Multiplicative noncommutative deformations of left invariant Riemannian metrics on Heisenberg groups

Amine Bahayou; Mohamed Boucetta

doi:10.1016/j.crma.2009.04.013

Differential Geometry

Multiplicative noncommutative deformations of left invariant Riemannian metrics on Heisenberg groups

Presented by: Charles-Michel Marle

Amine Bahayou ¹; Mohamed Boucetta ²

¹ Université Ouargla, BP 511, route Ghardaïa, 30000 Ouargla, Algeria
² Faculté des sciences et techniques Gueliz, BP 549, 40000 Marrakech, Morocco

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 791-796.

Abstract
Résumé

Let $H_{n}$ be the Heisenberg group of dimension $2 n + 1$ . We give a precise description of all $(π, 〈, 〉)$ , where π is a multiplicative Poisson tensor on $H_{n}$ and $〈, 〉$ is a left invariant metric on $H_{n}$ such that $(π, 〈, 〉)$ satisfies the necessary conditions, introduced by Eli Hawkins, to the existence of a noncommutative deformation of the spectral triple associated to $(H_{n}, 〈, 〉)$ .

Soit $H_{n}$ le groupe de Heisenberg de dimension $2 n + 1$ . Nous donnons une description complète des couples $(π, 〈, 〉)$ où π est un tenseur de Poisson multiplicatif et $〈, 〉$ une métrique invariante à gauche sur $H_{n}$ tels que $(π, 〈, 〉)$ vérifie les conditions nécessaires, introduites par Eli Hawkins, à l'existence d'une déformation non commutative du triple spectral associé à $(H_{n}, 〈, 〉)$ .

Received: 2008-12-29
Accepted: 2009-04-17
Published online: 2009-05-09

DOI: 10.1016/j.crma.2009.04.013

Author's affiliations:

Amine Bahayou ¹; Mohamed Boucetta ²

¹ Université Ouargla, BP 511, route Ghardaïa, 30000 Ouargla, Algeria
² Faculté des sciences et techniques Gueliz, BP 549, 40000 Marrakech, Morocco

@article{CRMATH_2009__347_13-14_791_0,
     author = {Amine Bahayou and Mohamed Boucetta},
     title = {Multiplicative noncommutative deformations of left invariant {Riemannian} metrics on {Heisenberg} groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {791--796},
     publisher = {Elsevier},
     volume = {347},
     number = {13-14},
     year = {2009},
     doi = {10.1016/j.crma.2009.04.013},
     language = {en},
}

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Amine Bahayou; Mohamed Boucetta. Multiplicative noncommutative deformations of left invariant Riemannian metrics on Heisenberg groups. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 791-796. doi : 10.1016/j.crma.2009.04.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.013/

References
Cited by

[1] M. Boucetta Compatibilité des structures pseudo-riemanniennes et des structures de Poisson, C. R. Acad. Sci. Paris, Ser. I, Volume 333 (2001), pp. 763-768

[2] J.P. Dufour; N.T. Zung Poisson Structures and Their Normal Forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, Basel, Boston, New York, 2005

[3] R.L. Fernandes Connections in Poisson geometry. I. Holonomy and invariants, J. Differential Geom., Volume 54 (2000) no. 2, pp. 303-365

[4] E. Hawkins Noncommutative rigidity, Commun. Math. Phys., Volume 246 (2004), pp. 211-235

[5] E. Hawkins The structure of noncommutative deformations, J. Differential Geom., Volume 77 (2007), pp. 385-424

[6] J.L. Koszul Crochet de Schouten–Nijenhuis et Cohomologie, Astérisque, Num´ero Hors Série (1985), pp. 257-271

[7] J. Milnor Curvature of Left invariant metrics on Lie groups, Adv. Math., Volume 21 (1976), pp. 293-329

[8] I. Szymczak; S. Zakrzemski Quantum deformations of the Heisenberg group obtained by geometric quantization, J. Geom. Phys., Volume 7 (1990), pp. 553-569

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