  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 ${\frak
g}_Q$.
