60 research outputs found
Bicommutant categories from fusion categories
Bicommutant categories are higher categorical analogs of von Neumann algebras
that were recently introduced by the first author. In this article, we prove
that every unitary fusion category gives an example of a bicommutant category.
This theorem categorifies the well known result according to which a finite
dimensional *-algebra that can be faithfully represented on a Hilbert space is
in fact a von Neumann algebra.Comment: Updated to the published version + fixed some small typo
Constructing spoke subfactors using the jellyfish algorithm
Using Jones' quadratic tangles formulas, we automate the construction of the
4442, 3333, 3311, and 2221 spoke subfactors by finding sets of 1-strand
jellyfish generators. The 4442 spoke subfactor is new, and the 3333, 3311, and
2221 spoke subfactors were previously known.Comment: 44 pages, many figures, to appear in Transactions of the American
Mathematical Societ
-algebras from planar algebras I: canonical -algebras associated to a planar algebra
From a planar algebra, we give a functorial construction to produce numerous
associated -algebras. Our main construction is a Hilbert -bimodule
with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner,
and generalized free semicircular -algebras. By compressing this system,
we obtain various canonical -algebras, including Doplicher-Roberts
algebras, Guionnet-Jones-Shlyakhtenko algebras, universal
(Toeplitz-)Cuntz-Krieger algebras, and the newly introduced free graph
algebras. This is the first article in a series studying canonical
-algebras associated to a planar algebra.Comment: 47 pages, many figure
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