Automorphisms of the bipartite graph planar algebra

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.