Tilting modules and support $\tau$-tilting modules over preprojective algebras associated with symmetrizable Cartan matrices

TL;DR

This paper explores tilting and support τ-tilting modules over generalized preprojective algebras linked to symmetrizable Cartan matrices, establishing bijections with Weyl groups that extend classical classifications.

Contribution

It introduces new bijections between tilting modules and Weyl groups for generalized preprojective algebras, broadening the scope of classical results.

Findings

Bijection between cofinite tilting ideals and Weyl group for non-Dynkin types.

Bijection between support τ-tilting modules and Weyl group for Dynkin types.

Generalizes classical classification results to broader algebraic structures.

Abstract

For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Gei\ss~ et al. (2016) introduced a generalized preprojective algebra $\Pi(C, D)$. We study tilting modules and support $\tau$-tilting modules for the generalized preprojective algebra $\Pi(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $\Pi(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $\tau$-tilting $\Pi(C,D)$-modules and the corresponding Weyl group $W(C)$. These results generalize the classification results of Buan et al. (Compos. Math. 145(4), 1035-1079, 2009) and Mizuno (Math. Zeit. 277(3), 665-690, 2014) over classical preprojective algebras.