Wigner-Weyl-Moyal formalism on algebraic structures

We first introduce the Wigner-Weyl-Moyal formalism for a theory whose phase space is an arbitrary Lie algebra. We also generalize to quantum Lie algebras and to supersymmetric theories. It turns out that the noncommutativity leads to a deformation of the classical phase space: instead of being a vector space, it becomes a manifold, the topology of which is given by the commutator relations. It is shown in fact that the classical phase space, for a semisimple Lie algebra, becomes a homogeneous symplectic manifold. The symplectic product is also deformed. We finally make some comments on how to generalise to C*-algebras and other operator algebras, too.