# K-theory of some AF-algebras from braided categories

### Andreas Aaserud (Cardiff University)

Thursday 25th October, 2018 16:00-17:00 Maths 311B

#### Abstract

In the 1980s, Renault, Wassermann, Handelman and Rossmann explicitly described the K-theory of the fixed point algebras of certain actions of compact groups on AF-algebras as polynomial rings. Similarly, Evans and Gould in 1994 explicitly described the K-theory of certain AF-algebras related to SU(2) as quotients of polynomial rings. In this talk, I will explain how, in all of these cases, the multiplication in the polynomial ring (quotient) is induced by a $*$-homomorphism $A\otimes A\to A$ (where $A$ denotes the AF-algebra whose K-theory is being computed) arising from a unitary braiding on an underlying C*-tensor category and essentially defined by Erlijman and Wenzl in 2007. I will also give new explicit descriptions of the K-theory of certain AF-algebras related to SU(3), Sp(4) and G$_2$ as quotients of polynomial rings. Finally, I will attempt to explain how this work is motivated by the Freed-Hopkins-Teleman formula for the fusion rings of WZW-models in conformal field theory. This is all based on joint work with David E. Evans.