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 $A^e=A{\otimes }_{\mathrm{l}\text{\hspace{0.17em}⁣}\mathrm{k}}{A}^{\mathrm{o}\mathrm{p}}$ its enveloping algebra. It is shown that $A^e$ is representation-finite if and only if $A\cong A_n\mathrm{/}{\mathrm{r}\mathrm{a}\mathrm{d}}^2{A}_n$, where $A_n$ is the path algebra $\mathrm{l}\text{\hspace{0.17em}⁣}\mathrm{k}\mathcal{Q}$ with $\mathcal{Q}=1⟶2⟶\cdots ⟶n$. Moreover, we show that the number of all isoclasses of indecomposable $(A_n\mathrm{/}{\mathrm{r}\mathrm{a}\mathrm{d}}^2{A}_n{)}^e$-modules is $\frac{4}{3}{n}^3+n^2-\frac{7}{3}n+1$, and classify all indecomposable modules over $(A_n\mathrm{/}{\mathrm{r}\mathrm{a}\mathrm{d}}^2{A}_n{)}^e$. Finally, the Clebsch-Gordon problem over $(A_n\mathrm{/}{\mathrm{r}\mathrm{a}\mathrm{d}}^2{A}_n{)}^e$ is studied.