# Journal of Noncommutative Geometry

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**Volume 9, Issue 3, 2015, pp. 877–937**

**DOI: 10.4171/JNCG/211**

Published online: 2015-10-29

Multiplicative structures on the twisted equivariant K-theory of finite groups

César Galindo^{[1]}, Ismael Gutiérrez

^{[2]}and Bernardo Uribe

^{[3]}(1) Universidad de los Andes, Bogota, Colombia

(2) Universidad del Norte, Barranquilla, Colombia

(3) Universidad del Norte, Barranquilla, Colombia

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.

*Keywords: *Twisted equivariant K-theory, twisted Drinfeld double, fusion category, multiplicative structure, fusion algebra

Galindo César, Gutiérrez Ismael, Uribe Bernardo: Multiplicative structures on the twisted equivariant K-theory of finite groups. *J. Noncommut. Geom.* 9 (2015), 877-937. doi: 10.4171/JNCG/211