Hopf algebras and root systems / István Heckenberger, Hans-Jürgen Schneider.

Format:

Book

Author/Creator:

Heckenberger, István, 1969- author.

Schneider, Hans-Jürgen, 1944- author.

Series:

Mathematical surveys and monographs ; no. 247.

Mathematical surveys and monographs, 0076-5376 ; volume 247

Language:

English

Subjects (All):

Hopf algebras.

Root systems (Algebra).

Weyl groups.

Physical Description:

xix, 582 pages : illustrations ; 27 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2020]

Summary:

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-- Publisher's description.

Contents:

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 and Weyl groupoids

The structure of Cartan graphs and root systems

Cartan graphs of Lie superalgebras

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

Nichols algebras of diagonal type

Nichols algebras of Cartan type

Nichols algebras over non-abelian groups.

Notes:

Includes bibliographical references (pages 565-574) and indexes.

ISBN:

9781470452322

1470452324

OCLC:

1147925299