Universal deformation rings of a special class of modules over generalized Brauer tree algebras

Let $\mathbb{k}$ be an algebraically closed field and $\mathrm{\Lambda }$ a generalized Brauer tree algebra over $\mathbb{k}$. We compute the universal deformation rings of the periodic string modules over $\mathrm{\Lambda }$. Moreover, for a specific class of generalized Brauer tree algebras $\mathrm{\Lambda }(n,\bar{m})$, we classify the universal deformation rings of the modules lying in $\mathrm{\Omega }$-stable components $\mathfrak{C}$ of the stable Auslander-Reiten quiver provided that $\mathfrak{C}$ contains at least one simple module. Our approach uses several tools and techniques from the representation theory of Brauer graph algebras. Notably, we leverage Duffield’s work on the Auslander-Reiten theory of these algebras and Opper-Zvonareva’s results on derived equivalences between Brauer graph algebras.