AG Lie-Gruppen: L. M. Lawson, Max Planck Institute for the Science of Light, Erlangen: Crossed Module of Semisimple Hopf Algebra

Crossed Module of Semisimple Hopf Algebra – Vortragender: Latévi M. Lawson, Max
Planck Institute for the Science of Light, Erlangen – Einladende: C. Meusburger

Abstract
Let (E ∂ −→ G,▷) be a crossed module of groups with ∂ : E → G, a group
morphism and ▷ a left group action of G on E by automorphism. Let X and Y
be finite groups on which G acts by automorphisms, and let f : Y → X be a G-equivariant
group morphism i.e g ∈ G and y ∈ Y , then f(g▷y) = g▷f(y). The goal
of this talk is to prove that ( FC(X) ⊗ CE ∂ −→ FC(Y ) ⋊ CG,▷ ) is a crossed module
of semisimple Hopf algebra where:

FC(X) and FC(Y ) are algebras of complex continuous functions on the finite
groups X and Y respectively
CE and CG are group algebras
⋊ is a semi-direct product.

To achieve this goal, we firstly prove that (FC(X) ⊗ CE ∂ −→ FC(Y ) ⋊ CG,▷) is a
crossed module Hopf algebra. Secondly, we prove that FC(Y )⋊CG and FC(X)⊗CE
are semisimple Hopf algebra.

Keywords : Hopf module; Hopf algebra; Haar integral; Semisimplicity and Cosemisimplicity;
Crossed module Hopf algebra; semi-direct product Hopf algebra

References
[1] David E. Radford. Hopf algebras., volume 49 of Ser. Knots Everything. Hackensack,
NJ: World Scientific, 2012.
[2] F. Quinn. Lectures on axiomatic topological quantum field theory. In D. Freed and
K. Uhlenbeck, editors, Geometry and Quantum Field Theory, volume 1 of IAS/Park
City Mathematics Series. AMS/IAS, 1995