Projective connections, group Vey cocycle and deformation quantization

0.1. In this paper we construct a number of geometrically meaningful 1– and 2– cocycles of infinite dimensional Lie groups and Lie algebras. The groups are groups of diffeomorphisms of smooth manifolds and their subgroups, such as the group of symplectomorphisms of a symplectic manifold; the Lie algebras are algebras of vector fields. Coefficient spaces of the cocycles are spaces of various tensor fields, naturally acted upon by diffeomorphisms and vector fields. Our constructions are closely related to deformation quantization. 0.2. Definition of cohomology. Recall the definition of Lie group and Lie algebra cohomology (see, e.g., [Fu]). For a group G and its left module A the space of cochains C(G, A) consists of continuous maps G× ...×G } {{ } q → A, and the differential d : C(G, A)→ C(G, A) is given by the formula: