Quiver Hecke superalgebras

Abstract

Abstract We introduce a new family of superalgebras which should be considered as a super version of the Khovanov–Lauda–Rouquier algebras. Let I be the set of vertices of a Dynkin diagram with a decomposition I = I even ⊔ I odd $I=I_{{\mathrm {even}}}\sqcup I_{{\mathrm {odd}}}$ . To this data, we associate a family of graded superalgebras R n , the quiver Hecke superalgebras. When I odd = ∅ $I_{{\mathrm {odd}}}=\emptyset $ , these algebras are nothing but the usual Khovanov–Lauda–Rouquier algebras. We then define another family of graded superalgebras RC n , the quiver Hecke–Clifford superalgebras, and show that the superalgebras R n and RC n are weakly Morita superequivalent to each other. Moreover, we prove that the affine Hecke–Clifford superalgebras, as well as their degenerate version, the affine Sergeev superalgebras, are isomorphic to quiver Hecke–Clifford superalgebras RC n after a completion.