On universal deformation rings of modules over a certain class of symmetric algebras of finite representation type

TL;DR

This paper classifies certain modules over symmetric algebras of finite representation type, identifies those with stable endomorphism ring isomorphic to the base field, and computes their universal deformation rings.

Contribution

It determines indecomposable modules with stable endomorphism ring isomorphic to the base field and calculates their universal deformation rings for a specific class of symmetric algebras.

Findings

Identified modules with stable endomorphism ring isomorphic to ieldield.

Computed universal deformation rings for these modules.

Extended understanding of deformation theory in symmetric algebras of finite representation type.

Abstract

Let $\mathbf{k}$ be an algebraically closed field. Recently, K. Erdmann classified the symmetric $\mathbf{k}$-algebras $\Lambda$ of finite representation type such that every non-projective module $M$ has period dividing four. The goal of this paper is to determine the indecomposable modules $M$ over these class of algebras $\Lambda$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$, and then calculate their corresponding universal deformation rings (in the sense of F. M. Bleher and the third author).