Phase space quantization, noncommutativity, and the gravitational field

In this paper we study the structure of the phase space in noncommutative geometry in the presence of a nontrivial frame. Our basic assumptions are that the underlying space is a symplectic and parallelizable manifold. Furthermore, we assume the validity of the Leibniz rule and the Jacobi identities. We consider noncommutative spaces due to the quantization of the symplectic structure and determine the momentum operators that guarantee a set of canonical commutation relations, appropriately extended to include the nontrivial frame. We stress the important role of left vs right acting operators and of symplectic duality. This enables us to write down the form of the full phase space algebra on these noncommutative spaces, both in the noncompact and in the compact case. We test our results against the class of four-dimensional and six-dimensional symplectic nilmanifolds, thus presenting a large set of nontrivial examples that realizes the general formalism.