Compact symmetric spaces, triangular factorization, and Poisson geometry

Details Compact symmetric spaces, triangular factorization, and Poisson geometry Compact symmetric spaces, triangular factorization, and Poisson geometry

2006-08-18

arxiv.org/abs/math/0608454v2

Mathematics

Symplectic Geometry

27 pages, 3 figures, exposition substantially revised, proof of main results is given in the general case, submitted to the Jo

Scientific paper

Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let \g denote the complexification of the Lie algebra of U, \g=\u^\C. Each \u-compatible triangular decomposition \g=\n_- + \h + \n_+ determines a Poisson Lie group structure \pi_U on U. The Evens-Lu construction produces a (U,\pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of \g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that the action is Hamiltonian and the momentum map is computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.

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