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.