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# Quantum Topology

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**Volume 7, Issue 2, 2016, pp. 329–433**

**DOI: 10.4171/QT/78**

Published online: 2016-02-08

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

Alexander P. Ellis^{[1]}and Aaron D. Lauda

^{[2]}(1) University of Oregon, Eugene, USA

(2) University of Southern California, Los Angeles, United States

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.

*Keywords: *Covering algebras, categorified quantum groups, cyclotomic quotients, odd nil-Hecke algebra, odd Khovanov homology

Ellis Alexander, Lauda Aaron: An odd categorification of $U_q (\mathfrak{sl}_2)$. *Quantum Topol.* 7 (2016), 329-433. doi: 10.4171/QT/78