  In order to understand the structure of the cohomologies involved in the
study of projectively equivariant quantizations, we introduce a notion of
affine representation of a Lie algebra.We show how it is related to linear
representations and 1-cohomology classes of the algebra. We classify the affine
representations of the Lie algebra of vector fields of a smooth manifold
associated to its action on symmetric tensor fields of type (1,2). Among them,
we recover the space of symmetric affine linear connections and that of
projective structures of the manifold. We compute some of the associated
cohomologies.
