In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:BMinn(C,H) ≅ B(C) ⊕ H2(H,K)In particular, if C is the symmetric closed category of C-modules with K a...

We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$, that we call type of the graph. We prove that for $p\gg k$ the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

Let $kG$ be a group algebra, and $D\left(kG\right)$ its quantum double. We first prove that the structure of the Grothendieck ring of $D\left(kG\right)$ can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of $G$. As a special case, we then give an application to the group algebra $k{D}_n$, where $k$ is a field of characteristic $2$ and $D_n$ is a dihedral group of order $2n$.

Nous considérons deux algèbres de Hopf graduées connexes en interaction, l’une étant un comodule-cogèbre sur l’autre. Nous montrons comment définir l’analogue du groupe de renormalisation et de la fonction Bêta de Connes-Kreimer lorsque la bidérivation de graduation est remplacée par une bidérivation provenant d’un caractère infinitésimal de la deuxième algèbre de Hopf.