On Weyl Groups and Artin Groups Associated to Orbifold Projective Lines

We associate a generalized root system in the sense of Kyoji Saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra. We generalize results by Saito--Takebayshi and Yamada for elliptic Weyl groups and elliptic Artin groups to the Weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems. Moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of $2$-Calabi--Yau completion of the bound quiver algebra.