Planar Algebras in Braided Tensor Categories / André Gil Henriques, David Penneys, and James E. Tener.

Format:

Book

Author/Creator:

Henriques, André G. (André Gil), 1977- author.

Penneys, David, author.

Tener, James E., author.

Series:

Memoirs of the American Mathematical Society ; Volume 282.

Memoirs of the American Mathematical Society Series ; Volume 282

Language:

English

Subjects (All):

Categories (Mathematics).

Tensor algebra.

Tensor products.

Physical Description:

1 online resource (112 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We generalize Jones' planar algebras by internalising the notion to a pivotal braided tensor category C. To formulate the notion, the planar tangles are now equipped with additional 'anchor lines' which connect the inner circles to the outer circle. We call the resulting notion an anchored planar algebra. If we restrict to the case when C is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in C and pivotal module tensor categories over C equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Chapter 2. Anchored planar algebras

2.1. Planar algebras

2.2. Anchored planar algebras

2.3. Generators for the anchored planar operad

Chapter 3. The main theorem and examples

3.1. Module tensor categories

3.2. The main theorem

3.2.1. Generalisations

3.3. Examples

3.3.1. Near group categories

3.3.2. Temperley-Lieb-Jones module categories

Chapter 4. Constructing anchored planar algebras

4.1. Generic tangles and tangles in standard form

4.2. The ribbon braid group

4.3. Assigning maps to generic tangles

4.4. Proof of Theorem 4.1

4.5. Induction along a lax functor

Chapter 5. Anchored planar algebras from module tensor categories

5.1. Diagram cheat sheet from [HPT16]

5.2. Constructing the anchored planar algebra

5.3. Functoriality

Chapter 6. Module tensor categories from anchored planar algebras

6.1. Reconstructing the category \cM

6.2. The adjoint pair Φ⊣\Tr_{\cC}

6.3. The tensor structure on \cM

6.4. The pivotal structure

6.5. The central functor

6.6. Functoriality

Chapter 7. Equivalence of categories

7.1. Module tensor categories to planar algebras and back

7.2. Naturality of Ψ

7.3. Planar algebras to module tensor categories and back

Appendix A. An associativity type relation

Appendix B. Anchored planar tangles with coupons

Appendix C. The tube string calculus for the categorified trace

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Henriques, André Gil Planar Algebras in Braided Tensor Categories

ISBN:

9781470473488

1470473488