Applications of spherical twist functors to Lie algebras associated to root categories of preprojective algebras

Abstract

Let ${\Lambda }_Q$ be the preprojective algebra of a finite acyclic quiver $Q$ of non-Dynkin type and $D^b({\mathrm{rep}}^n{\Lambda }_Q)$ be the bounded derived category of finite dimensional nilpotent ${\Lambda }_Q$-modules. We define spherical twist functors over the root category ${\mathcal{R}}_{{\Lambda }_Q}$ of $D^b({\mathrm{rep}}^n{\Lambda }_Q)$ and then realize the Weyl group associated to $Q$ as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra $\mathfrak{g}({\mathcal{R}}_{{\Lambda }_Q})$ of ${\mathcal{R}}_{{\Lambda }_Q}$ induced by spherical twist functors. We also present a conjectural relation between certain Lie subalgebras of $\mathfrak{g}({\mathcal{R}}_{{\Lambda }_Q})$ and $\mathfrak{g}({\mathcal{R}}_Q)$, where $\mathfrak{g}({\mathcal{R}}_Q)$ is the Ringe-Hall Lie algebra associated to the root category ${\mathcal{R}}_Q$ of $Q$.

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