Invariant Deformations of the Poisson Lie Algebra of a Symplectic Manifold and Star-Products

Abstract

Let (M, F) be a symplectic manifold and $\mathbb{G}$ a subalgebra of its Lie algebra of symplectic vector fields. It is shown that if M admits a $\mathbb{G}$-invariant linear connection, every differential deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to $\mathbb{G}$, extends to an invariant differential deformation of infinite order. A similar result holds true for star-products. In particular, if M admits a $\mathbb{G}$-invariant linear connection, there always exists a $\mathbb{G}$-invariant star-product.