The planar algebra of a fixed point subfactor

TL;DR

This paper investigates the structure of planar algebras associated with fixed point subfactors arising from compact quantum group actions on II_1 factors and finite-dimensional inclusions, revealing a specific algebraic form.

Contribution

It characterizes the planar algebra of fixed point subfactors as a natural action on bipartite graph algebras in the case of commuting inclusions.

Findings

Planar algebra of fixed point subfactors has a specific algebraic form.

The form involves the bipartite graph algebra with quantum group action.

Results apply to all known examples with commuting inclusions.

Abstract

We consider inclusions of type $(P\otimes A)^G\subset(P\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\subset B$. In the case $[A,B]=0$, which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form $P(A\subset B)^G$, with $G$ acting in some natural sense on the bipartite graph algebra $P(A\subset B)$.