Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data

TL;DR

This paper provides a geometric construction of Khovanov-Lauda-Rouquier algebras linked to Borcherds-Cartan data using quiver varieties, establishing a correspondence with canonical bases in quantum groups.

Contribution

It introduces a geometric realization of these algebras and proves a bijective correspondence with canonical bases under certain conditions.

Findings

Established a geometric model for KLR algebras with Borcherds-Cartan matrices.

Proved a one-to-one correspondence between indecomposable projective modules and canonical bases.

Applied the construction to relate algebraic and geometric structures in quantum groups.

Abstract

We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra $R$ associated with a symmetric Borcherds-Cartan matrix $A=(a_{ij})_{i,j\in I}$ via quiver varieties. As an application, if $a_{ii} \ne 0$ for any $i\in I$, we prove that there exists a 1-1 correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of $U_\A^-(\g)$ (resp.\ $V_\A(\lambda)$) and the set of isomorphism classes of indecomposable projective graded modules over $R$ (resp.\ $R^\lambda$).