Hopf Algebras And Generalizations

Author : Gabriella Böhm
Publisher : Springer
Page : 165 pages
File Size : 43,7 Mb
Release : 2018-11-01
Category : Mathematics
ISBN : 9783319981376

Get Book

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 : Louis H. Kauffman,David E. Radford,Fernando José Oliveira Souza
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 40,8 Mb
Release : 2007
Category : Hopf algebras
ISBN : 9780821838204

Get Book

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 : István Heckenberger,Hans-Jürgen Schneider
Publisher : American Mathematical Soc.
Page : 582 pages
File Size : 43,9 Mb
Release : 2020-06-19
Category : Education
ISBN : 9781470452322

Get Book

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 : Daniel Bulacu,Stefaan Caenepeel,Florin Panaite,Freddy Van Oystaeyen
Publisher : Cambridge University Press
Page : 545 pages
File Size : 46,5 Mb
Release : 2019-02-21
Category : Mathematics
ISBN : 9781108427012

Get Book

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 : Nicolás Andruskiewitsch,Walter Ricardo Ferrer Santos,Hans-Jürgen Schneider
Publisher : American Mathematical Soc.
Page : 376 pages
File Size : 49,7 Mb
Release : 2000
Category : Hopf algebras
ISBN : 9780821821268

Get Book

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 : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 53,6 Mb
Release : 2024-06-28
Category : Electronic
ISBN : 9789814464161

Get Book

Group Theory and Hopf Algebras by Anonim Pdf

Author : A. P. Balachandran,S. G. Jo,Giuseppe Marmo
Publisher : World Scientific
Page : 270 pages
File Size : 47,5 Mb
Release : 2010
Category : Science
ISBN : 9789814322201

Get Book

Group Theory and Hopf Algebras by A. P. Balachandran,S. G. Jo,Giuseppe Marmo Pdf

This book is addressed to graduate students and research workers in theoretical physics who want a thorough introduction to group theory and Hopf algebras. It is suitable for a one-semester course in group theory or a two-semester course which also treats advanced topics. Starting from basic definitions, it goes on to treat both finite and Lie groups as well as Hopf algebras. Because of the diversity in the choice of topics, which does not place undue emphasis on finite or Lie groups, it should be useful to physicists working in many branches. A unique aspect of the book is its treatment of Hopf algebras in a form accessible to physicists. Hopf algebras are generalizations of groups and their concepts are acquiring importance in the treatment of conformal field theories, noncommutative spacetimes, topological quantum computation and other important domains of investigation. But there is a scarcity of treatments of Hopf algebras at a level and in a manner that physicists are comfortable with. This book addresses this need superbly. There are illustrative examples from physics scattered throughout the book and in its set of problems. It also has a good bibliography. These features should enhance its value to readers. The authors are senior physicists with considerable research and teaching experience in diverse aspects of fundamental physics. The book, being the outcome of their combined efforts, stands testament to their knowledge and pedagogical skills.

Author : Pierre Cartier,Frédéric Patras
Publisher : Springer Nature
Page : 277 pages
File Size : 51,7 Mb
Release : 2021-09-20
Category : Mathematics
ISBN : 9783030778453

Get Book

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 : Florin Felix Nichita
Publisher : MDPI
Page : 239 pages
File Size : 46,8 Mb
Release : 2019-01-31
Category : Mathematics
ISBN : 9783038973249

Get Book

Hopf Algebras, Quantum Groups and Yang-Baxter Equations by Florin Felix Nichita Pdf

This book is a printed edition of the Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations" that was published in Axioms

Author : Nicolás Andruskiewitsch,Juan Cuadra,Blas Torrecillas
Publisher : American Mathematical Soc.
Page : 347 pages
File Size : 48,6 Mb
Release : 2013-02-21
Category : Mathematics
ISBN : 9780821875643

Get Book

Hopf Algebras and Tensor Categories by Nicolás Andruskiewitsch,Juan Cuadra,Blas Torrecillas Pdf

This volume contains the proceedings of the Conference on Hopf Algebras and Tensor Categories, held July 4-8, 2011, at the University of Almeria, Almeria, Spain. The articles in this volume cover a wide variety of topics related to the theory of Hopf algebras and its connections to other areas of mathematics. In particular, this volume contains a survey covering aspects of the classification of fusion categories using Morita equivalence methods, a long comprehensive introduction to Hopf algebras in the category of species, and a summary of the status to date of the classification of Hopf algebras of dimensions up to 100. Among other topics discussed in this volume are a study of normalized class sum and generalized character table for semisimple Hopf algebras, a contribution to the classification program of finite dimensional pointed Hopf algebras, relations to the conjecture of De Concini, Kac, and Procesi on representations of quantum groups at roots of unity, a categorical approach to the Drinfeld double of a braided Hopf algebra via Hopf monads, an overview of Hom-Hopf algebras, and several discussions on the crossed product construction in different settings.

Author : Lindsay Childs
Publisher : American Mathematical Soc.
Page : 118 pages
File Size : 49,6 Mb
Release : 1998
Category : Mathematics
ISBN : 9780821810774

Get Book

Hopf Algebras, Polynomial Formal Groups, and Raynaud Orders by Lindsay Childs Pdf

This book gives two new methods for constructing $p$-elementary Hopf algebra orders over the valuation ring $R$ of a local field $K$ containing the $p$-adic rational numbers. One method constructs Hopf orders using isogenies of commutative degree 2 polynomial formal groups of dimension $n$, and is built on a systematic study of such formal group laws. The other method uses an exponential generalization of a 1992 construction of Greither. Both constructions yield Raynaud orders as iterated extensions of rank $p$ Hopf algebras; the exponential method obtains all Raynaud orders whose invariants satisfy a certain $p$-adic condition.

Author : Robert G. Underwood
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 51,8 Mb
Release : 2011-08-30
Category : Mathematics
ISBN : 9780387727653

Get Book

An Introduction to Hopf Algebras by Robert G. Underwood Pdf

Only book on Hopf algebras aimed at advanced undergraduates

Author : Lindsay Childs
Publisher : American Mathematical Soc.
Page : 225 pages
File Size : 50,6 Mb
Release : 2000
Category : Field extensions (Mathematics).
ISBN : 9780821821312

Get Book

Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory by Lindsay Childs Pdf

This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory. Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree p and p2; and Kummer theory of formal groups. Beyond a general background in graduate-level algebra, some chapters assume an acquaintance with some algebraic number theory. From there, this exposition serves as an excellent resource and motivation for further work in the field.

Author : William Joseph Haboush
Publisher : American Mathematical Soc.
Page : 429 pages
File Size : 53,6 Mb
Release : 1994
Category : Differential algebra
ISBN : 9780821815410

Get Book

Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods by William Joseph Haboush Pdf

Proceedings of a research institute held at Pennsylvania State University, July 1991, focusing on quantum and infinite-dimensional methods of algebraic groups. Topics include perverse sheaves, finite Chevalley groups, the general theory of algebraic groups, representations, invariant theory, general

Author : Sorin Dascalescu,Constantin Nastasescu,Serban Raianu
Publisher : CRC Press
Page : 420 pages
File Size : 53,6 Mb
Release : 2000-09-15
Category : Mathematics
ISBN : 9781482270747

Get Book

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.