Quantum Lévy processes on dual groups

Abstract

We present a theory of quantum (non-commutative) Levy processes on dual groups which generalizes the theory of Levy processes on bialgebras. It follows from a result of N. Muraki that there exist exactly 5 notions of non-commutative ‘positive’ stochastic independence. We show that one can associate a commutative bialgebra with each pair consisting of a dual group and one of the 5 notions of independence. This construction is related to a construction of U. Franz. Our construction has the advantage that the important case of free independence is included. We show that Levy processes are given by their generators which are precisely the conditonally positive linear functionals on the dual group.