The ribbon category of representations of a crossed module

Abstract

The theory of representations of a crossed module is a direct generalization of the theory of representations of groups. For a finite group G, the Drinfeld quantum double of the group G is a Hopf algebra that represents a special case of crossed module of finite groups. Here we study how to extend the construction of the Drinfeld quantum double for any other kind of crossed module of finite groups. This leads to a Hopf algebra D(G,H) that presents similarities with a Drinfeld double. We then study simple subalgebras of D(G,H) and give two isomorphisms for the decomposition into a product of simple subalgebras. We then study the category D(G,H)-mod^Fd of finite dimensional modules over D(G,H), which turns out to be isomorphic to the category of finite dimensional representations of finite crossed modules of groups. These categories being monoidal, we also study links between direct sums of simple objects and tensor products of simple objects and give some results for a Clebsch-Gordan formula. We, in this context, present and develop the character theory for representations of crossed modules of finite groups, and detail the proofs. We then study the category itself, which leads to some ribbon invariants.