Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Abstract

Let $k$ be an arbitrary field, $\Lambda$ be a $k$-algebra and $V$ be a $\Lambda$-module. When it exists, the universal deformation ring $R(\Lambda,V)$ of $V$ is a $k$-algebra whose local homomorphisms to $R$ parametrize the lifts of $V$ up to $R\otimes_k \Lambda$, where $R$ is any complete, local commutative Noetherian $k$-algebra with residue field $k$. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type $p$-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let $\Lambda$ be any 1-domestic, symmetric special biserial algebra. By viewing $\Lambda$ as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those $\Lambda$-modules $V$ with stable endomorphism ring isomorphic to $k$. The latter is a natural condition, since it guarantees the existence of the universal deformation ring $R(\Lambda,V)$.