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

TL;DR

The paper computes universal deformation rings for periodic string modules over generalized Brauer tree algebras by exploiting derived equivalences to star graphs, yielding explicit classifications in terms of the tube distance $d_M$ and vertex multiplicities. It extends the results to Ω-stable components containing simple modules in non-polynomial growth cases, providing complete deformation-ring data that is invariant under stable Morita-type equivalences. An application to standard Koszul symmetric special biserial algebras shows that periods in the Koszul setting yield deformation rings $\Bbbk$ or $\Bbbk[[x]]$ depending on the boundary distance, with explicit representatives. The methods combine Krause’s canonical maps for string modules, Duffield’s AR-quiver analysis, and Opper–Zvonareva’s derived equivalences to reduce complex Brauer-graph situations to tractable star configurations, thereby enriching deformation-theoretic invariants for Brauer graph algebras.

Abstract

Let $\Bbbk$ be an algebraically closed field and $Λ$ a generalized Brauer tree algebra over $\Bbbk$. We compute the universal deformation rings of the periodic string modules over $Λ$. Moreover, for a specific class of generalized Brauer tree algebras $Λ(n,\overline{m})$, we classify the universal deformation rings of the modules lying in $Ω$-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.

Figures (6)

• Figure 1: The star $\mathcal{W}_{n,\overline{m}}$ and its associated quiver $Q$
• Figure 2: Radical series of projective indecomposable modules in $\mathcal{W}_{n,\overline{m}}$.
• Figure 3: The star $\mathcal{W}_{n,\overline{m}}$ associated to $\Lambda_{\lambda}$
• Figure 4: The paths from conditions $i)$ and $ii)$, respectively.
• Figure 5: Component $\mathfrak{C}_{S(t)}$

Theorems & Definitions (86)

• Theorem 2.1: rv
• Theorem 2.2: bv1
• Definition 2.1: syb
• Definition 2.2: sybbr
• Definition 2.3: syb,oz
• Definition 2.4: syboz
• Definition 2.5: syb
• Theorem 2.3: sib
• Definition 2.6: syb
• Definition 2.7: oz