The aim of this paper is to give an alternative proof of Kac’s theorem for weighted projective lines over the complex field. The geometric realization of complex Lie algebras arising from derived categories is essentially used.

A class of piecewise linear paths, as a generalization of Littelmann’s paths, are introduced, and some operators, acting on the above paths with fixed parametrization, are defined. These operators induce the ordinary Littelmann’s root operators’ action on the equi-alence classes of paths. With these induced operators, an explicit realization of B(∞) is given in terms of equivalence classes of paths, where B(∞) is the crystal base of the negative part of a quantum group U _q(g). Furthermore, we conjecture that there is a complete set of representatives for the above model by fixing a parametrization, and we prove the case when g is of finite type.