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

Let $\mathrm{c}\mathrm{o}\mathrm{h}\mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\mathrm{c}\mathrm{o}\mathrm{h}\mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\mathrm{c}\mathrm{o}\mathrm{h}\mathbb{X})$ 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 ${\frak g}_Q$.