Some extension algebras for standard modules over KLR algebras of type $A$

Abstract

Khovanov-Lauda-Rouquier algebras $R_{\theta }$ of finite Lie type are affine quasihereditary with standard modules $\Delta (\pi )$ labeled by Kostant partitions of $\theta$. Let $\Delta$ be the direct sum of all standard modules. It is known that the Yoneda algebra ${\mathcal{E}}_{\theta }:={\mathrm{Ext}}_{R_{\theta }}^{\ast }(\Delta ,\Delta )$ carries a structure of an $A_{\infty }$-algebra which can be used to reconstruct the category of standardly filtered $R_{\theta }$-modules. In this paper, we explicitly describe ${\mathcal{E}}_{\theta }$ in two special cases: (1) when $\theta$ is a positive root in type $\mathtt{A}$, and (2) when $\theta$ is of Lie type ${\mathtt{A}}_2$. In these cases, ${\mathcal{E}}_{\theta }$ turns out to be torsion free and intrinsically formal. We provide an example to show that the $A_{\infty }$-algebra ${\mathcal{E}}_{\theta }$ is non-formal in general.