Quiver varieties and crystals in symmetrizable type via modulated graphs

TL;DR

This paper extends the geometric construction of crystal bases from symmetric to symmetrizable Kac-Moody algebras by generalizing Lusztig's quiver varieties using Dlab and Ringel's algebra, accommodating non-symmetric types.

Contribution

It introduces a new framework for constructing crystals in symmetrizable types using modulated graphs, broadening the scope beyond symmetric cases.

Findings

Generalization of Lusztig's quiver varieties to symmetrizable types

Construction of crystals via irreducible components of these varieties

Handling non-algebraically-closed fields in non-symmetric types

Abstract

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztig's preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.