Dirac method and symplectic submanifolds in the cotangent bundle of a factorizable Lie group

Abstract

In this work we study some symplectic submanifolds in the cotangent bundle of a factorizable Lie group defined by second class constraints. By applying the Dirac method, we study many issues of these spaces as fundamental Dirac brackets, symmetries, and collective dynamics. This last item allows to study integrability as inherited from a system on the whole cotangent bundle, leading in a natural way to the AKS theory for integrable systems.

Dirac approach to constrained submanifolds in a double loop group: From Wess-Zumino-Novikov-Witten to Poisson-Lie σ-model

“…In this section we adapt the Dirac bracket construction of ref. [6] to the case where canonical symplectic form on G × g * is modified by adding a R-valued 2-cocycle on g and we write the Dirac bracket on the submanifolds N c (g − , η − ). Let C : G −→ g * be a coadjoint 1-cocycle, that is, for g, h ∈ G it satisfy…”

“…In this first part we study the fibration Ψ : G × g * −→ G − × g * − , for a double Lie group G = G + G − , as the phase space of systems constrained to the fibers Ψ −1 (g − , η − ). We adapt the Dirac's machinery developed in [6] to the framework of G × g * equipped with the 2-cocycle extended symplectic form, pointing to the loop groups stage. We also address the left translation action of G on G × g * and its restriction to Ψ −1 (g − , η − ), finding out the fibers on which it turns a phase space symmetry with Ad-equivariant momentum maps.…”

“…In Part I we concentrate in developing the Dirac machinery and symmetry issues for the 2-cocycle extended symplectic form on T * G. Thus, in Section 1 we describe the phase space on a double Poisson-Lie Lie group and build the fibration of constrained submanifolds. In the Section 2 we adapt the scheme developed in [6] to the case where the canonical symplectic form is modified by adding a 2-cocycle. Section 3 is devoted to specialize the construction to loop groups.…”

Integrable systems on semidirect product Lie groups

“…Integral curves of this projected vector field are the trajectories of the constrained hamiltonian system. In the rest of the current section we adapt the approach developed in reference [4] to the framework of semidirect product and factorization as introduced above.…”

“…can be supplied with a nondegenerate Dirac bracket constructed following ref. [4]. We shall use the linear bijection σ : h 1 −→ h * 1 allows to translate those results to the current framework.…”