Deformation Quantization of Poisson Structures Associated to Lie Algebroids
In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E*, where E → M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's forma...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149144 |
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| Cite this: | Deformation Quantization of Poisson Structures Associated to Lie Algebroids / N. Neumaier, S. Waldmann // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 44 назв. — англ. |
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Neumaier, N. Waldmann, S. 2019-02-19T17:39:43Z 2019-02-19T17:39:43Z 2009 Deformation Quantization of Poisson Structures Associated to Lie Algebroids / N. Neumaier, S. Waldmann // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 44 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53D55; 53D17 https://nasplib.isofts.kiev.ua/handle/123456789/149144 In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E*, where E → M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concreteness of the construction allows to obtain explicit formulas even for a wide class of derivations and self-equivalences of the products. Moreover, we can show that some of our products are in direct relation to the universal enveloping algebra associated to the Lie algebroid. Finally, we show that for a certain class of star products on E* the integration with respect to a density with vanishing modular vector field defines a trace functional. This paper is a contribution to the Special Issue on Deformation Quantization. We would like to thank Janusz Grabowski, Simone Gutt, Yvette Kosmann-Schwarzbach and Alan Weinstein for valuable discussions and remarks. Moreover, we thank the referees for many interesting suggestions and remarks. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformation Quantization of Poisson Structures Associated to Lie Algebroids Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Deformation Quantization of Poisson Structures Associated to Lie Algebroids |
| spellingShingle |
Deformation Quantization of Poisson Structures Associated to Lie Algebroids Neumaier, N. Waldmann, S. |
| title_short |
Deformation Quantization of Poisson Structures Associated to Lie Algebroids |
| title_full |
Deformation Quantization of Poisson Structures Associated to Lie Algebroids |
| title_fullStr |
Deformation Quantization of Poisson Structures Associated to Lie Algebroids |
| title_full_unstemmed |
Deformation Quantization of Poisson Structures Associated to Lie Algebroids |
| title_sort |
deformation quantization of poisson structures associated to lie algebroids |
| author |
Neumaier, N. Waldmann, S. |
| author_facet |
Neumaier, N. Waldmann, S. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E*, where E → M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concreteness of the construction allows to obtain explicit formulas even for a wide class of derivations and self-equivalences of the products. Moreover, we can show that some of our products are in direct relation to the universal enveloping algebra associated to the Lie algebroid. Finally, we show that for a certain class of star products on E* the integration with respect to a density with vanishing modular vector field defines a trace functional.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149144 |
| citation_txt |
Deformation Quantization of Poisson Structures Associated to Lie Algebroids / N. Neumaier, S. Waldmann // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 44 назв. — англ. |
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AT neumaiern deformationquantizationofpoissonstructuresassociatedtoliealgebroids AT waldmanns deformationquantizationofpoissonstructuresassociatedtoliealgebroids |
| first_indexed |
2025-12-07T15:48:21Z |
| last_indexed |
2025-12-07T15:48:21Z |
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1850865091995500544 |