$K$-theory of AF-algebras from braided C*-tensor categories

TL;DR

This paper links the K-theory of AF-algebras derived from braided C*-tensor categories to polynomial rings, providing explicit calculations for certain Lie groups and aiming to simplify the realization of Verlinde rings in K-theory.

Contribution

It demonstrates that the multiplication in the K-theory of AF-algebras is induced by a *-homomorphism from a braided tensor category, and provides new explicit calculations for rank two Lie groups.

Findings

K-theory of AF-algebras can be described via polynomial rings with multiplication from braided tensor categories.

Explicit K-theoretic calculations performed for SU(3), Sp(4), and G2.

Verlinde rings can be recovered in specific cases through this approach.

Abstract

Renault, Wassermann, Handelman and Rossmann (early 1980s) and Evans and Gould (1994) explicitly described the $K$-theory of certain unital AF-algebras $A$ as (quotients of) polynomial rings. In this paper, we show that in each case the multiplication in the polynomial ring (quotient) is induced by a $*$-homomorphism $A\otimes A\to A$ arising from a unitary braiding on a C*-tensor category and essentially defined by Erlijman and Wenzl (2007). We also present some new explicit calculations based on the work of Gepner, Fuchs and others. Specifically, we perform computations for the rank two compact Lie groups SU(3), Sp(4) and G$_2$ that are analogous to the Evans-Gould computation for the rank one compact Lie group SU(2). The Verlinde rings are the fusion rings of Wess-Zumino-Witten models in conformal field theory or, equivalently, of certain related C*-tensor categories. Freed, Hopkins and Teleman (early 2000s) realized these rings via twisted equivariant $K$-theory. Inspired by this, our long-term goal is to realize these rings in a simpler $K$-theoretical manner, avoiding the technicalities of loop group analysis. As a step in this direction, we note that the Verlinde rings can be recovered as above in certain special cases.