Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data (Q2856120)

The Khovanov-Lauda-Rouquier algebras (or quiver Hecke algebras) were introduced independently by \textit{M. Khovanov} and \textit{A. D. Lauda} [Represent. Theory 13, 309--347 (2009; Zbl 1188.81117)] and [\textit{R. Rouquier}, ``2-Kac-Moody algebras'', preprint, \url{arXiv:0812.5023}] to construct a categorification of quantum groups associated with symmetrizable Cartan data. For a dominant integral weight \(\lambda\in P^+\), Khovanov and Lauda conjectured that the cyclotomic quotient \(R^\lambda\) of the KLR algebra \(R\) gives a categorification of the irreducible highest weight module \(V(\lambda)\), which was proved recently by \textit{S.-J. Kang} and \textit{M. Kashiwara} [Invent. Math. 190, No. 3, 699--742 (2012; Zbl 1280.17017)].NEWLINENEWLINEWhen the Cartan datum is symmetric, \textit{M. Varagnolo} and \textit{E. Vasserot} [J. Reine Angew. Math. 659, 67--100 (2011; Zbl 1229.17019)] and \textit{R. Rouquier} [Algebra Colloq. 19, No. 2, 359--410 (2012; Zbl 1247.20002)] gave a geometric realization of KLR algebra via quiver varieties and proved that the isomorphism classes of projective indecomposable modules corresponds to Kashiwara's lower global basis (or Lusztig's canonical basis).NEWLINENEWLINELater, \textit{S.-J. Kang}, \textit{S.-J. Oh} and \textit{E. Park} [Int. J. Math. 23, No. 11, Paper No. 1250116, 51 p. (2012; Zbl 1283.17014)] introduced a family of KLR algebra \(R\) associated with symmetrizable Borcherds-Cartan data and showed that they provide a categorification of quantum generalized Kac-Moody algebras and their crystals.NEWLINENEWLINEIn this paper, the authors follow the framework of the work of Varagnolo-Vasserot, they construct a geometric realization of KLR algebras associated with symmetric Borcherds-Cartan data via quivers possibly with loops. One of the main ingredients is Steinberg-type varieties arising from quivers. As an application, when \(a_{ii}\not= 0\) for any \(i\in I\), the authors prove that there exists a one-to-one correspondence between Kashiwara's lower basis (or Lusztig's canonical basis) of \(U^{-}(\mathfrak{g})\) (resp., \(V(\lambda)\)) and the set of isomorphism classes of indecomposable projective graded modules mover \(R\) (resp. the cyclotomic quotient \(R^\lambda\)).