Universal deformation rings of strings modules over a certain symmetric special biserial algebra. (Q2256995)

Let \(k\) be an algebraically closed field. \textit{K. Erdmann} [in: Blocks of tame representation type and related algebras. Lect. Notes Math. 1428. Berlin: Springer-Verlag (1990; Zbl 0696.20001)] introduced \(k\)-algebras of dihedral, semidihedral and quaternion type to describe all possible Morita equivalence classes of blocks of group algebras of infinite tame representation type. In [\textit{F. M. Bleher} and \textit{J. A. Vélez-Marulanda}, J. Algebra 367, 176-202 (2012; Zbl 1301.16013)], the reviewer and the author of the paper under review showed that Mazur's deformation theory can be translated to arbitrary finite dimensional \(k\)-algebras, with some additional properties when the algebras are self-injective or Frobenius. Moreover, one particular algebra of dihedral type, which cannot be Morita equivalent to any block of a group ring, was investigated with respect to modules with stable endomorphism ring \(k\) and their universal deformation rings. This algebra is the smallest member of an infinite family of algebras of dihedral type. In the paper under review, the author studies the whole family of these algebras of dihedral type, which he denotes by \(\Lambda_{\underline r}\) where \(\underline r=(r_0,r_1,r_2,k)\) for \(k\geq 1\) and \(r_0,r_1,r_2\geq 2\). He first determines all \(\Lambda_{\underline r}\)-modules \(V\) whose endomorphism rings are isomorphic to \(k\). He then finds all modules that belong to the same component of the stable Auslander-Reiten quiver of \(\Lambda_{\underline r}\) as \(V\) and whose \textbf{stable} endomorphism rings are isomorphic to \(k\). He moreover determines the universal deformation rings of all of these modules.