Abstract
We show that given a finite-dimensional real Lie algebra \({\mathcal{G}}\) acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on \({\mathcal{G}}\) , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this contravariant connection vanishes if the isotropy Lie subalgebras of the action are trivial. Those results permit to get a large class of smooth manifolds satisfying the necessary conditions, introduced by Eli Hawkins, to the existence of noncommutative deformations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benson C. and Gordon C. (1988). Kähler and symplectic structures on nilmanifolds. Topology 27(4): 513–518
Bordemann M., Medina A. and Ouadfel A. (1993). Le groupe affine comme variété symplectique. Tôhoku Math. J. 45: 423–436
Boucetta M. (2001). Compatibilité des structures pseudo-riemanniennes et des structures de Poisson. C. R. Acad. Sci. Paris, t. 333, Série I: 763–768
Drinfeld V.G. (1983). On constant quasiclassical solutions of the Yang–Baxter quantum equation. Sov. Math. Dokl. 28: 667–671
Fernandes R.L. (2000). Connections in Poisson Geometry1: holonomy and invariants. J. Differ. Geom. 54: 303–366
Gurevich D., Rubtsov V. and Zobin N. (1992). Quantization of Poisson pairs: the R-matix approach. J. Geom. Phys 9: 25–44
Hawkins E. (2004). Noncommutative Rigidity. Commun. Math. Phys. 246: 211–235 math.QA/0211203
Hawkins, E.: The structure of noncommutative deformations, arXiv:math.QA/0504232
Medina A. and Revoy Ph. (1985). Les groupes oscillateurs et leurs réseaux. Manuscripta Math. 52: 81–95
Medina, A., Revoy, Ph.: Groupes de Lie à structure symplectique invariante. Symplectic geometry, groupoids and integrable systems. In: “Séminaire Sud Rhodanien”, M.S.R.I., Springer-Verlag, New York, 247–266 (1991)
Milnor J. (1976). Curvature of left invariant metrics on Lie groups. Adv. Math. 21: 283–329
Reshetikhin N., Voronov A. and Weinstein A. (1996). Semiquantum geometry, algebraic geometry. J. Math. Sci. 82(1): 3255–3267
Vaisman, I.: Lecture on the geometry of Poisson manifolds. Prog. Math., vol. 118, Birkhausser, Berlin (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Recherche menée dans le cadre du Programme Thématique d’Appui à la Recherche Scientifique PROTARS III.
Rights and permissions
About this article
Cite this article
Boucetta, M. Solutions of the Classical Yang–Baxter Equation and Noncommutative Deformations. Lett Math Phys 83, 69–81 (2008). https://doi.org/10.1007/s11005-007-0197-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-007-0197-4