Nonabelian 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 nonabelian. Quantisation further introduces noncommutativity as a deformation in powers of Planck’s constant ~. 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 nonabelian Poisson manifold. The latter provides a natural framework for a matrix–valued classical dynamics.