The ribbon category of representations of a crossed module
Ony Aubril
TL;DR
The paper generalizes the Drinfeld quantum double to finite crossed modules by constructing a Hopf algebra $D(G,H)$ whose module category is monoidally equivalent to the category of representations of the crossed module $X=(G,H, extmu, extgamma)$. It develops the semisimple structure, centers, and a Clebsch–Gordan formula for tensor products via both algebraic and character-theoretic methods, and establishes that $ ext{M}(X)_{ extbf{Fd}}$ is a braided ribbon fusion category with explicit ribbon data, enabling ribbon invariants. The work provides a complete character theory for crossed-module representations, including character tables and fusion rules, and situates these structures within a fusion/ribbon framework with potential applications to quantum topology and conformal field theories. The explicit Hopf-algebra realization $D(G,H)$ unifies crossed-module representations with module categories over Hopf algebras, extending the reach of Drinfeld doubles beyond finite groups.
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)-modFd 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.