Traces for Star Products on the Dual of a Lie Algebra

Bieliavsky, Pierre;Gutt, Simone;Bordemann, Martin;Waldmann, Stefan
(2003) Reviews in Mathematical Physics : a journal for survey and expository articles in the field of mathematical physics — Vol. 15, n° 05, p. 425-445 (2003)

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  • Bieliavsky, PierreDépartement de Mathématique, Université Libre de Bruxelles, Campus Plaine, C. P. 218, Boulevard du Triomphe, B-1050 Bruxelles, Belgique
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  • Gutt, SimoneDépartement de Mathématique, Université Libre de Bruxelles, Campus Plaine, C. P. 218, Boulevard du Triomphe, B-1050 Bruxelles, Belgique
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  • Bordemann, MartinLaboratoire de Mathématiques, Université de Haute-Alsace Mulhouse, 4, Rue des Frères Lumière, F.68093 Mulhouse, France
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  • Waldmann, StefanFakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann Herder Straße 3, D 79104 Freiburg, Germany
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Abstract
<jats:p> In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on. </jats:p>
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Bieliavsky, P., Gutt, S., Bordemann, M., & Waldmann, S. (2003). Traces for Star Products on the Dual of a Lie Algebra. Reviews in Mathematical Physics : a journal for survey and expository articles in the field of mathematical physics, 15(05), 425-445. https://doi.org/10.1142/S0129055X03001643 (Original work published 2003)