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 ${\frak g}_Q$.

中文翻译：

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变异在加权射影线的派生类别中在李代数和量子代数中的应用

令 $\rm{coh}\mathbb{X}$ 是加权投影线 $\mathbb{X}$ 上的相干滑轮的范畴，令 $D^​​b(\rm{coh}\mathbb{X})$是它的有界派生范畴。本论文的重点是研究在 $D^b(\rm{coh}\mathbb{X})$ 附加到某些线束中出现的左右变异函子。作为应用，我们首先证明这些突变函子对与 $\mathbb{X}$ 相关的星形箭袋 $Q$ 的 Weyl 群产生了简单的反射。通过进一步处理 $\mathbb{X}$ 的 Ringel--Hall 代数，我们表明这些函子为 Kac--Moody 代数 $\frak{g}_Q$ 的 Tits 自同构提供了实现$，以及 ${\frak g}_Q$ 的量子包络代数的 Lusztig 对称性。