Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces

Abstract

Abstract. This paper is a sequel to [Caine A., Pickrell D., Int. Math. Res. Not., to appear, arXiv:0710.4484], where we studied the Hamiltonian systems which arise from the EvensLu 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.

Loops in SU(2) and Factorization

“…This Theorem basically follows from results in [4], but it is possible to give a direct argument (not involving Lie theory). We will present this, and functional analytic generalizations, in Section 2.…”

“…This paper is a sequel to [4]. The main purpose of the paper is to prove functional analytic generalizations of Theorems 0.1 and 0.2 below.…”

Loops in SU(2), Riemann Surfaces, and Factorization, I

“…Even in the classical case, our proof of this is far more illuminating than the one in [14]. For example we will prove the following much stronger statement, at the level of operators: Theorem 1.6.…”

“…There are other senses in which the factors k 1 , exp(χ) and k 2 are expected to be "independent". For example in the classical case we have previously conjectured that these factors are independent random variables with respect to the large temperature limit for Wiener measure on the loop group, and that they Poisson commute with respect to the Evens-Lu homogeneous Poisson structure on LSU(2) (see [14]). To properly formulate nonclassical analogues of these conjectures, we need to prove the existence of a factorization for a generic loop, as we discussed at the end of the previous subsection.…”

Homogeneous Poisson Structures on Symmetric Spaces

“…The following theorem is a reformulation of results in [Lu] on the standard Poisson structure. This reformulation is of importance in connection with infinite dimensional generalizations (see [Pi2]). We denote the symplectic form on Ň − simply by ω (in our earlier notation this is ω 1 , from the noncompact point of view, and Π −1 1 , from the compact point of view).…”