On monomial algebras with representation-finite enveloping algebras

The present paper mainly considers the representation type of the enveloping algebra of monomial algebra. Let $A$ be a monomial algebra and $A^e= A\otimes_{\mathrm{l}\!\mathrm{k}} A^{\mathrm{op}}$ its enveloping algebra. It is shown that $A^e$ is representation-finite if and only if $A \cong \pmb{A}_n/\mathrm{rad}^2 \pmb{A}_n$, where $\pmb{A}_n$ is the path algebra $\mathrm{l}\!\mathrm{k}\mathcal{Q}$ with $\mathcal{Q} = 1 \longrightarrow 2 \longrightarrow \cdots \longrightarrow n$. Moreover, we show that the number of all isoclasses of indecomposable $(\pmb{A}_n/ \mathrm{rad}^2\pmb{A}_n)^e$-modules is $\frac{4}{3}n^3 + n^2-\frac{7}{3}n+1$, and classify all indecomposable modules over $(\pmb{A}_n/ \mathrm{rad}^2\pmb{A}_n)^e$. Finally, the Clebsch-Gordon problem over $(\pmb{A}_n/ \mathrm{rad}^2\pmb{A}_n)^e$ is studied.