An odd categorification of $U_q({\mathfrak{sl}}_2)$

Abstract

We define a 2-category that categorifies the covering Kac–Moody algebra for ${\mathfrak{sl}}_2$ introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a $\mathbb{Z}\times {\mathbb{Z}}_2$-grading giving its Grothendieck group the structure of a free module over the group algebra of $\mathbb{Z}\times {\mathbb{Z}}_2$. By specializing the ${\mathbb{Z}}_2$-action to +1 or to −1, the construction specializes to an “odd” categorification of ${\mathfrak{sl}}_2$ and to a supercategorification of ${\mathfrak{osp}}_{1|2}$, respectively.