On representation-finite gendo-symmetric biserial algebras

TL;DR

This paper explores the structure and classification of representation-finite gendo-symmetric biserial algebras, revealing their connection to Brauer tree algebras and their properties under derived equivalences.

Contribution

It classifies these algebras using Brauer tree combinatorics and studies their homological dimensions and derived equivalences.

Findings

Associated symmetric algebras are Brauer tree algebras.

Classified algebras up to almost ν-stable derived equivalence.

Found that these algebras are always Iwanaga-Gorenstein.

Abstract

Gendo-symmetric algebras were introduced by Fang and Koenig as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost $\nu$-stable derived equivalences, introduced by Hu and Xi, between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost $\nu$-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.