# Rigid modules and Schur roots

## TL;DR

This paper explores the representation theory of algebras associated with symmetrizable Cartan matrices, establishing bijections between tilting modules and parametrizations of rigid modules via real Schur roots.

## Contribution

It introduces a Noetherian algebra over a power series ring linking the representation theories of different algebra types and classifies indecomposable rigid modules using real Schur roots.

## Key findings

- Bijections between tilting modules over different algebras.
- Parametrization of rigid modules by real Schur roots.
- Connection between module categories via reduction and localization functors.

## Abstract

Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $\Omega$. 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 $(C,\Omega)$. 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 $(C^T,\Omega)$.