TY - JOUR

T1 - Planar Algebras in Braided Tensor Categories

AU - Henriques, André Gil

AU - Penneys, David

AU - Tener, James

N1 - Publisher Copyright:
© 2023 by the American Mathematical Society. All rights reserved.

PY - 2023/2

Y1 - 2023/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85152103360&partnerID=8YFLogxK

U2 - 10.1090/memo/1392

DO - 10.1090/memo/1392

M3 - Article

SN - 0065-9266

VL - 282

SP - 1

EP - 112

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

IS - 1392

ER -