The wondrous world of hyperfinite subfactors
MATHEMATICAL PICTURE LANGUAGE
Dietmar Bisch - Vanderbilt University
The hyperfinite II1 factor contains a wealth of subfactors, many of which give rise to new and fascinating mathematical structures. For instance, the standard representation of a subfactor generates a certain unitary tensor category that Jones described as (what he called) a "planar algebra." It is a complete invariant for amenable, hyperfinite subfactors due to a deep result of Popa. However, generic subfactors are not amenable, and one typically does not know how to distinguish them. I will discuss a notion of "noncommutativity'' for a subfactor that provides an invariant that is complementary to the planar algebra. Bare hand constructions of hyperfinite subfactors generally lead to "commutative'' examples, and I will explain a theorem that allows us to produce "very noncommutative'' ones as well. It involves actions of suitable groups on the hyperfinite II1 factor.