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Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups
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  • Published: September 1997

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

  • U. Franz1,
  • D. Neuenschwander3 &
  • R. Schott1 

Probability Theory and Related Fields volume 109, pages 101–127 (1997)Cite this article

Summary.

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 difference 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 infinitely 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.

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Authors and Affiliations

  1. Institut Elie Cartan and CRIN-CNRS, BP 239, Université H. Poincaré-Nancy I, F-54506 Vandoeuvre-lès-Nancy, France, , , , , , FR

    U. Franz & R. Schott

  2. Université de Lausanne, Ecole des Hautes Etudes Commerciales, Institut de Sciences Actuarielles, CH-1015 Lausanne, Switzerland, , , , , , CH

    D. Neuenschwander

Authors
  1. U. Franz
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  2. D. Neuenschwander
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  3. R. Schott
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Received: 30 October 1996 / In revised form: 1 April 1997

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Franz, U., Neuenschwander, D. & Schott, R. Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups. Probab Theory Relat Fields 109, 101–127 (1997). https://doi.org/10.1007/s004400050127

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  • Issue Date: September 1997

  • DOI: https://doi.org/10.1007/s004400050127

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  • Mathematics Subject Classification (1991): 60B99
  • 16W30
  • 81R50
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