On representation-finite gendo-symmetric algebras with only one non-injective projective module

TL;DR

This paper classifies a specific class of representation-finite gendo-symmetric algebras with at most one non-injective projective module and explores their derived equivalence classes.

Contribution

It provides a complete classification of these algebras and identifies their almost ν-stable derived equivalence classes, extending previous work on gendo-symmetric algebras.

Findings

Classification of the algebras as quotients of symmetric algebras by socles.

Identification of their derived equivalence classes.

Connection to representation-finite symmetric algebras.

Abstract

Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig addressed some behaviour of the endomorphism algebra of a generator over a symmetric algebra, which they called gendo-symmetric algebra. Continuing this line of works, we classify in this article the representation-finite gendo-symmetric algebras that have at most one isomorphism class of indecomposable non-injective projective module. We also determine their almost {\nu}-stable derived equivalence classes in the sense of Wei Hu and Changchang Xi. It turns out that a representative can be chosen as the quotient of a representation-finite symmetric algebra by the socle of a certain indecomposable projective module.