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

Identifieur interne : 00BC48 ( Main/Curation ); précédent : 00BC47; suivant : 00BC49

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

Auteurs : U. Franz ; D. Neuenschwander [Suisse] ; R. Schott

Source :

Probability Theory and Related Fields [ 0178-8051 ] ; 1997-09-01.

RBID : ISTEX:3A6DAD0E0D285425C7075B37937DD3129C5488EF

Descripteurs français

Pascal (Inist)
- 81R50, Algèbre Hopf, Fonctionnelle Gauss, Groupe Heisenberg, Groupe Lie, Groupe Weyl, Groupe connexe, Groupe convolution, Groupe nilpotent, Groupe quantique, Groupe tressé, Loi Gauss, Loi probabilité, Plongement, Propriété Bernstein, Propriété Poincré Birkhoff Witt, Semigroupe, Suite variable aléatoire, Unicité plongement.

English descriptors

KwdEn :
- 16W30, 81R50, Braided group, Convolution group, Embedding, Gauss law, Gaussian functional, Heisenberg group, Hopf algebra, Lie group, Mathematics Subject Classification (1991): 60B99, Probability distribution, Quantum group, Random variable sequence, Semigroup, Weyl group.

Abstract

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.

Url:

https://api.istex.fr/ark:/67375/VQC-7NJ3BBP1-V/fulltext.pdf

DOI: 10.1007/s004400050127

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U. Franz

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</affiliation>

R. Schott

<affiliation><wicri:noCountry code="subField">FR</wicri:noCountry>
</affiliation>

Le document en format XML

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<front><div type="abstract" xml:lang="en">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.</div>
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Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

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

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