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arXiv · 1906.11380

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

Doeke Buursma·Alexander Kleshchev·David J. Steinberg
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Abstract

Khovanov-Lauda-Rouquier algebras $R_θ$ of finite Lie type are affine quasihereditary with standard modules $Δ(π)$ labeled by Kostant partitions of $θ$. Let $Δ$ be the direct sum of all standard modules. It is known that the Yoneda algebra $\mathcal{E}_θ:=\operatorname{Ext}_{R_θ}^*(Δ, Δ)$ carries a structure of an $A_\infty$-algebra which can be used to reconstruct the category of standardly filtered $R_θ$-modules. In this paper, we explicitly describe $\mathcal{E}_θ$ in two special cases: (1) when $θ$ is a positive root in type $\mathtt{A}$, and (2) when $θ$ is of Lie type $\mathtt{A_2}$. In these cases, $\mathcal{E}_θ$ turns out to be torsion free and intrinsically formal. We provide an example to show that the $A_\infty$-algebra $\mathcal{E}_θ$ is non-formal in general.

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BibTeXRIS

Doeke Buursma, Alexander Kleshchev, David J. Steinberg. 2019-06-26. Some extension algebras for standard modules over KLR algebras of type $A$. https://arxiv.org/abs/1906.11380

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