Quiver Hecke superalgebras

TL;DR

This paper introduces quiver Hecke superalgebras, a super version of Khovanov-Lauda-Rouquier algebras, and explores their relationships with affine Hecke-Clifford superalgebras, expanding the algebraic framework for superalgebra representations.

Contribution

The paper defines quiver Hecke superalgebras and establishes their weak Morita superequivalence with quiver Hecke-Clifford superalgebras, linking them to affine Hecke-Clifford superalgebras.

Findings

Quiver Hecke superalgebras generalize Khovanov-Lauda-Rouquier algebras to the super setting.

Quiver Hecke-Clifford superalgebras are weakly Morita superequivalent to quiver Hecke superalgebras.

Affine Hecke-Clifford superalgebras are isomorphic to quiver Hecke-Clifford superalgebras after completion.

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 parity. To this data, we associate a family of graded superalgebras, the quiver Hecke superalgebras. When there are no odd vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras, the quiver Hecke-Clifford superalgebras, and show that they are weakly Morita superequivalent to the quiver Hecke superalgebras. Moreover, we prove that the affine Hecke-Clifford superalgebras, as well as their degenerate version, the affine Sergeev superalgebras, are isomorphic to the quiver Hecke-Clifford superalgebras after a completion.