springer Science & Business Media
Page : 167 pages
File Size : 11,98 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1461262178
Ah Love! Could you and I with Him consl?ire To grasp this sorry Scheme of things entIre' KHAYYAM People investigating algebraic groups have studied the same objects in many different guises. My first goal thus has been to take three different viewpoints and demonstrate how they offer complementary intuitive insight into the subject. In Part I we begin with a functorial idea, discussing some familiar processes for constructing groups. These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con struct new examples. Study of their representations shows that they are closely related to groups of matrices, and closed sets in matrix space give us a geometric picture of some of the objects involved. This interplay of methods continues as we turn to specific results. In Part II, a geometric idea (connectedness) and one from classical matrix theory (Jordan decomposition) blend with the study of separable algebras. In Part III, a notion of differential prompted by the theory of Lie groups is used to prove the absence of nilpotents in certain Hopf algebras. The ring-theoretic work on faithful flatness in Part IV turns out to give the true explanation for the behavior of quotient group functors. Finally, the material is connected with other parts of algebra in Part V, which shows how twisted forms of any algebraic structure are governed by its automorphism group scheme.
