Morita Equivalence of Formal Poisson Structures

Abstract

We extend the notion of Morita equivalence of Poisson manifolds to the setting of formal Poisson structures, that is, formal power series of bivector fields $\pi =\pi _0 + \lambda \pi _1 +\cdots $ satisfying the Poisson integrability condition $[\pi ,\pi ]=0$. Our main result gives a complete description of Morita equivalent formal Poisson structures deforming the zero structure ($\pi _0=0$) in terms of $B$-field transformations, relying on a general study of formal deformations of Poisson morphisms and dual pairs. Combined with previous work on Morita equivalence of star products [ 5], our results link the notions of Morita equivalence in Poisson geometry and noncommutative algebra via deformation quantization.

Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry

“…If (M, θ) is integrable, there is a notion of Morita self-equivalence, see e.g. [15]. We will return to the question of uniqueness, as well as the meaning of the symplectic structure on T * M away from the zero section, later on where we will find that they have natural interpretations in terms of gauge algebras (see remark 5.13 and proposition 3.7, respectively).…”

“…In this paper we will work in the setting of formal Poisson structures and formal symplectic structures, see e.g. [15]. Definition 2.6.…”