Angle Operators of Commuting Square Subfactors

Abstract

Complex Hadamard matrices are biunitaries for spin model commuting squares. The corresponding subfactor standard invariant can be identified with the $1$-eigenspace of the angle operator defined by Vaughan Jones. We identify the angle operator as an element of the symmetric enveloping algebra and compute its trace. We then show the angle operator spectrum coincides with the principal graph spectrum up to a constant iff the subfactor is amenable. We use this to show Paley type $II$ Hadamard matrices and Petrescu's $7 \times 7$ family of complex Hadamard matrices yield infinite depth subfactors. We determine the connected components of the profile matrix to show that $4n \times 4n$ Hadamard matrices, where $n \geq 3$ is odd, yield at least two-supertransitive subfactors. Finally, we define a colored graph planar algebra to study symmetric commuting square subfactors. In the colored graph planar algebra associated to a symmetric commuting square, biunitaries satisfy type $II$ Reidemeister relations. We then generalize the angle operator for symmetric commuting squares and prove a criterion that implies subfactors from symmetric commuting squares are infinite depth.