# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.11380