  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.

Representation Theory