Categorification of quantum Kac-Moody superalgebras

TL;DR

This paper develops a new categorification framework for quantum Kac-Moody superalgebras without isotropic odd roots, utilizing spin quiver Hecke algebras and a super sign as spin.

Contribution

It introduces a novel characterization of these superalgebras and demonstrates their categorification via spin quiver Hecke algebras, connecting super signs to spin.

Findings

Categorification of half the quantum Kac-Moody superalgebras achieved

New characterization of superalgebras using a bilinear form

Super sign categorified as spin (parity-shift functor)

Abstract

We introduce a non-degenerate bilinear form and use it to provide a new characterization of quantum Kac-Moody superalgebras with no isotropic odd simple roots. We show that the spin quiver Hecke algebras introduced by Kang-Kashiwara-Tsuchioka provide a categorification of half the quantum Kac-Moody superalgebras, using the recent work of Ellis-Khovanov-Lauda. A new idea here is that a super sign is categorified as spin (i.e., the parity-shift functor).