On Finite GK-Dimensional Nichols Algebras over Abelian Groups.

Format:

Book

Author/Creator:

Andruskiewitsch, Nicolás.

Contributor:

Angiono, Iván.

Heckenberger, István.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.271

Language:

English

Subjects (All):

Hopf algebras.

Physical Description:

1 online resource (142 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim if and only if the size of the block is 2 and the eigenvalue is 1; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim. Consequently we present several new examples of Nichols algebras with finite GKdim, including two not in the class alluded to above. We determine which among these Nichols algebras are domains"-- Provided by publisher.

Contents:

Cover

Title page

List of Tables

Chapter 1. Introduction

1.1. Antecedents

1.2. Points and blocks

1.3. The main result

1.3.1. The class of braided vector spaces

1.3.2. Diagonal type

1.3.3. Flourished graphs

1.3.4. Organization of the paper and scheme of the proof

1.3.5. About the proofs

1.3.6. The Poseidon Nichols algebras

1.4. Applications

1.4.1. Examples of Hopf algebras

1.4.2. Domains

1.4.3. Co-Frobenius Hopf algebras

Chapter 2. Preliminaries

2.1. Conventions

2.2. Nichols algebras of diagonal type

2.3. On the Gelfand-Kirillov dimension

2.3.1. Basic facts

2.3.2. A criterium for infinite \GK

Chapter 3. Yetter-Drinfeld modules of dimension 2

3.1. Indecomposable modules and blocks

3.2. The Jordan plane

3.3. The super Jordan plane

3.4. Filtrations of Nichols algebras

3.5. Proof of Theorem 3.1.2

Chapter 4. Yetter-Drinfeld modules of dimension 3

4.1. The setting

4.1.1. A block and a point

4.1.2. A pale block and a point

4.1.3. Indecomposable of dimension 3

4.1.4. Notations

4.1.5. Strong interaction

4.2. Weak interaction

4.2.1. Preparations

4.2.2. Proof of Theorem 4.1.3

4.2.3. Proof of Theorem 4.1.1, weak interaction

4.3. The Nichols algebras with finite \GK

4.3.1. The Nichols algebra \cB(\lstr(1,\ghost))

4.3.2. The Nichols algebra \cB(\lstr(-1,\ghost))

4.3.3. The Nichols algebra \cB(\lstr₋(1,\ghost))

4.3.4. The Nichols algebra \cB(\lstr₋(-1,\ghost))

4.3.5. The Nichols algebra \cB(\lstr( ,1))

4.4. Mild interaction

4.4.1. The Nichols algebra \cB(\cyc₁)

Chapter 5. One block and several points

5.1. The setting

5.2. Proof of Theorem 5.1.1 ( =1)

5.2.1. Weak interaction and the algebra

5.2.2. | |=2

5.2.3. | |&gt

2

5.3. The Nichols algebras with finite \GK, _{\diag} connected.

5.3.1. The Nichols algebra \cB(\lstr( (1|0)₁

)), ∈\G_{ }', ≥3

5.3.2. The Nichols algebra \cB(\lstr( (1|0)₁

)), ∉\G_{∞}

5.3.3. The Nichols algebra \cB(\lstr( (1|0)₂

))

5.3.4. The Nichols algebra \cB(\lstr( (1|0)₃

5.3.5. The Nichols algebra \cB(\lstr( (2|0)₁

5.3.6. The Nichols algebra \cB(\lstr( (2|1)

5.3.7. The Nichols algebra \cB(\lstr( ₂,2))

5.3.8. The Nichols algebra \cB(\lstr( _{ -1}))

5.4. Proof of Theorem 5.1.2 ( =-1)

5.4.1. Connected components of _{\diag}

5.4.2. The Nichols algebra \cB(\cyc₂)

5.4.3. Several components

5.4.4. The Nichols algebras with finite \GK, several connected components in _{\diag}

Chapter 6. Two blocks

6.1. The setting

6.2. ₁=1

6.3. ₁= ₂=-1

Chapter 7. Several blocks, several points

7.1. Notations

7.2. Several blocks, one point

7.3. The Nichols algebras \pos(\bq,\ghost)

7.4. Several blocks, several points

Chapter 8. Appendix

8.1. Nichols algebras over abelian groups

8.1.1. The context

8.1.2. A pale block and a point

8.1.3. The block has =1

8.1.4. The block has =-1

8.1.5. The block has = ∈\G₃'

8.2. Admissible flourished diagrams

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470466367

1470466368

OCLC:

1266906978