Applications of mutations in the derived categories of weighted projective lines to Lie and quantum algebras

TL;DR

This paper explores how mutation functors in derived categories of weighted projective lines relate to Lie and quantum algebra symmetries, providing new geometric realizations of algebraic automorphisms.

Contribution

It demonstrates that mutation functors induce Weyl group reflections and realize Tits' automorphisms and Lusztig's symmetries in associated Lie and quantum algebras.

Findings

Mutation functors correspond to simple reflections in Weyl groups.

They realize Tits' automorphisms of Kac--Moody algebras.

They provide geometric realizations of Lusztig's symmetries.

Abstract

Let $\rm{coh}\mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\rm{coh}\mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\rm{coh}\mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver $Q$ associated with $\mathbb{X}$. By further dealing with the Ringel--Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits' automorphisms of the Kac--Moody algebra $\frak{g}_Q$ associated with $Q$, as well as for Lusztig's symmetries of the quantum enveloping algebra of ${\frak g}_Q$.