Lévy Processes on $U_q(g)$ as Infinitely Divisible Representations

13 pages, LATEX file, ASI-TPA/13/99 (TU Clausthal); 6/99 (Preprint-Reihe Mathmatik, Univ. Greifswald);Lévy processes on bialgebras are families of infinitely divisible representations. We classify the generators of Lévy processes on the compact forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then we show how the processes themselves can be reconstructed from their generators and study several classical stochastic processes that can be associated to these processes