An odd categorification of quantum sl(2)

TL;DR

This paper introduces a super-2-category that categorifies the covering Kac-Moody algebra for sl(2), incorporating a (Z x Z_2)-grading and specializing to odd and supercategorifications of sl(2) and osp(1|2).

Contribution

It constructs a novel super-2-category framework that extends existing categorifications to include odd and supercategorifications of sl(2) and osp(1|2).

Findings

Defines a super-2-category for covering Kac-Moody algebra of sl(2).

Introduces a (Z x Z_2)-grading structure in the categorification.

Specializes to odd and supercategorifications of sl(2) and osp(1|2).

Abstract

We define a 2-category that categorifies the covering Kac-Moody algebra for 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 (Z x Z_2)-grading giving its Grothendieck group the structure of a free module over the group algebra of Z x Z_2. By specializing the Z_2-action to +1 or to -1, the construction specializes to an "odd" categorification of sl(2) and to a supercategorification of osp(1|2), respectively.