GAUSS LAWS IN THE SENSE OF BERNSTEIN AND UNIQUENESS OF EMBEDDING INTOCONVOLUTION 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 nilpoten t. First we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as standard example. We introduce Gaussian function als on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that the sum and difference of independent random Variables are also independent. The correspondin g functionals on the braided line, braided plane and a braided q-Heise nberg-Weyl group are determined. Section 5 deals with continuous convo lution semigroups on nilpotent quantum groups and braided groups. We e xtend recent results proving the uniqueness of the embedding of an inf initely divisible probability law into st continuous convolution semig roup for simply connected nilpotent Lie groups to nilpotent quantum gr oups and braided groups. Finally, in Section 6 we give some indication s how the semigroup approach of Heyer and Hazod to the Bernstein theor em on groups can be extended to quantum groups and braided groups.