On Representation-Finite Gendo-Symmetric Biserial Algebras

Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) 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 ν-stable derived equivalences, introduced in Hu and Xi (I. Nagoya Math. J., 200:107–152, 2010), between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-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.