Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Hase-Liu, Matthew, Zhou, Fan
Institutions: Massachusetts Institute of Technology, Columbia University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213223305