Journal of Lie Theory

We consider purely algebraic data generalizing the notion of a smooth differentiable manifold. It is given by a triple X, R, W where X is a set, R a commutative associative algebra over the ground field, W a Lie subalgebra and an R-submodule in the derivation algebra of R. Geometric structures studied in differential geometry can be defined on such triples. The main result answers the question about the existence and the uniqueness of an L-invariant unimodular, hamiltonian, contact, or pseudo-riemannian structure in terms of the isotropy subalgebras of points of X. The second major result generalizes a classical fact which says that the Lie algebra of infinitesimal automorphisms of a Riemann metric on a connected manifold is finite dimensional.