Multiplicative Structures on the Twisted Equivariant K-theory of Finite Groups

TL;DR

This paper classifies all possible multiplicative and monoidal structures on twisted equivariant K-theory and bundles over finite groups, using cohomological methods and explicit examples, with applications to the Twisted Drinfeld Double.

Contribution

It provides a comprehensive classification of multiplicative and monoidal structures in twisted equivariant K-theory and bundle categories for finite groups, linking cohomology and algebraic structures.

Findings

Classification of multiplicative and monoidal structures achieved.

Explicit examples using cohomology calculations provided.

Connection established between categorical structures and the Twisted Drinfeld Double.

Abstract

Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted $G$-equivariant K-theory of $K$, and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and $G$-equivariant bundles over $K$. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semi-direct product $K \rtimes G$; we use known calculations of the cohomology of $K$, $G$ and $K \rtimes G$ to produce concrete examples of our classification. In the case in which $K=G$ and $G$ acts by conjugation, the multiplication map $G \rtimes G \to G$ is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of $G$. We use the pullback of the multiplication map in cohomology to classify the possible ring structures that the Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.