arXiv subjects
Publications and source records attributed to David J. Steinberg.
Khovanov-Lauda-Rouquier algebras $R_θ$ of finite Lie type are affine quasihereditary with standard modules $Δ(π)$ labeled by Kostant partitions $π$ of $θ$. In type $A$, we construct explicit projective resolutions of standard modules $Δ(π)$.
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.
Khovanov-Lauda-Rouquier algebras of finite Lie type come with families of standard modules, which under the Khovanov-Lauda-Rouquier categorification correspond to PBW-bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and all non-zero endomorphisms of a standard module are injective. We obtain applications to extensions between standard modules and modular representation theory of KLR algebras.