Quivers with relations for symmetrizable Cartan matrices I : Foundations

TL;DR

This paper introduces a new class of Iwanaga-Gorenstein algebras derived from quivers with relations linked to symmetrizable Cartan matrices, expanding the understanding of their representation theory and root systems.

Contribution

It generalizes path algebras for symmetric Cartan matrices to symmetrizable cases and defines associated generalized preprojective algebras, providing new algebraic realizations of root systems.

Findings

New class of Iwanaga-Gorenstein algebras introduced

Representation-theoretic realizations of all finite root systems achieved

Generalization of path algebras for symmetrizable Cartan matrices

Abstract

We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. Without any assumption on the ground field, we obtain new representation-theoretic realizations of all finite root systems.