Nous définissons le produit de deux variétés de Dirac et la notion de groupe de Dirac–Lie de type Poisson. Cette notion est équivalente à celle de structure de Dirac multiplicative et tout groupe de Lie réel simplement connexe, de dimension au moins 2 porte une structure de Dirac multiplicative non triviale.
We define the product of two Dirac manifolds and introduce the notion of a Dirac–Lie group of Poisson type. This notion is equivalent to that of multiplicative Dirac structure and any real simply-connected Lie group carries a no trivial multiplicative Dirac structure when its dimension is at least 2.
Accepté le :
Publié le :
@article{CRMATH_2009__347_21-22_1299_0,
author = {Affane, Atallah},
title = {Caract\'erisation et existence de structures de {Dirac} multiplicatives},
journal = {Comptes Rendus. Math\'ematique},
pages = {1299--1304},
publisher = {Elsevier},
volume = {347},
number = {21-22},
year = {2009},
doi = {10.1016/j.crma.2009.10.002},
language = {fr},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.10.002/}
}
TY - JOUR AU - Affane, Atallah TI - Caractérisation et existence de structures de Dirac multiplicatives JO - Comptes Rendus. Mathématique PY - 2009 SP - 1299 EP - 1304 VL - 347 IS - 21-22 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.10.002/ DO - 10.1016/j.crma.2009.10.002 LA - fr ID - CRMATH_2009__347_21-22_1299_0 ER -
%0 Journal Article %A Affane, Atallah %T Caractérisation et existence de structures de Dirac multiplicatives %J Comptes Rendus. Mathématique %D 2009 %P 1299-1304 %V 347 %N 21-22 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.10.002/ %R 10.1016/j.crma.2009.10.002 %G fr %F CRMATH_2009__347_21-22_1299_0
Affane, Atallah. Caractérisation et existence de structures de Dirac multiplicatives. Comptes Rendus. Mathématique, Tome 347 (2009) no. 21-22, pp. 1299-1304. doi : 10.1016/j.crma.2009.10.002. http://www.numdam.org/articles/10.1016/j.crma.2009.10.002/
[1] Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier, Volume 53 (2003), pp. 309-337
[2] Lie–Poisson groups: Remarks and examples, Lett. Math. Phys., Volume 19 (1990), pp. 343-353
[3] Dirac manifolds, Trans. Amer. Math. Soc., Volume 319 (1990), pp. 631-661
[4] Hamiltonian structures on Lie groups, Lie bialgebras and geometric meaning of the classical Yang–Baxter equations, Soviet Math. Dokl., Volume 27 (1983) no. 1, pp. 68-71
[5] Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978
[6] K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Lecture Notes London Math. Soc., vol. 213, 2005
[7] Multiplicative Dirac structures on Lie groups, 12 June 2009 | arXiv
Cité par Sources :
