Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 069, 33 pages      arXiv:0801.3277      http://dx.doi.org/10.3842/SIGMA.2008.069
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces

Doug Pickrell
Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

Received June 14, 2008, in final form September 27, 2008; Published online October 07, 2008

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 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.

Key words: Poisson structure; loop space; symmetric space; Toeplitz determinant.

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References

  1. Brezis H., New questions related to topological degree, in The Unity of Mathematics, Prog. Math., Vol. 244, Birkhäuser, Boston, MA, 2006, 137-154.
  2. Caine A., Compact symmetric spaces, triangular factorization, and Poisson geometry, J. Lie Theory 18 (2008), 273-294, math.SG/0608454.
  3. Caine A., Pickrell D., Homogeneous Poisson structures on symmetric spaces, Int. Math. Res. Not., to appear, arXiv:0710.4484.
  4. Evens S., Lu J.-H., On the variety of Lagrangian subalgebras. I, Ann. Sci. École Norm. Sup. (4) 34 (2001), 631-668, math.DG/9909005.
  5. Kac V., Infinite-dimensional Lie algebras. An introduction, Birkhäuser, Boston, MA, 1983.
  6. Kac V., Constructing groups from infinite-dimensional Lie algebras, in Infinite-Dimensional Groups with Applications (Berkeley, Calif., 1984), Editor V. Kac, Math. Sci. Res. Inst. Publ., Vol. 4, Springer, New York, 1985, 167-216.
  7. Lu J.-H., Coordinates on Schubert cells, Kostant's harmonic forms, and the Bruhat-Poisson structure on G/B, Transform. Groups 4 (1999), 355-374, dg-ga/9610009.
  8. Odzijewicz A., Ratiu T., Banach Lie-Poisson spaces and reduction, Comm. Math. Phys. 243 (2003), 1-54, math.SG/0210207.
  9. Pickrell D., An invariant measure for the loop space of a simply connected compact symmetric space, J. Funct. Anal. 234 (2006), 321-363, math-ph/0409013.
  10. Pickrell D., A survey of conformally invariant measures on Hm(D), math.PR/0702672.
  11. Pressley A., Segal G., Loop groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986.
  12. Widom H., Asymptotic behavior of block Toeplitz matrices and determinants. II, Adv. Math. 21 (1976), 1-29.

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