modules and Projective Modules over $H(\lambda, \mu)$ {#3}
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Throughout this section, assume $p>2$. Let $n=p^st$ with $p\nmid t$ and $s\geqslant 1$. Let $\xi$ be a $t$-$th$ primitive root of unity in $k$. Let $\lambda, \mu\in k$. We will investigate simple modules and projective modules over $H(\lambda, \mu)$ in this section. Note that $kG$ is the coradical of $H(\lambda, \mu)$.

Since $p|n$, we know that $kG$ is not semisimple. It has $t$ non-isomorphic simple modules, which are all 1-dimensional and given by the corresponding algebra homomorphisms $\rho_i: kG\rightarrow k$, $\rho_i(g)=\xi^i$, $0\leqslant i\leqslant t-1$. Moreover, $kG$ has $n$ non-isomorphic indecomposable modules, which can be described by the matrix representations as follows: $$\rho_{r,i}(g)=\left(\begin{array}{ccccc}
\xi^i&1&\cdots&0&0\\
0&\xi^i&\cdots&0&0\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
0&0&\cdots&\xi^i&1\\
0&0&\cdots&0&\xi^i
\end{array}\right)_{r\times r}$$ where $1\leqslant r\leqslant p^s$ and $0\leqslant i\leqslant t-1$ (see [@Don09]).

\[3.1\]If $t=1$, there is only one simple module $S$ over $H(\lambda,\mu)$, which is $1$-dimensional and given by $g\cdot v=v$, $a\cdot v=0$ and $b\cdot v=0$ for all $v\in S$. In particular, $H(\lambda,\mu)$ is a local algebra in this case.

Assume $t=1$. Then by Lemma \[2.4\](1), we know that $a,b\in J$, the Jacobson radical of $H(\lambda, \mu)$, and $H(\lambda,\mu)/\langle a, b\rangle\cong kG$, where $\langle a,
b\rangle$ is the ideal of $H(\lambda, \mu)$ generated by 