Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups
Identifieur interne : 00BC48 ( Main/Exploration ); précédent : 00BC47; suivant : 00BC49Gauss 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. SchottSource :
- Probability Theory and Related Fields [ 0178-8051 ] ; 1997-09-01.
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 :
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:
DOI: 10.1007/s004400050127
Affiliations:
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Franz</name><affiliation><wicri:noCountry code="subField">FR</wicri:noCountry></affiliation></author><author><name sortKey="Neuenschwander, D" sort="Neuenschwander, D" uniqKey="Neuenschwander D" first="D." last="Neuenschwander">D. Neuenschwander</name><affiliation wicri:level="4"><orgName type="university">Université de Lausanne</orgName><country>Suisse</country><placeName><settlement type="city">Lausanne</settlement><region nuts="3" type="region">Canton de Vaud</region></placeName></affiliation></author><author><name sortKey="Schott, R" sort="Schott, R" uniqKey="Schott R" first="R." last="Schott">R. Schott</name><affiliation><wicri:noCountry code="subField">FR</wicri:noCountry></affiliation></author></analytic><monogr></monogr><series><title level="j">Probability Theory and Related Fields</title><title level="j" type="abbrev">Probab Theory Relat Fields</title><idno type="ISSN">0178-8051</idno><idno type="eISSN">1432-2064</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>Berlin/Heidelberg</pubPlace><date type="published" when="1997-09-01">1997-09-01</date><biblScope unit="volume">109</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="101">101</biblScope><biblScope unit="page" to="127">127</biblScope></imprint><idno type="ISSN">0178-8051</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0178-8051</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>16W30</term><term>81R50</term><term>Braided group</term><term>Convolution group</term><term>Embedding</term><term>Gauss law</term><term>Gaussian functional</term><term>Heisenberg group</term><term>Hopf algebra</term><term>Lie group</term><term>Mathematics Subject Classification (1991): 60B99</term><term>Probability distribution</term><term>Quantum group</term><term>Random variable sequence</term><term>Semigroup</term><term>Weyl group</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>81R50</term><term>Algèbre Hopf</term><term>Fonctionnelle Gauss</term><term>Groupe Heisenberg</term><term>Groupe Lie</term><term>Groupe Weyl</term><term>Groupe connexe</term><term>Groupe convolution</term><term>Groupe nilpotent</term><term>Groupe quantique</term><term>Groupe tressé</term><term>Loi Gauss</term><term>Loi probabilité</term><term>Plongement</term><term>Propriété Bernstein</term><term>Propriété Poincré Birkhoff Witt</term><term>Semigroupe</term><term>Suite variable aléatoire</term><term>Unicité plongement</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><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. 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