Weak Hopf Algebras, Smash Products and Applications to Adjoint-Stable Algebras

Abstract

For a semisimple quasi-triangular Hopf algebra $\left(H,R\right)$ over a field $k$ of characteristic zero, and a strongly separable quantum commutative $H$-module algebra $A$ over which the Drinfeld element of $H$ acts trivially, we show that $A\#H$ is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra $\mathrm{End}{A}^{\ast }\otimes H$. With these structure, ${}_{A\#H}\mathrm{Mod}$ is the monoidal category introduced by Cohen and Westreich, and ${}_{\mathrm{End}{A}^{\ast }\otimes H}\mathcal{M}$ is tensor equivalent to ${}_H\mathcal{M}$. If $A$ is in the M{ü}ger center of ${}_H\mathcal{M}$, then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter-Drinfeld modules for a finite group algebra.