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Comptes Rendus Mathématique
Volume 348, n° 15-16
pages 919-922 (août 2010)
Doi : 10.1016/j.crma.2010.07.005
Received : 22 July 2009 ;  accepted : 6 July 2010
Lie bialgebroids of generalized CRF -manifolds
Bi-algébroïdes de Lie des variétés CRF généralisées
 

Yat Sun Poon a , Aïssa Wade b
a Department of Mathematics, University of California at Riverside, CA 92521, USA 
b Department of Mathematics, Penn State University, University Park, PA 16802, USA 

Abstract

The notion of a generalized CRF -structure on a smooth manifold was recently introduced and studied by Vaisman (2008) [[6]]. An important class of generalized CRF -structures on an odd dimensional manifold M consists of CRF -structures having complementary frames of the form  , where ξ is a vector field and η is a 1-form on M with  . It turns out that these kinds of CRF -structures give rise to a special class of what we called strong generalized contact structures in Poon and Wade [[5]]. More precisely, we show that to any CRF -structures with complementary frames of the form  , there corresponds a canonical Lie bialgebroid. Finally, we explain the relationship between generalized contact structures and another generalization of the notion of a Cauchy–Riemann structure on a manifold.

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Résumé

La notion de structure CRF généralisée sur une variété lisse a été récemment introduite et étudiée par Vaisman (2008) [[6]]. Une classe importante de structures CRF généralisées sur une variété M de dimension impaire est constituée de structures CRF généralisées ayant des repères supplémentaires de la forme  , où ξ est un champ de vecteurs et η est une 1-forme differentielle sur M avec  . Il s'avère que ces types de structures CRF généralisées donnent lieu à une classe spéciale de structures que nous avons appelées des structures de contact généralisées fortes dans Poon et Wade [[5]]. Plus précisément, nous montrons qu'à toute structure CRF généralisée ayant des repères supplémentaires de la forme  , il correspond un bi-algèbroïde de Lie canonique. Finalement, nous expliquons la relation entre les structures de contact généralisées et une autre généralisation de la notion de structure-CR (Cauchy–Riemann) sur une variété.

The full text of this article is available in PDF format.


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