Faithfulness of simple 2-representations of ${\mathfrak{sl}}_2$

Abstract

Let $\mathcal{U}$ be the 2-category associated with ${\mathfrak{sl}}_2$. We prove that a complex of 1-morphisms of $\mathcal{U}$ is null-homotopic if and only if its image in every simple 2-representation is null-homotopic. Under mild boundedness assumptions, we prove that it actually suffices for the image in the simple 2-representations to be acyclic. We apply this result to the study of the Rickard complex $\Theta$ categorifying the action of the simple reflection of ${\mathrm{SL}}_2$. We prove that $\Theta$ is invertible in the homotopy category of $\U$, and that there is a homotopy equivalence $\Theta E\simeq F\Theta [-1]$.

24 pages

References in corpus (2)

• 2-Kac-Moody algebras
• A categorification of quantum sl(2)