Rigid modules and Schur roots

Abstract

Let C be a symmetrizable generalized Cartan matrix with symmetrizer D and orientation $\Omega$. In Geiß et al. (Invent Math 209(1):61–158, 2017) we constructed for any field $\mathbb {F}$ an $\mathbb {F}$-algebra $H := H_\mathbb {F}(C,D,\Omega )$, defined in terms of a quiver with relations, such that the locally free H-modules behave in many aspects like representations of a hereditary algebra ${\widetilde{H}}$ of the corresponding type. We define a Noetherian algebra ${\widehat{H}}$ over a power series ring, which provides a direct link between the representation theory of H and of ${\widetilde{H}}$. We define and study a reduction and a localization functor relating the module categories of ${\widehat{H}}$, ${\widetilde{H}}$ and H. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over the three algebras ${\widehat{H}}$, ${\widetilde{H}}$ and H. We show that the indecomposable rigid locally free H-modules are parametrized, via their rank vectors, by the real Schur roots associated to $(C,\Omega )$. Moreover, the left finite bricks of H, in the sense of Asai, are parametrized, via their dimension vectors, by the real Schur roots associated to $(C^T,\Omega )$.