
Journal of Lie Theory 18 (2008), No. 2, 273294 Copyright Heldermann Verlag 2008 Compact Symmetric Spaces, Triangular Factorization, and Poisson Geometry Arlo Caine MaxPlanckInstitut für Mathematik, Postfach 7280, 53111 Bonn, Germany caine@mpimbonn.mpg.de [Abstractpdf] \def\C{{\Bbb C}} \def\g{{\frak g}} \def\h{{\frak h}} \def\n{{\frak n}} \def\u{{\frak u}} 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 EvensLu 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 EvensLu 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 EvensLu Poisson structure are displayed in several examples. Keywords: Homogeneous poisson structures, symmetric spaces, momentum map. MSC: 53D17, 53D20, 53C35 [ Fulltextpdf (298 KB)] for subscribers only. 