Constrained Hamiltonian systems on Lie groups, moment map reductions, and central extensions

The canonical symplectic structure on the cotangent bundle T*G of a Lie group, whose algebra gc admits a central extension gc, is modified so that the moment map associated to right translations generates a representation of gc. Zero moment map constraints are applied to a class of Hamiltonian systems on T*G that are not necessarily right invariant to yield reduced systems on coadjoint orbits in gc*. The Hamiltonian formulation of the WZW model is developed as an example in which G = LG0 is a loop group and the reduced space is the group of based loops ΩG0, identified as a coadjoint orbit of the affine Kac–Moody algebra [Formula: see text]. An extension of the geometric quantization method leads to representations of [Formula: see text] on Hilbert spaces of sections of holomorphic vector bundles over ΩG0.