On monomial algebras with representation-finite enveloping algebras

TL;DR

This paper characterizes when the enveloping algebra of a monomial algebra is representation-finite, classifies indecomposable modules in this case, and explores related module theory problems.

Contribution

It provides a complete characterization of monomial algebras with representation-finite enveloping algebras and classifies their indecomposable modules.

Findings

A characterization of monomial algebras with finite representation type of their enveloping algebra.

Explicit formula for the number of indecomposable modules over the enveloping algebra.

Classification of all indecomposable modules over the specific algebra $(A_n/rad^2A_n)^e$.

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 $A \cong \pmb{A}_n/\mathrm{rad}^2 \pmb{A}_n$, 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.