The coherent KLR sheaf

Abstract

We construct the KLR algebra (in type $A$) as affine sections of a coherent sheaf $\mathcal{E}$ on a product of projective spaces ${\mathbb{P}}^n$. It was shown by Elias-Qi that the Lie algebra ${\mathfrak{sl}}_2$ acts on such KLR algebras; our construction geometrically explains this action as stemming from the ${\mathfrak{sl}}_2$-action on the twisting sheaves $\mathcal{O}(k)$ of ${\mathbb{P}}^1$ and gives the maximal finite-dimensional ${\mathfrak{sl}}_2$-submodule of KLR as global sections of $\mathcal{E}$. We furthermore extend the Elias-Qi action to a Witt action. Kleshchev-Loubert-Miemietz showed that the KLR of type $A$ has a graded affine cellular structure; our construction also gives a geometric incarnation of this.

arXiv commentSubmitted shortly after `White Dew'