Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups
Identifieur interne : 000C29 ( PascalFrancis/Checkpoint ); précédent : 000C28; suivant : 000C30Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups
Auteurs : U. Franz [France, Allemagne] ; D. Neuenschwander [Suisse] ; R. Schott [France]Source :
- Probability theory and related fields [ 0178-8051 ] ; 1997.
Descripteurs français
- Pascal (Inist)
- Plongement, Semigroupe, Groupe quantique, Loi probabilité, Groupe Heisenberg, Groupe Weyl, Suite variable aléatoire, Groupe Lie, Algèbre Hopf, 81R50, Unicité plongement, Propriété Bernstein, Groupe nilpotent, Groupe connexe, Propriété Poincré Birkhoff Witt, Loi Gauss, Groupe convolution, Groupe tressé, Fonctionnelle Gauss.
English descriptors
- KwdEn :
Abstract
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.
Affiliations:
- Allemagne, France, Suisse
- Canton de Vaud, Grand Est, Lorraine (région)
- Lausanne, Vandœuvre-lès-Nancy
- Université de Lausanne
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Physik, Technische Universität Clausthal, Leibnitzstr. 10</s1><s2>38678 Clausthal-Zellerfeld</s2><s3>DEU</s3><sZ>1 aut.</sZ></inist:fA14><country>Allemagne</country><wicri:noRegion>38678 Clausthal-Zellerfeld</wicri:noRegion><wicri:noRegion>Leibnitzstr. 10</wicri:noRegion><wicri:noRegion>38678 Clausthal-Zellerfeld</wicri:noRegion></affiliation></author><author><name sortKey="Neuenschwander, D" sort="Neuenschwander, D" uniqKey="Neuenschwander D" first="D." last="Neuenschwander">D. Neuenschwander</name><affiliation wicri:level="4"><inist:fA14 i1="03"><s1>Université de Lausanne, Ecole des Hautes Etudes Commerciales, Institut de Sciences Actuarielles</s1><s2>1015 Lausanne</s2><s3>CHE</s3><sZ>2 aut.</sZ></inist:fA14><country>Suisse</country><placeName><settlement type="city">Lausanne</settlement><region nuts="3" type="region">Canton de Vaud</region></placeName><orgName type="university">Université de Lausanne</orgName><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:level="3"><inist:fA14 i1="01"><s1>Institut Elie Cartan and CRIN-CNRS, BP 239, Université H. Poincaré-Nancy I</s1><s2>54506 Vandoeuvre-lès-Nancy</s2><s3>FRA</s3><sZ>1 aut.</sZ><sZ>3 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">INIST</idno><idno type="inist">97-0494440</idno><date when="1997">1997</date><idno type="stanalyst">PASCAL 97-0494440 INIST</idno><idno type="RBID">Pascal:97-0494440</idno><idno type="wicri:Area/PascalFrancis/Corpus">000C32</idno><idno type="wicri:Area/PascalFrancis/Curation">000C47</idno><idno type="wicri:Area/PascalFrancis/Checkpoint">000C29</idno><idno type="wicri:explorRef" wicri:stream="PascalFrancis" wicri:step="Checkpoint">000C29</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a">Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups</title><author><name sortKey="Franz, U" sort="Franz, U" uniqKey="Franz U" first="U." last="Franz">U. Franz</name><affiliation wicri:level="3"><inist:fA14 i1="01"><s1>Institut Elie Cartan and CRIN-CNRS, BP 239, Université H. Poincaré-Nancy I</s1><s2>54506 Vandoeuvre-lès-Nancy</s2><s3>FRA</s3><sZ>1 aut.</sZ><sZ>3 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Arnold Sommerfeld Institut für math. Physik, Technische Universität Clausthal, Leibnitzstr. 10</s1><s2>38678 Clausthal-Zellerfeld</s2><s3>DEU</s3><sZ>1 aut.</sZ></inist:fA14><country>Allemagne</country><wicri:noRegion>38678 Clausthal-Zellerfeld</wicri:noRegion><wicri:noRegion>Leibnitzstr. 10</wicri:noRegion><wicri:noRegion>38678 Clausthal-Zellerfeld</wicri:noRegion></affiliation></author><author><name sortKey="Neuenschwander, D" sort="Neuenschwander, D" uniqKey="Neuenschwander D" first="D." last="Neuenschwander">D. Neuenschwander</name><affiliation wicri:level="4"><inist:fA14 i1="03"><s1>Université de Lausanne, Ecole des Hautes Etudes Commerciales, Institut de Sciences Actuarielles</s1><s2>1015 Lausanne</s2><s3>CHE</s3><sZ>2 aut.</sZ></inist:fA14><country>Suisse</country><placeName><settlement type="city">Lausanne</settlement><region nuts="3" type="region">Canton de Vaud</region></placeName><orgName type="university">Université de Lausanne</orgName><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:level="3"><inist:fA14 i1="01"><s1>Institut Elie Cartan and CRIN-CNRS, BP 239, Université H. Poincaré-Nancy I</s1><s2>54506 Vandoeuvre-lès-Nancy</s2><s3>FRA</s3><sZ>1 aut.</sZ><sZ>3 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation></author></analytic><series><title level="j" type="main">Probability theory and related fields</title><title level="j" type="abbreviated">Probab. theory relat. fields</title><idno type="ISSN">0178-8051</idno><imprint><date when="1997">1997</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Probability theory and related fields</title><title level="j" type="abbreviated">Probab. theory relat. fields</title><idno type="ISSN">0178-8051</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><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>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>Plongement</term><term>Semigroupe</term><term>Groupe quantique</term><term>Loi probabilité</term><term>Groupe Heisenberg</term><term>Groupe Weyl</term><term>Suite variable aléatoire</term><term>Groupe Lie</term><term>Algèbre Hopf</term><term>81R50</term><term>Unicité plongement</term><term>Propriété Bernstein</term><term>Groupe nilpotent</term><term>Groupe connexe</term><term>Propriété Poincré Birkhoff Witt</term><term>Loi Gauss</term><term>Groupe convolution</term><term>Groupe tressé</term><term>Fonctionnelle Gauss</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">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></front></TEI><inist><standard h6="B"><pA><fA01 i1="01" i2="1"><s0>0178-8051</s0></fA01><fA02 i1="01"><s0>PTRFEU</s0></fA02><fA03 i2="1"><s0>Probab. theory relat. fields</s0></fA03><fA05><s2>109</s2></fA05><fA06><s2>1</s2></fA06><fA08 i1="01" i2="1" l="ENG"><s1>Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups</s1></fA08><fA11 i1="01" i2="1"><s1>FRANZ (U.)</s1></fA11><fA11 i1="02" i2="1"><s1>NEUENSCHWANDER (D.)</s1></fA11><fA11 i1="03" i2="1"><s1>SCHOTT (R.)</s1></fA11><fA14 i1="01"><s1>Institut Elie Cartan and CRIN-CNRS, BP 239, Université H. Poincaré-Nancy I</s1><s2>54506 Vandoeuvre-lès-Nancy</s2><s3>FRA</s3><sZ>1 aut.</sZ><sZ>3 aut.</sZ></fA14><fA14 i1="02"><s1>Arnold Sommerfeld Institut für math. Physik, Technische Universität Clausthal, Leibnitzstr. 10</s1><s2>38678 Clausthal-Zellerfeld</s2><s3>DEU</s3><sZ>1 aut.</sZ></fA14><fA14 i1="03"><s1>Université de Lausanne, Ecole des Hautes Etudes Commerciales, Institut de Sciences Actuarielles</s1><s2>1015 Lausanne</s2><s3>CHE</s3><sZ>2 aut.</sZ></fA14><fA20><s1>101-127</s1></fA20><fA21><s1>1997</s1></fA21><fA23 i1="01"><s0>ENG</s0></fA23><fA43 i1="01"><s1>INIST</s1><s2>9653</s2><s5>354000069352750060</s5></fA43><fA44><s0>0000</s0><s1>© 1997 INIST-CNRS. All rights reserved.</s1></fA44><fA45><s0>21 ref.</s0></fA45><fA47 i1="01" i2="1"><s0>97-0494440</s0></fA47><fA60><s1>P</s1></fA60><fA61><s0>A</s0></fA61><fA64 i1="01" i2="1"><s0>Probability theory and related fields</s0></fA64><fA66 i1="01"><s0>DEU</s0></fA66><fC01 i1="01" l="ENG"><s0>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.</s0></fC01><fC02 i1="01" i2="X"><s0>001A02H01C</s0></fC02><fC02 i1="02" i2="X"><s0>001A02C07</s0></fC02><fC03 i1="01" i2="X" l="FRE"><s0>Plongement</s0><s5>01</s5></fC03><fC03 i1="01" i2="X" l="ENG"><s0>Embedding</s0><s5>01</s5></fC03><fC03 i1="01" i2="X" l="SPA"><s0>Inmersión</s0><s5>01</s5></fC03><fC03 i1="02" i2="X" l="FRE"><s0>Semigroupe</s0><s5>02</s5></fC03><fC03 i1="02" i2="X" l="ENG"><s0>Semigroup</s0><s5>02</s5></fC03><fC03 i1="02" i2="X" l="SPA"><s0>Semigrupo</s0><s5>02</s5></fC03><fC03 i1="03" i2="X" l="FRE"><s0>Groupe quantique</s0><s5>03</s5></fC03><fC03 i1="03" i2="X" l="ENG"><s0>Quantum group</s0><s5>03</s5></fC03><fC03 i1="03" i2="X" l="SPA"><s0>Grupo cuántico</s0><s5>03</s5></fC03><fC03 i1="04" i2="X" l="FRE"><s0>Loi probabilité</s0><s5>04</s5></fC03><fC03 i1="04" i2="X" l="ENG"><s0>Probability distribution</s0><s5>04</s5></fC03><fC03 i1="04" i2="X" l="SPA"><s0>Ley probabilidad</s0><s5>04</s5></fC03><fC03 i1="05" i2="X" l="FRE"><s0>Groupe Heisenberg</s0><s5>05</s5></fC03><fC03 i1="05" i2="X" l="ENG"><s0>Heisenberg group</s0><s5>05</s5></fC03><fC03 i1="05" i2="X" l="SPA"><s0>Grupo Heisenberg</s0><s5>05</s5></fC03><fC03 i1="06" i2="X" l="FRE"><s0>Groupe Weyl</s0><s5>06</s5></fC03><fC03 i1="06" i2="X" l="ENG"><s0>Weyl group</s0><s5>06</s5></fC03><fC03 i1="06" i2="X" l="SPA"><s0>Grupo Weyl</s0><s5>06</s5></fC03><fC03 i1="07" i2="X" l="FRE"><s0>Suite variable aléatoire</s0><s5>07</s5></fC03><fC03 i1="07" i2="X" l="ENG"><s0>Random variable sequence</s0><s5>07</s5></fC03><fC03 i1="07" i2="X" l="SPA"><s0>Sucesión variable aleatoria</s0><s5>07</s5></fC03><fC03 i1="08" i2="X" l="FRE"><s0>Groupe Lie</s0><s5>08</s5></fC03><fC03 i1="08" i2="X" l="ENG"><s0>Lie group</s0><s5>08</s5></fC03><fC03 i1="08" i2="X" l="SPA"><s0>Grupo Lie</s0><s5>08</s5></fC03><fC03 i1="09" i2="X" l="FRE"><s0>Algèbre Hopf</s0><s5>09</s5></fC03><fC03 i1="09" i2="X" l="ENG"><s0>Hopf algebra</s0><s5>09</s5></fC03><fC03 i1="09" i2="X" l="SPA"><s0>Algebra Hopf</s0><s5>09</s5></fC03><fC03 i1="10" i2="X" l="FRE"><s0>81R50</s0><s4>INC</s4><s5>28</s5></fC03><fC03 i1="11" i2="X" l="FRE"><s0>Unicité plongement</s0><s4>INC</s4><s5>75</s5></fC03><fC03 i1="12" i2="X" l="FRE"><s0>Propriété Bernstein</s0><s4>INC</s4><s5>76</s5></fC03><fC03 i1="13" i2="X" l="FRE"><s0>Groupe nilpotent</s0><s4>INC</s4><s5>77</s5></fC03><fC03 i1="14" i2="X" l="FRE"><s0>Groupe connexe</s0><s4>INC</s4><s5>78</s5></fC03><fC03 i1="15" i2="X" l="FRE"><s0>Propriété Poincré Birkhoff Witt</s0><s4>INC</s4><s5>79</s5></fC03><fC03 i1="16" i2="X" l="FRE"><s0>Loi Gauss</s0><s4>CD</s4><s5>96</s5></fC03><fC03 i1="16" i2="X" l="ENG"><s0>Gauss law</s0><s4>CD</s4><s5>96</s5></fC03><fC03 i1="17" i2="X" l="FRE"><s0>Groupe convolution</s0><s4>CD</s4><s5>97</s5></fC03><fC03 i1="17" i2="X" l="ENG"><s0>Convolution group</s0><s4>CD</s4><s5>97</s5></fC03><fC03 i1="18" i2="X" l="FRE"><s0>Groupe tressé</s0><s4>CD</s4><s5>98</s5></fC03><fC03 i1="18" i2="X" l="ENG"><s0>Braided group</s0><s4>CD</s4><s5>98</s5></fC03><fC03 i1="19" i2="X" l="FRE"><s0>Fonctionnelle Gauss</s0><s4>CD</s4><s5>99</s5></fC03><fC03 i1="19" i2="X" l="ENG"><s0>Gaussian functional</s0><s4>CD</s4><s5>99</s5></fC03><fN21><s1>300</s1></fN21></pA></standard></inist><affiliations><list><country><li>Allemagne</li><li>France</li><li>Suisse</li></country><region><li>Canton de Vaud</li><li>Grand Est</li><li>Lorraine (région)</li></region><settlement><li>Lausanne</li><li>Vandœuvre-lès-Nancy</li></settlement><orgName><li>Université de Lausanne</li></orgName></list><tree><country name="France"><region name="Grand Est"><name sortKey="Franz, U" sort="Franz, U" uniqKey="Franz U" first="U." last="Franz">U. Franz</name></region><name sortKey="Schott, R" sort="Schott, R" uniqKey="Schott R" first="R." last="Schott">R. Schott</name></country><country name="Allemagne"><noRegion><name sortKey="Franz, U" sort="Franz, U" uniqKey="Franz U" first="U." last="Franz">U. Franz</name></noRegion></country><country name="Suisse"><region name="Canton de Vaud"><name sortKey="Neuenschwander, D" sort="Neuenschwander, D" uniqKey="Neuenschwander D" first="D." last="Neuenschwander">D. Neuenschwander</name></region></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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