Hopf algebras and root systems
著者
書誌事項
Hopf algebras and root systems
（Mathematical surveys and monographs, v. 247）
American Mathematical Society, c2020
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注記
Includes bibliographical references (p. 565-574) and indexes
内容説明・目次
内容説明
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.
目次
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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