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

Abstract

We construct a geometric realization of the Khovanov–Lauda–Rouquier algebra R associated with a symmetric Borcherds–Cartan matrix A = (aij)i, j∈I via quiver varieties. As an application, if aii ≠ 0 for any i ∈ I, we prove that there exists a one-to-one correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of U𝔸−(𝔤) (respectively, V𝔸(λ)) and the set of isomorphism classes of indecomposable projective graded modules over R (respectively, Rλ).