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

TL;DR

This paper explicitly describes the extension algebras of standard modules over KLR algebras of type A, showing they are torsion free and formal in specific cases, but not in general.

Contribution

It provides explicit descriptions of the Yoneda algebra for certain types, revealing conditions for formality and torsion freeness in these extension algebras.

Findings

Extension algebra is torsion free in special cases.

Extension algebra is intrinsically formal in these cases.

Counterexample shows non-formality in general.

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 $\mathcal{E}_\theta:=\operatorname{Ext}_{R_\theta}^*(\Delta, \Delta)$ 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.