Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we will call nilpotent. First we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as standard example. We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that the sum and diierence of independent random variables are also independent. The corresponding functionals on the braided line, braided plane and a braided q-Heisenberg-Weyl group are determined. Section 5 deals with continuous convolution semigroups on nilpotent quantum groups and braided groups. We extend recent results proving the uniqueness of the embedding of an innnitely divisible probability law into a continuous convolution semigroup for simply connected nilpotent Lie groups to nilpotent quantum groups and braided groups. Finally , in Section 6 we give some indications how the semigroup approach of Heyer and Hazod to the Bernstein theorem on groups can be extended to quantum groups and braided groups.