Non-Abelian Poisson manifolds from D-branes

Superimposed D-branes have matrix-valued functions as their transverse coordinates, since the latter take values in the Lie algebra of the gauge group inside the stack of coincident branes. This leads to considering a classical dynamics where the multiplication law for coordinates and/or momenta, being given by matrix multiplication, is non-Abelian. Quantization further introduces noncommutativity as a deformation in powers of Planck's constant h. Given an arbitrary simple Lie algebra g and an arbitrary Poisson manifold M, both finite-dimensional, we define a corresponding C*-algebra that can be regarded as a non-Abelian Poisson manifold. The latter provides a natural framework for a matrix-valued classical dynamics.