Drinfeld center of planar algebra

TL;DR

This paper establishes a connection between the Drinfeld center of a subfactor planar algebra and Hilbert affine representations, proving Kevin Walker's conjecture for planar algebras.

Contribution

It introduces fusion, contragradient, and braiding structures for Hilbert affine representations of planar algebras and proves their equivalence to the Drinfeld center, confirming a conjecture.

Findings

Proved the equivalence of the Drinfeld center with Hilbert affine representations under certain conditions.

Extended the theory to planar algebras not necessarily of finite depth.

Confirmed Kevin Walker's conjecture for planar algebras.

Abstract

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra $P$ (not necessarily having finite depth). We prove that if $N \subset M$ is a subfactor realization of $P$, then the Drinfeld center of the $N$-$N$-bimodule category generated by $_N L^2 (M)_M$, is equivalent to the category of Hilbert affine representations of $P$ satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.