Graph Planar Algebra Embeddings and New A-infinity Subfactors
Caceres Gonzales, Julio
0000-0001-5693-1187
:
2024-05-09
Abstract
A commuting square is a square of inclusions of finite dimensional C*-algebras that satisfy a notion of orthogonality. Commuting square subfactors are subfactors that can be constructed by iterating the basic construction on a commuting square, they are always hyperfinite. We prove that the subfactor planar algebra of a commuting square subfactor embeds into the graph planar algebra of the first vertical inclusion graph of the commuting square. We use this to show that a commuting square subfactor does not have finite depth, provided the first vertical inclusion is not a module graph for any finite depth subfactor with the same index. We then construct commuting squares that produce irreducible hyperfinite subfactors with indices $\approx 4.37220$, the index of the Extended Haagerup subfactor, and $\frac{5+\sqrt{17}}{2}$, the index of the Asaeda-Haagerup subfactor. Since the vertical inclusion graphs for these commuting squares are not module graphs, the subfactors must have infinite depth and by classification, they are $A_\infty$-subfactors.
Alternatively, we construct 1-parameter families of non-equivalent commuting squares based on the 4-stars $S(i,i,j,j)$ for all $i,j\geq 1$. Using Kawahigashi's characterization of finite-dimensional commuting squares, we show that each family must produce at least one infinite depth subfactor. Therefore we have constructed hyperfinite irreducible subfactors with indices $\frac{5+\sqrt{17}}{2}$, $3+\sqrt{3}$, $\frac{5+\sqrt{21}}{2}$, $5$ and $3+\sqrt{5}$. By classification, all but the last one must be $A_\infty$-subfactors.