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

TL;DR

This paper explores how spherical twist functors in the root category of preprojective algebras relate to Weyl groups and Lie algebras, revealing new symmetries and conjectural algebraic connections.

Contribution

It introduces spherical twist functors for the root category of preprojective algebras and links Weyl groups to automorphisms of the Ringel-Hall Lie algebra, proposing a new algebraic relationship.

Findings

Weyl group realized as subquotient of automorphism group

Spherical twist functors induce symmetries in the Lie algebra

Conjectural relations between Lie subalgebras of different root categories

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 $Q$.