Rigid modules and Schur roots

Abstract

Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $Ω$. In previous work we associated an algebra $H$ to this data, such that the locally free $H$-modules behave in many aspects like representations of a hereditary algebra $\tilde{H}$ of the corresponding type. We define a Noetherian algebra $\hat{H}$ over a power series ring, which provides a direct link between the representation theory of $H$ and of $\tilde{H}$. We define and study a reduction and a localization functor relating the module categories of these three types of algebras. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over $H$, $\hat{H}$ and $\tilde{H}$. We show that the indecomposable rigid locally free modules over $H$ and $\hat{H}$ are parametrized, via their rank vector, by the real Schur roots associated to $(C,Ω)$. Moreover, the left finite bricks of $H$, in the sense of Asai, are parametrized, via their dimension vector, by the real Schur roots associated to the dual datum $(C^T,Ω)$.