# Quivers with relations for symmetrizable Cartan matrices IV: Crystal graphs and semicanonical functions

## TL;DR

This paper extends the geometric construction of crystal graphs and semicanonical functions from symmetric to symmetrizable Cartan matrices, aiming to deepen understanding of Kac-Moody algebra representations.

## Contribution

It generalizes Lusztig's nilpotent varieties and constructs semicanonical functions for symmetrizable cases, proposing new bases for Kac-Moody algebra enveloping algebras.

## Key findings

- Generalization of crystal graph construction to symmetrizable case
- Construction of semicanonical functions in generalized preprojective algebras
- Conjecture of these functions forming semicanonical bases

## Abstract

We generalize Lusztig's nilpotent varieties, and Kashiwara and Saito's geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac-Moody algebras.