## Angle Operators of Commuting Square Subfactors

 dc.contributor.advisor Bisch, Dietmar dc.creator Montgomery, Michael Ray dc.date.accessioned 2022-07-12T16:45:36Z dc.date.available 2022-07-12T16:45:36Z dc.date.created 2022-06 dc.date.issued 2022-05-18 dc.date.submitted June 2022 dc.identifier.uri http://hdl.handle.net/1803/17523 dc.description.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. dc.format.mimetype application/pdf dc.language.iso en dc.subject subfactors dc.subject planar algebras dc.title Angle Operators of Commuting Square Subfactors dc.type Thesis dc.date.updated 2022-07-12T16:45:36Z dc.type.material text thesis.degree.name PhD thesis.degree.level Doctoral thesis.degree.discipline Mathematics thesis.degree.grantor Vanderbilt University Graduate School dc.creator.orcid 0000-0002-7389-5007 dc.contributor.committeeChair Bisch, Dietmar
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