Categorical representations, KLR algebras and Koszul duality

Abstract

The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of ${\widetilde{\mathfrak{sl}}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of components of the functor $F$ for the category $\mathbf{A}$ at level $-N-e-1$. To prove this, we use the approach of categorical representations. We prove a general fact about categorical representations: a category with an action of ${\widetilde{\mathfrak{s}l}}_{e+1}$ contains a subcategory with an action of ${\widetilde{\mathfrak{s}l}}_e$. To prove this claim, we construct an isomorphism between the KLR algebra associated with the quiver $A_{e-1}^{(1)}$ and a subquotient of the KLR algebra associated with the quiver $A_e^{(1)}$.

This preprint is split into two papers: arXiv:1901.11026 (published in Algebra & Number Theory) and arXiv:2007.11267. However, the original preprint arXiv:1512.04878v1 may be interesting for the reader because it shows more explicitly how the second part uses the first part

References in corpus (1)

• 2-Kac-Moody algebras

Cited by in corpus (1)

• Categorical representations and KLR algebras