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

  • César Galindo

    Universidad de los Andes, Bogota, Colombia
  • Ismael Gutiérrez

    Universidad del Norte, Barranquilla, Colombia
  • Bernardo Uribe

    Universidad del Norte, Barranquilla, Colombia

Abstract

Let be a finite group and let be a finite group acting on 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 -equivariant K-theory of , and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and -equivariant bundles over . 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 ; we use known calculations of the cohomology of , and to produce concrete examples of our classification.

In the case in which and acts by conjugation, the multiplication map 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 . 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.

Cite this article

César Galindo, Ismael Gutiérrez, Bernardo Uribe, Multiplicative structures on the twisted equivariant K-theory of finite groups. J. Noncommut. Geom. 9 (2015), no. 3, pp. 877–937

DOI 10.4171/JNCG/211