Overview

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.

Full Product Details

Author:   Istvan Heckenberger ,  Hans-Jurgen Schneider
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Weight:   1.290kg
ISBN:  

9781470452322

ISBN 10:   1470452324
Pages:   582
Publication Date:   30 August 2020
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Hopf algebras, Nichols algebras, braided monoidal categories, and quantized enveloping algebras: A quick introduction to Nichols algebras Basic Hopf algebra theory Braided monoidal categories Yetter-Drinfeld modules over Hopf algebras Gradings and filtrations Braided structures Nichols algebras Quantized enveloping algebras and generalizations Cartan graphs, Weyl groupoids, and root systems: Cartan graphs and Weyl groupoids The structure of Cartan graphs and root systems Cartan graphs of Lie superalgebras Weyl groupoids and root systems of Nichols algebras: A braided monoidal isomorphism of Yetter-Drinfeld modules Nichols systems, and semi-Cartan graph of Nichols algebras Right coideal subalgebras of Nichols systems, and Cartan graph of Nichols algebras Applications: Nichols algebras of diagonal type Nichols algebras of Cartan type Nichols algebras over non-abelian groups Bibliography Index of symbols Subject index.

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Author Information

Istvan Heckenberger, Philipps Universitat Marburg, Germany. Hans-Jurgen Schneider, Ludwig-Maximilians-Universität München, Mathematisches Institut, München, Germany.

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