On endomorphism algebras of string almost gentle algebras

TL;DR

This paper investigates the relationship between the representation types of string almost gentle algebras, their associated endomorphism algebras, and Cohen-Macaulay Auslander algebras, revealing deep structural connections.

Contribution

It establishes the equivalence of representation types among various algebras derived from string gentle algebras, providing new insights into their structural properties.

Findings

Representation type of a string gentle algebra is equivalent to that of its $ ext{R}$-endomorphism algebras.

All $ ext{R}$-endomorphism algebras share the same representation type.

The Cohen-Macaulay Auslander algebra has the same representation type as the original algebra.

Abstract

For any arbitrary string almost gentle algebra, we consider specific subsets of its quiver's arrow set, denoted by $\mathcal{R}$. For each such $\mathcal{R}$, we introduce the finitely generated module $M_{\mathcal{R}}$ and define its associated $\mathcal{R}$-endomorphism algebra $A_{\mathcal{R}}$. In this paper, we show that the representation type of a string gentle algebra $A$, the representation type of the $\mathcal{R}$-endomorphism algebra $A_{\mathcal{R}}$ for some $\mathcal{R}$, the representation types of all $\mathcal{R}$-algebras, and the representation type of the Cohen-Macaulay Auslander algebra $A^{\mathrm{CMA}}$ of $A$ are equivalent. The results presented here reveal a deep structural connection between different classes of algebras derived from string gentle algebras. By showing the equivalence of representation types, this work offers new insights into the nature of endomorphism algebras and Cohen-Macaulay Auslander algebras, contributing to a broader understanding of their algebraic properties and classification.