Fundamentals Of Hopf Algebras

Author : Robert G. Underwood
Publisher : Springer
Page : 164 pages
File Size : 40,5 Mb
Release : 2015-06-10
Category : Mathematics
ISBN : 9783319189918

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Fundamentals of Hopf Algebras by Robert G. Underwood Pdf

This text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields and linearly recursive sequences. The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforward applications of the theory to problems that are devised to challenge the reader. Questions for further study are provided after selected exercises. Most proofs are given in detail, though a few proofs are omitted since they are beyond the scope of this book.

Author : David E. Radford
Publisher : World Scientific
Page : 584 pages
File Size : 53,6 Mb
Release : 2011
Category : Mathematics
ISBN : 9789814338660

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Hopf Algebras by David E. Radford Pdf

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 : Sorin Dascalescu,Constantin Nastasescu,Serban Raianu
Publisher : CRC Press
Page : 428 pages
File Size : 55,8 Mb
Release : 2000-09-15
Category : Mathematics
ISBN : 0824704819

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Hopf Algebra by Sorin Dascalescu,Constantin Nastasescu,Serban Raianu Pdf

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 : Sorin Dascalescu,Constantin Nastasescu,Serban Raianu
Publisher : CRC Press
Page : 420 pages
File Size : 43,6 Mb
Release : 2000-09-15
Category : Mathematics
ISBN : 9781482270747

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Hopf Algebra by Sorin Dascalescu,Constantin Nastasescu,Serban Raianu Pdf

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 : Gabriella Böhm
Publisher : Springer
Page : 165 pages
File Size : 49,9 Mb
Release : 2018-11-01
Category : Mathematics
ISBN : 9783319981376

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Hopf Algebras and Their Generalizations from a Category Theoretical Point of View by Gabriella Böhm Pdf

These lecture notes provide a self-contained introduction to a wide range of generalizations of Hopf algebras. Multiplication of their modules is described by replacing the category of vector spaces with more general monoidal categories, thereby extending the range of applications. Since Sweedler's work in the 1960s, Hopf algebras have earned a noble place in the garden of mathematical structures. Their use is well accepted in fundamental areas such as algebraic geometry, representation theory, algebraic topology, and combinatorics. Now, similar to having moved from groups to groupoids, it is becoming clear that generalizations of Hopf algebras must also be considered. This book offers a unified description of Hopf algebras and their generalizations from a category theoretical point of view. The author applies the theory of liftings to Eilenberg–Moore categories to translate the axioms of each considered variant of a bialgebra (or Hopf algebra) to a bimonad (or Hopf monad) structure on a suitable functor. Covered structures include bialgebroids over arbitrary algebras, in particular weak bialgebras, and bimonoids in duoidal categories, such as bialgebras over commutative rings, semi-Hopf group algebras, small categories, and categories enriched in coalgebras. Graduate students and researchers in algebra and category theory will find this book particularly useful. Including a wide range of illustrative examples, numerous exercises, and completely worked solutions, it is suitable for self-study.

Author : Jeffrey Bergen,Stefan Catoiu,William Chin
Publisher : CRC Press
Page : 200 pages
File Size : 55,6 Mb
Release : 2004-01-28
Category : Mathematics
ISBN : 9781482293159

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Hopf Algebras by Jeffrey Bergen,Stefan Catoiu,William Chin Pdf

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

Author : Stefaan Caenepeel
Publisher : CRC Press
Page : 128 pages
File Size : 49,8 Mb
Release : 2020-09-30
Category : Mathematics
ISBN : 9781000116731

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Rings, Hopf Algebras, and Brauer Groups by Stefaan Caenepeel Pdf

"Based on papers presented at a recent international conference on algebra and algebraic geometry held jointly in Antwerp and Brussels, Belgium. Presents both survey and research articles featuring new results from the intersection of algebra and geometry. "

Author : Robert G. Underwood
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 43,5 Mb
Release : 2011-08-30
Category : Mathematics
ISBN : 9780387727653

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An Introduction to Hopf Algebras by Robert G. Underwood Pdf

Only book on Hopf algebras aimed at advanced undergraduates

Author : Robert G. Underwood
Publisher : Springer
Page : 0 pages
File Size : 41,7 Mb
Release : 2011-08-31
Category : Mathematics
ISBN : 0387566155

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An Introduction to Hopf Algebras by Robert G. Underwood Pdf

Only book on Hopf algebras aimed at advanced undergraduates

Author : Rinat Kashaev
Publisher : Unknown
Page : 0 pages
File Size : 44,5 Mb
Release : 2023
Category : Electronic
ISBN : 3031263073

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A Course on Hopf Algebras by Rinat Kashaev Pdf

This textbook provides a concise, visual introduction to Hopf algebras and their application to knot theory, most notably the construction of solutions of the Yang-Baxter equations. Starting with a reformulation of the definition of a group in terms of structural maps as motivation for the definition of a Hopf algebra, the book introduces the related algebraic notions: algebras, coalgebras, bialgebras, convolution algebras, modules, comodules. Next, Drinfel'd's quantum double construction is achieved through the important notion of the restricted (or finite) dual of a Hopf algebra, which allows one to work purely algebraically, without completions. As a result, in applications to knot theory, to any Hopf algebra with invertible antipode one can associate a universal invariant of long knots. These constructions are elucidated in detailed analyses of a few examples of Hopf algebras. The presentation of the material is mostly based on multilinear algebra, with all definitions carefully formulated and proofs self-contained. The general theory is illustrated with concrete examples, and many technicalities are handled with the help of visual aids, namely string diagrams. As a result, most of this text is accessible with minimal prerequisites and can serve as the basis of introductory courses to beginning graduate students.

Author : Stefaan Caenepeel,Freddy Van Oystaeyen
Publisher : CRC Press
Page : 332 pages
File Size : 45,9 Mb
Release : 2019-05-07
Category : Mathematics
ISBN : 9780429529986

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Hopf Algebras and Quantum Groups by Stefaan Caenepeel,Freddy Van Oystaeyen Pdf

This volume is based on the proceedings of the Hopf-Algebras and Quantum Groups conference at the Free University of Brussels, Belgium. It presents state-of-the-art papers - selected from over 65 participants representing nearly 20 countries and more than 45 lectures - on the theory of Hopf algebras, including multiplier Hopf algebras and quantum g

Author : Moss E. Sweedler
Publisher : Unknown
Page : 360 pages
File Size : 42,8 Mb
Release : 1969
Category : Hopf algebras
ISBN : UOM:39015011447524

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Hopf Algebras by Moss E. Sweedler Pdf

Author : István Heckenberger,Hans-Jürgen Schneider
Publisher : American Mathematical Soc.
Page : 582 pages
File Size : 55,8 Mb
Release : 2020-06-19
Category : Education
ISBN : 9781470452322

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Hopf Algebras and Root Systems by István Heckenberger,Hans-Jürgen Schneider Pdf

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 : H. Yanagihara
Publisher : Springer
Page : 315 pages
File Size : 53,7 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540370772

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Theory of Hopf Algebras Attached to Group Schemes by H. Yanagihara Pdf

Author : Susan Montgomery,Hans-Jurgen Schneider
Publisher : Cambridge University Press
Page : 502 pages
File Size : 42,6 Mb
Release : 2002-05-06
Category : Mathematics
ISBN : 0521815126

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New Directions in Hopf Algebras by Susan Montgomery,Hans-Jurgen Schneider Pdf

Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.