On monomial algebras with representation-finite enveloping algebras

Abstract

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 $A\cong A_n/{\mathrm{rad}}^2{A}_n$, where $A_n$ is the path algebra $\mathrm{l}\mathrm{k}\mathcal{Q}$ with $\mathcal{Q}=1⟶2⟶\cdots ⟶n$. Moreover, we show that the number of all isoclasses of indecomposable $(A_n/{\mathrm{rad}}^2{A}_n{)}^e$-modules is $\frac{4}{3}{n}^3+n^2-\frac{7}{3}n+1$, and classify all indecomposable modules over $(A_n/{\mathrm{rad}}^2{A}_n{)}^e$. Finally, the Clebsch-Gordon problem over $(A_n/{\mathrm{rad}}^2{A}_n{)}^e$ is studied.

arXiv commentVery sorry, we have received an email from Mazorchuk and found that the results of our paper are consistent with the conclusion of Mazorchuk's paper "Bimodules over uniformly oriented An quivers with radial square zero" (doi: 10.1215/21562261-2019-0052). Therefore, we plan to make a decision to withdraw the manuscript. We apologize again