Logarithmic Poisson cohomology: example of calculation and application to prequantization

• [1] Université de Maroua, Ecole Normale Supérieure, Département de Mathématiques, BP 55 Maroua au Cameroun
• Volume: 21, Issue: 4, page 623-650
• ISSN: 0240-2963

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Abstract

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In this paper we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure. We prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kähler differentials. Therefore it induces a differential complex from which we derive the notion of logarithmic Poisson cohomology. We prove that Poisson cohomology and logarithmic Poisson cohomology are equal when the Poisson structure is log symplectic. We give an example of non log symplectic but logarithmic Poisson structure for which these cohomology spaces are equal. We give an example for which these cohomologies are different. We discuss and modify the K. Saito definition of logarithmic differential forms. This note ends with an application to a prequantization of the logarithmic Poisson algebra: $\left(ℂ\left[x,y\right],\left\{x,y\right\}=x\right).$

How to cite

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Dongho, Joseph. "Logarithmic Poisson cohomology: example of calculation and application to prequantization." Annales de la faculté des sciences de Toulouse Mathématiques 21.4 (2012): 623-650. <http://eudml.org/doc/251001>.

@article{Dongho2012,
abstract = {In this paper we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure. We prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kähler differentials. Therefore it induces a differential complex from which we derive the notion of logarithmic Poisson cohomology. We prove that Poisson cohomology and logarithmic Poisson cohomology are equal when the Poisson structure is log symplectic. We give an example of non log symplectic but logarithmic Poisson structure for which these cohomology spaces are equal. We give an example for which these cohomologies are different. We discuss and modify the K. Saito definition of logarithmic differential forms. This note ends with an application to a prequantization of the logarithmic Poisson algebra: $(\{\mathbb\{C\}\}[x,y],\lbrace x,y\rbrace =x).$},
affiliation = {Université de Maroua, Ecole Normale Supérieure, Département de Mathématiques, BP 55 Maroua au Cameroun},
author = {Dongho, Joseph},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Poisson geometry; cohomology; logarithmic Poisson structures; prequantization},
language = {eng},
month = {10},
number = {4},
pages = {623-650},
publisher = {Université Paul Sabatier, Toulouse},
title = {Logarithmic Poisson cohomology: example of calculation and application to prequantization},
url = {http://eudml.org/doc/251001},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Dongho, Joseph
TI - Logarithmic Poisson cohomology: example of calculation and application to prequantization
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/10//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 4
SP - 623
EP - 650
AB - In this paper we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure. We prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kähler differentials. Therefore it induces a differential complex from which we derive the notion of logarithmic Poisson cohomology. We prove that Poisson cohomology and logarithmic Poisson cohomology are equal when the Poisson structure is log symplectic. We give an example of non log symplectic but logarithmic Poisson structure for which these cohomology spaces are equal. We give an example for which these cohomologies are different. We discuss and modify the K. Saito definition of logarithmic differential forms. This note ends with an application to a prequantization of the logarithmic Poisson algebra: $({\mathbb{C}}[x,y],\lbrace x,y\rbrace =x).$
LA - eng
KW - Poisson geometry; cohomology; logarithmic Poisson structures; prequantization
UR - http://eudml.org/doc/251001
ER -

References

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1. Alekseevsky (D.), Michor (P.), Ruppert (W.).— Extensions of Lie Algebras. Erwin Schrödinger Institut fut Mathematische Physik Boltzmanngasse, 9, A-1090 Wien, Austria.
2. Braconnier (J.).— Algèbres de Poisson, C. R. Acad. Sci. Paris Sér. A-B 284, no. 21, A1345-A1348 (1977). Zbl0356.17007MR443000
3. Chevalley (C.) and Eilenberg (S.).— Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63, p. 85-124 (1948). Zbl0031.24803MR24908
4. Deligne (P.).— Équations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematics. Berlin. Heidelberg. New York. Zbl0244.14004
5. Goto (R.).— Rozansky-Witten Invariants of log symplectic Manifolds, Contemporary Mathematics, volume 309, p. 69-84 (2002). Zbl1043.53039MR1953353
6. Hochschild (G.), Kostant (B.) and Rosenberg (A.).— Differential Forms On Regular Affine Algebras, Trans. Amer. Math. Soc. 102, p. 383-408 (1962). Zbl0102.27701MR142598
7. Huebschmann (J.).— Poisson Cohomology and quantization, J.Reine Angew. Math. 408 p. 57-113 (1990). Zbl0699.53037MR1058984
8. Krasilshchik (I.).— Hamiltonian cohomology of canonical algebras, (Russian), Dokl. Akad. Nauk SSSR 251, no.6, p. 1306-1309 (1980). Zbl0454.58020MR570152
9. Lichnerowicz (A.).— Les variétés de Poisson et leurs algèbres de Lie associées. (French) J. Differential Geometry 12, no. 2., p. 253-300 (1977). Zbl0405.53024MR501133
10. Lie (S.).— Theorie der Transformations gruppen (Zweiter Abschnitt, unter Mitwirkung von Prof. Dr. Friederich Engel). Teubner, Leipzig (1890).
11. Palais (R.).— The cohomology of Lie ring, Proc. Symp. Pure Math. 3. p. 130-137 (1961). Zbl0126.03404MR125867
12. Rinehart (G.).— Differential forms on general commutative algebras, Trans. Amer. Math. Soc., Vol. 108, no.2, p. 195-222 (1963). Zbl0113.26204MR154906
13. Saito (K.).— Theory of logarithmic differential forms and logarithmic vector fields, Sec. IA, J.Fac.Sci. Univ. Tokyo. 27 p. 265-291 (1980). Zbl0496.32007MR586450
14. Vaisman (I.).— On the geometric quantization of Poisson manifolds, J. Math. Phys 32, p. 3339-3345 (1991). Zbl0749.58023MR1137387
15. Vinogradov (A.), Krasilshchik (I.).— What is Hamiltonian formalism?, (Russian), Uspehi Mat. Nauk, Vol. 30, no.1, p. 1059-1062 (1975). Zbl0327.70006MR650307
16. Weinstein (A.).— The local structure of Poisson manifolds, J. Differential Geometry 18, p. 523-557 (1983). Zbl0524.58011MR723816
17. Woodhouse (N.M.J.).— Geometric quantization, Oxford Mathematiccal Monograph, Claredon Press. Oxford, Second edition (1992). Zbl0747.58004MR1183739

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