An odd categorification of \(U_q(\mathfrak{sl}_2)\) (Q260212)

The present work continues the program of identification of categorifications of quantized enveloping algebras. That is, one aims to find a category whose Grothendieck group is isomorphic to the algebra one is interested in. This category can be obtained as a category of modules or sheaves, or presented diagrammatically.NEWLINENEWLINEThis work concerns a categorification \(\dot{\mathcal{U}}_{\pi}\) of an algebra \(U_{q,\pi}\) defined by \textit{S. Clark} et al. [Transform. Groups 18, No. 4, 1019--1053 (2013; Zbl 1359.17023)] and subsequent papers), which is a simultaneous generalisation of \(U_{q}(\mathfrak{sl}_{2})\) (via \(\pi\to 1\)) and \(U_{q}(\mathfrak{osp}_{1|2})\) (via \(\pi\to -1\)), the latter being the quantized enveloping superalgebra of an orthosymplectic Lie superalgebra. This categorification is a super-2-categorification, building on work of\textit{S.-J. Kang} et al. [Adv. Math. 242, 116--162 (2013; Zbl 1304.17012); Adv. Math. 265, 169--240 (2014; Zbl 1304.17012)]).NEWLINENEWLINEIn addition to a detailed construction and analysis of the categorification itself, the authors prove the more general result that any strong supercategorical action of \(\mathfrak{sl}_{2}\) on a category \(\mathcal{C}\) can be extended to a 2-functor \(\dot{\mathcal{U}}_{\pi} \to \mathcal{C}\), yielding a 2-representation.NEWLINENEWLINE(We note for those less familiar with this area that all of the terminology above and several important inputs from the literature are explained in detail, so that the present work is mostly self-contained.)