Weak quantization of Poisson structures

Damien Calaque (1) , Gilles Halbout (2)

1 ICJ - Institut Camille Jordan
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier

In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira. We begin the paper with a recollection of known facts about deformation theory of cosimplicial differential graded Lie algebras.

Domaines: Algèbres quantiques [math.QA]

Damien Calaque, Gilles Halbout. Weak quantization of Poisson structures. Journal of Geometry and Physics, 2011, 61 (8), pp.1401-1414. ⟨10.1016/j.geomphys.2011.03.004⟩. ⟨hal-00162755⟩