Automorphisms of the bipartite graph planar algebra

TL;DR

This paper classifies the automorphism group of bipartite graph planar algebras and constructs new subfactor examples by taking fixed points under these automorphisms, enriching the understanding of subfactor planar algebras.

Contribution

It provides a complete classification of automorphisms for bipartite graph planar algebras and introduces a method to generate new subfactors via fixed points.

Findings

Automorphism groups of bipartite graph planar algebras are fully classified.

New subfactor examples are constructed using fixed points under automorphism groups.

Provides new descriptions for known planar algebras.

Abstract

For any abstract subfactor planar algebra $P$, there exists a finite index extremal subfactor $M_0 \subset M_1$ with $P$ as its standard invariant. In this paper, we classify the automorphism group of a bipartite graph planar algebra, and obtain subfactor planar subalgebras by taking fixed points under groups of automorphisms. This construction provides both new examples of subfactors and new descriptions of the planar algebras of previously known examples.