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

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 [15,16] 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 precisely one isomorphism class of indecomposable non-injective projective module. We also determine their almost ν-stable derived equivalence classes in the sense of Wei Hu and Changchang Xi [21]. 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.