Hardcover ISBN:  9781470452322 
Product Code:  SURV/247 
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eBook ISBN:  9781470456801 
Product Code:  SURV/247.E 
List Price:  $140.00 
MAA Member Price:  $126.00 
AMS Member Price:  $112.00 
Hardcover ISBN:  9781470452322 
eBook: ISBN:  9781470456801 
Product Code:  SURV/247.B 
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MAA Member Price:  $252.00 $189.00 
AMS Member Price:  $224.00 $168.00 
Hardcover ISBN:  9781470452322 
Product Code:  SURV/247 
List Price:  $140.00 
MAA Member Price:  $126.00 
AMS Member Price:  $112.00 
eBook ISBN:  9781470456801 
Product Code:  SURV/247.E 
List Price:  $140.00 
MAA Member Price:  $126.00 
AMS Member Price:  $112.00 
Hardcover ISBN:  9781470452322 
eBook ISBN:  9781470456801 
Product Code:  SURV/247.B 
List Price:  $280.00 $210.00 
MAA Member Price:  $252.00 $189.00 
AMS Member Price:  $224.00 $168.00 

Book DetailsMathematical Surveys and MonographsVolume: 247; 2020; 582 ppMSC: Primary 16
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 PBWbases 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.
ReadershipGraduate students and researchers interested in category theory, Hopf algebras, and applications.

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

YetterDrinfeld 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 YetterDrinfeld modules

Nichols systems, and semiCartan 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 nonabelian groups


Additional Material

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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 PBWbases 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.
Graduate students and researchers interested in category theory, Hopf algebras, and applications.

Hopf algebras, Nichols algebras, braided monoidal categories, and quantized enveloping algebras

A quick introduction to Nichols algebras

Basic Hopf algebra theory

Braided monoidal categories

YetterDrinfeld 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 YetterDrinfeld modules

Nichols systems, and semiCartan 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 nonabelian groups