Bicrossed products with the Taft algebra (Q2424591)

Let $A$ and $B$ be two Hopf algebras over a field $K$. A bicrossed product of $A$ and $B$ is any Hopf algebra structure on $A\otimes B$, equal to $A\otimes B$ as a coalgebra, and such that $A\otimes K$ and $K\otimes B$ are subalgebras, trivially isomorphic to $A$ and $B$. When $A$ and $B$ are given, all bicrossed products of $A$ and $B$ can be described in terms of actions of $A$ over $B$ and of $B$ over $A$. When one of these actions is trivial, the bicrossed product is said to be a smash product. \par In this paper, bicrossed product of the Taft algebra $T_{m^2}(q)$ and of a group algebra $K[G]$ are studied, where $q$ is a $m$-th primitive root of unity and $G$ is a group which is generated by elements of finite order. It is proved that any bicrossed product between $T_{m^2}(q)$ and $K[G]$ is in fact a smash product. All bicrossed products are then given when $G$ belongs to a family of semidirect products of cyclic groups, including the dihedral groups. \par Generally, the classification of these smash products is strongly related to the automorphism group of $G$. This classification is done when $G$ is a dihedral group $D_{2n}$, leading to arithmetical considerations on $m$ and $n$.