Classical Hopf Algebras And Their Applications

Author : Pierre Cartier,Frédéric Patras
Publisher : Springer Nature
Page : 277 pages
File Size : 41,6 Mb
Release : 2021-09-20
Category : Mathematics
ISBN : 9783030778453

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Classical Hopf Algebras and Their Applications by Pierre Cartier,Frédéric Patras Pdf

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 : Gabriella Böhm
Publisher : Springer
Page : 165 pages
File Size : 54,9 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 : Susan Montgomery,Hans-Jurgen Schneider
Publisher : Cambridge University Press
Page : 502 pages
File Size : 45,7 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.

Author : Abdenacer Makhlouf
Publisher : John Wiley & Sons
Page : 336 pages
File Size : 48,7 Mb
Release : 2021-12-07
Category : Mathematics
ISBN : 9781119880905

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Algebra and Applications 2 by Abdenacer Makhlouf Pdf

This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras. The chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory. The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and Rota–Baxter algebras are explored. Algebra and Applications 2 is of great interest to graduate students and researchers. Each chapter combines some of the features of both a graduate level textbook and of research level surveys.

Author : Stefaan Caenepeel,Fred Van Oystaeyen
Publisher : CRC Press
Page : 344 pages
File Size : 52,7 Mb
Release : 2019-05-07
Category : Mathematics
ISBN : 9781482276718

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Hopf Algebras in Noncommutative Geometry and Physics by Stefaan Caenepeel,Fred Van Oystaeyen Pdf

This comprehensive reference summarizes the proceedings and keynote presentations from a recent conference held in Brussels, Belgium. Offering 1155 display equations, this volume contains original research and survey papers as well as contributions from world-renowned algebraists. It focuses on new results in classical Hopf algebras as well as the

Author : István Heckenberger,Hans-Jürgen Schneider
Publisher : American Mathematical Soc.
Page : 582 pages
File Size : 49,7 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 : Robert G. Underwood
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 52,9 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 : 164 pages
File Size : 47,7 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 : Nicolás Andruskiewitsch,Walter Ricardo Ferrer Santos,Hans-Jürgen Schneider
Publisher : American Mathematical Soc.
Page : 376 pages
File Size : 47,5 Mb
Release : 2000
Category : Hopf algebras
ISBN : 9780821821268

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New Trends in Hopf Algebra Theory by Nicolás Andruskiewitsch,Walter Ricardo Ferrer Santos,Hans-Jürgen Schneider Pdf

This volume presents the proceedings from the Colloquium on Quantum Groups and Hopf Algebras held in Cordoba (Argentina) in 1999. The meeting brought together researchers who discussed recent developments in Hopf algebras, one of the most important being the influence of quantum groups. Articles offer introductory expositions and surveys on topics of current interest that, to date, have not been available in the current literature. Surveys are included on characteristics of Hopf algebras and their generalizations, biFrobenius algebras, braided Hopf algebras, inner actions and Galois theory, face algebras, and infinitesimal Hopf algebras. The following topics are also covered: existence of integrals, classification of semisimple and pointed Hopf algebras, *-Hopf algebras, dendriform algebras, etc. Non-classical topics are also included, reflecting its applications both inside and outside the theory.

Author : Louis H. Kauffman,David E. Radford,Fernando José Oliveira Souza
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 47,7 Mb
Release : 2007
Category : Hopf algebras
ISBN : 9780821838204

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Hopf Algebras and Generalizations by Louis H. Kauffman,David E. Radford,Fernando José Oliveira Souza Pdf

Hopf algebras have proved to be very interesting structures with deep connections to various areas of mathematics, particularly through quantum groups. Indeed, the study of Hopf algebras, their representations, their generalizations, and the categories related to all these objects has an interdisciplinary nature. It finds methods, relationships, motivations and applications throughout algebra, category theory, topology, geometry, quantum field theory, quantum gravity, and also combinatorics, logic, and theoretical computer science. This volume portrays the vitality of contemporary research in Hopf algebras. Altogether, the articles in the volume explore essential aspects of Hopf algebras and some of their best-known generalizations by means of a variety of approaches and perspectives. They make use of quite different techniques that are already consolidated in the area of quantum algebra. This volume demonstrates the diversity and richness of its subject. Most of its papers introduce the reader to their respective contexts and structures through very expository preliminary sections.

Author : Daniel Bulacu,Stefaan Caenepeel,Florin Panaite,Freddy Van Oystaeyen
Publisher : Cambridge University Press
Page : 545 pages
File Size : 52,8 Mb
Release : 2019-02-21
Category : Mathematics
ISBN : 9781108427012

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Quasi-Hopf Algebras by Daniel Bulacu,Stefaan Caenepeel,Florin Panaite,Freddy Van Oystaeyen Pdf

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

Author : A. V. Zelevinsky
Publisher : Springer
Page : 189 pages
File Size : 45,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540387114

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Representations of Finite Classical Groups by A. V. Zelevinsky Pdf

Author : Rinat Kashaev
Publisher : Unknown
Page : 0 pages
File Size : 46,9 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 : Robert G. Underwood
Publisher : Springer
Page : 0 pages
File Size : 54,5 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 : Abdenacer Makhlouf
Publisher : John Wiley & Sons
Page : 338 pages
File Size : 45,7 Mb
Release : 2021-12-29
Category : Mathematics
ISBN : 9781789450187

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Algebra and Applications 2 by Abdenacer Makhlouf Pdf

This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras. The chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory. The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and Rota–Baxter algebras are explored. Algebra and Applications 2 is of great interest to graduate students and researchers. Each chapter combines some of the features of both a graduate level textbook and of research level surveys.