Unfurling Khovanov-Lauda-Rouquier algebras

Abstract

In this paper, we study the behavior of categorical actions of a Lie algebra $\mathfrak{g}$ under the deformation of their spectra. We give conditions under which the general point of a family of categorical actions of $\mathfrak{g}$ carry an action of a larger Lie algebra $\mathfrak{\tilde{g}}$, which we call an {\bf unfurling} of $\mathfrak{g}$. This is closely related to the folding of Dynkin diagrams, but to avoid confusion, we think it is better to use a different term. Our motivation for studying this topic is the difficulty of proving that explicitly presented algebras and categories in the theory of higher representation theory have the "expected size." Deformation is a powerful technique for showing this because of the upper semicontinuity of dimension under deformation. In particular, we'll use this to show the non-degeneracy (in the sense of Khovanov-Lauda) of the 2-quantum group $\mathcal U$ for an arbitrary Cartan datum and any homogeneous choice of parameters.