A generic classification of locally free representations of affine GLS algebras

Abstract

Throughout, let $K$ be an algebraically closed field of characteristic $0$. We provide a generic classification of locally free representations of Geiss-Leclerc-Schr\"oer's algebras $H_K(C,D,\Omega)$ associated to affine Cartan matrices $C$ with minimal symmetrizer $D$ and acyclic orientation $\Omega$. Affine GLS algebras are "smooth" degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several "tame" phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type $\tilde{BC}_1$, we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a $1$-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular $\tau$-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free $H$-module is isomorphic to a direct sum of $\tau$-rigid modules and modules from our $1$-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types, and we obtain such a decomposition by "folding" the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are $E$-tame in the sense of Derksen-Fei and Asai-Iyama.