Dirac approach to constrained submanifolds in a double loop group: from WZNW to Poisson-Lie $\sigma $-model

摘要

We study the restriction to a family of second class constrained submanifoldsin the cotangent bundle of a double Lie group, equipped with a 2-cocycleextended symplectic form, building the corresponding Dirac brackets. It isshown that, for a 2-cocycle vanishing on each isotropic subspaces of theassociated Manin triple, the Dirac bracket contains no traces of the cocycle.We also investigate the restriction of the left translation action of thedouble Lie group on its cotangent bundle, where it fails in to be a symmetry acanonical transformation. However, the hamiltonian symmetry is restored on somespecial submanifolds. The main application is on loop groups, showing that aWZNW-type model on the double Lie group with a quadratic Hamilton function inthe momentum maps associated with the left translation action on the cotangentbundle with the canonical symplectic form, restricts to a collective system onsome special submanifolds. There, the lagrangian version coincides with socalled Poisson-Lie $\sigma $-model.