Homological description of crystal structures on Lusztig's quiver varieties

TL;DR

This paper establishes an explicit homological isomorphism between two different crystal base realizations for quantum groups associated with simply-laced Lie algebras, linking geometric and algebraic approaches.

Contribution

It provides a homological algebra method to explicitly connect geometric and algebraic crystal base constructions for quantum groups.

Findings

Explicit crystal isomorphism between geometric and algebraic realizations

Homological algebra techniques relate quiver varieties and Hall algebra representations

Deepens understanding of the structure of crystal bases in quantum groups

Abstract

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions are closely related by studying sufficiently generic representations of the preprojective algebra.