Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces

TL;DR

This paper extends the study of homogeneous Poisson structures to loop spaces of symmetric spaces, providing new insights and applications such as integral formulas and Toeplitz determinant factorizations.

Contribution

It generalizes previous finite-dimensional results to infinite-dimensional loop spaces, clarifies their interpretation, especially for SU(2), and introduces applications in Toeplitz determinants.

Findings

Extension of Poisson structures to loop spaces

New integral formulas for Toeplitz determinants

Factorization results for Toeplitz determinants

Abstract

This paper is a sequel to [Caine A., Pickrell D., arXiv:0710.4484], where we studied the Hamiltonian systems which arise from the Evens-Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. In this paper we consider loop space analogues. Many of the results extend in a relatively routine way to the loop space setting, but new issues emerge. The main point of this paper is to spell out the meaning of the results, especially in the SU(2) case. Applications include integral formulas and factorizations for Toeplitz determinants.