Planar algebras in braided tensor categories / André Gil Henriques, David Penneys, James Tener.

Format:

Book

Author/Creator:

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

Penneys, David, author.

Tener, James, author.

Series:

Memoirs of the American Mathematical Society ; no. 1392.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1392

Language:

English

Subjects (All):

Categories (Mathematics).

Tensor products.

Tensor algebra.

Algebra.

Physical Description:

v, 100 pages : illustrations ; 26 cm

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:

Chapter 1. Introduction

Chapter 2. Anchored planar algebras

Chapter 3. The main theorem and examples

Chapter 4. Constructing anchored planar algebras

Chapter 5. Anchored planar algebras from modular tensor categories

Chapter 6. Modular tensor categories from anchored planar algebras

Chapter 7. Equivalence of categories

Appendix A. An associativity type relation

Appendix B. Anchored planar tangles with coupons

Appendix C. The tube string calculus for the categorified trace

Bibliography.

Notes:

Includes bibliographical references.

ISBN:

9781470455408

1470455404

OCLC:

1371970297