Loop groups and twisted K-theory I

TL;DR

This paper develops foundational tools for twisted equivariant K-theory of compact Lie groups, linking it to loop group representations and setting the stage for further exploration of the Verlinde ring connection.

Contribution

It establishes the basic properties of twisted equivariant K-theory for Lie groups and computes these groups for certain cases, connecting to loop group representation theory.

Findings

Computed twisted equivariant K-groups for compact connected Lie groups with torsion-free fundamental groups.

Linked twisted K-theory computations to the representation theory of loop groups.

Established foundational properties enabling effective calculations in twisted K-theory.

Abstract

This is the first in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the "Verlinde ring" of its loop group. In this paper we set up the foundations of twisted equivariant K-groups, and more generally twisted K-theory of groupoids. We establish enough basic properties to make effective computations. Using the Mayer-Vietoris spectral sequence we compute the twisted equivariant K-groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting.