Applications of Mutations in the Derived Categories of Weighted Projective Lines to Lie and Quantum Algebras

Abstract

Abstract Let $\textrm{coh}\ \mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\textrm{coh}\ \mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\textrm{coh}\ \mathbb{X})$ 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 ${\mathfrak g}_{Q}$ associated with ${Q}$, as well as for Lusztig’s symmetries of the quantum enveloping algebra of ${\mathfrak g}_{Q}$.