${}_A\!A\cong
\bigoplus_{S\in\widehat{A}}P(S)^{{\rm dim} S}$ as left $A$-modules by Wedderburn-Artin theorem.

Let $H$ be a finite-dimensional Hopf algebra. A left integral in $H$ is an element $t\in H$ such that $ht=\varepsilon(h)t$ for all $h\in H$. A right integral in $H$ is an element $t'\in H$ such that $t'h=\varepsilon(h)t'$ for all $h\in H$. $\int_H^l$ denotes the space of left integrals, and $\int_H^r$ denotes the space of right integrals. $H$ is called unimodular if $\int_H^l=\int_H^r$. Note that $\int_H^l$ and $\int_H^r$ are each one-dimensional (see [@Mon93]).

A $k$-algebra $A$ is called symmetric if there exists a nondegenerate $k$-bilinear form $\beta:A\times A\rightarrow k$, which is associative and symmetric. A symmetric algebra $A$ is self-injective, that is, the left regular module $A$ is injective. A finite dimensional Hopf algebra $H$ is a symmetric algebra if and only if $H$ is unimodular and $S^2$ is inner, where $S$ is the antipode of $H$ [@Lorenz; @ObSchn].

Throughout this paper, we work over an algebraically closed field $k$ with a positive characteristic $p$. All algebras, Hopf algebras and modules are finite dimensional over $k$. Unless otherwise stated, all maps are $k$-linear, dim and $\otimes$ stand for dim$_k$ and $\otimes_k$, respectively.

The Hopf Algebras $\mathcal{B}(V)\# kG$ and $H(\lambda, \mu)$ {#2}
