New Directions in Hopf Algebras
New Directions in Hopf Algebras

Author: Susan Montgomery

Publisher: Cambridge University Press

Published: 2002-05-06

Total Pages: 502

ISBN-13: 9780521815123

DOWNLOAD EBOOK

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.

Hopf Algebras, Tensor Categories and Related Topics
Hopf Algebras, Tensor Categories and Related Topics

Author: Nicolás Andruskiewitsch

Publisher: American Mathematical Soc.

Published: 2021-07-06

Total Pages: 359

ISBN-13: 1470456249

DOWNLOAD EBOOK

The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras. Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories.

Hopf Algebras and Their Generalizations from a Category Theoretical Point of View
Hopf Algebras and Their Generalizations from a Category Theoretical Point of View

Author: Gabriella Böhm

Publisher: Springer

Published: 2018-11-01

Total Pages: 171

ISBN-13: 3319981374

DOWNLOAD EBOOK

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.

New Developments in Lie Theory and Its Applications
New Developments in Lie Theory and Its Applications

Author: Carina Boyallian

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 169

ISBN-13: 0821852590

DOWNLOAD EBOOK

Focuses on representation theory, harmonic analysis in Lie groups, and mathematical physics related to Lie theory. The papers give a broad overview of these subjects and also of the recent developments in research.

Hopf Algebras in Noncommutative Geometry and Physics
Hopf Algebras in Noncommutative Geometry and Physics

Author: Stefaan Caenepeel

Publisher: CRC Press

Published: 2019-05-07

Total Pages: 344

ISBN-13: 1482276712

DOWNLOAD EBOOK

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

Tensor Categories and Hopf Algebras
Tensor Categories and Hopf Algebras

Author: Nicolás Andruskiewitsch

Publisher: American Mathematical Soc.

Published: 2019-04-18

Total Pages: 210

ISBN-13: 147044321X

DOWNLOAD EBOOK

This volume contains the proceedings of the scientific session “Hopf Algebras and Tensor Categories”, held from July 27–28, 2017, at the Mathematical Congress of the Americas in Montreal, Canada. Papers highlight the latest advances and research directions in the theory of tensor categories and Hopf algebras. Primary topics include classification and structure theory of tensor categories and Hopf algebras, Gelfand-Kirillov dimension theory for Nichols algebras, module categories and weak Hopf algebras, Hopf Galois extensions, graded simple algebras, and bialgebra coverings.




New Developments in Lie Theory and Geometry
New Developments in Lie Theory and Geometry

Author: Carolyn Gordon

Publisher: American Mathematical Soc.

Published: 2009

Total Pages: 363

ISBN-13: 0821846515

DOWNLOAD EBOOK

This volume is an outgrowth of the Sixth Workshop on Lie Theory and Geometry, held in the province of Cordoba, Argentina in November 2007. The representation theory and structure theory of Lie groups play a pervasive role throughout mathematics and physics. Lie groups are tightly intertwined with geometry and each stimulates developments in the other. The aim of this volume is to bring to a larger audience the mutually beneficial interaction between Lie theorists and geometers that animated the workshop. Two prominent themes of the representation theoretic articles are Gelfand pairs and the representation theory of real reductive Lie groups. Among the more geometric articles are an exposition of major recent developments on noncompact homogeneous Einstein manifolds and aspects of inverse spectral geometry presented in settings accessible to readers new to the area.

Handbook of Algebra
Handbook of Algebra

Author: M. Hazewinkel

Publisher: Elsevier

Published: 2008-04-18

Total Pages: 577

ISBN-13: 0080569412

DOWNLOAD EBOOK

Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest. In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc. The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published. A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed. - Thorough and practical source of information - Provides in-depth coverage of new topics in algebra - Includes references to relevant articles, books and lecture notes

Hopf Algebras and Tensor Categories
Hopf Algebras and Tensor Categories

Author: Nicolás Andruskiewitsch

Publisher: American Mathematical Soc.

Published: 2013-02-21

Total Pages: 347

ISBN-13: 0821875647

DOWNLOAD EBOOK

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.