Logarithmic Poisson cohomology: example of calculation and application to prequantization

Joseph Dongho[1]

  • [1] Université de Maroua, Ecole Normale Supérieure, Département de Mathématiques, BP 55 Maroua au Cameroun

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: 4, page 623-650
  • ISSN: 0240-2963

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: ( ℂ [ x , y ] , { x , y } = x ) .

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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