GAUSS LAWS IN THE SENSE OF BERNSTEIN AND UNIQUENESS OF EMBEDDING INTOCONVOLUTION SEMIGROUPS ON QUANTUM GROUPS AND BRAIDED GROUPS

The aim of this note is to characterize certain probability laws on a class of quantum groups or braided groups that will be called nilpoten t. First, we introduce a braided analogue of the Heisenberg-Weyl group , which shall serve as a standard example. We determine functionals on the braided line and on this group satisfying an analogue of the Bern stein property (see [3]), i.e. that the sum and difference of independ ent Gaussian random variables are also independent. We also study cont inuous convolution semigroups on nilpotent quantum groups and braided groups. We extend to nilpotent quantum groups and braided groups recen t results proving the uniqueness of the embedding of an infinitely div isible probability law in a continuous convolution semigroup for simpl y connected nilpotent Lie groups.