A family of Poisson non-compact symmetric spaces

A Poisson Structure on Compact Symmetric Spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.

Compact Symmetric Spaces, Triangular Factorization, and Poisson Geometry

LetX 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. Each u-compatible triangular decomposition g = n − + h + n+ determines a Poisson Lie group structure πU on U . The Evens-Lu construction ([EL]) produces a (U, πU )-homoge...

A Poisson Structure on Compact Symmetric Spacesphilip

A Note on Poisson Symmetric Spaces

We introduce the notion of a Poisson symmetric space and the associated infinitesimal object, a symmetric Lie bialgebra. They generalize corresponding notions for Lie groups due to V. G. Drinfel’d. We use them to give some geometric insight to certain Poisson brackets that have appeared before in the literature. 1 Motivation Let us recall briefly the best-known examples of Poisson manifolds. Th...

Unitarity of Strings and Non-compact Hermitian Symmetric Spaces

If G is a simple non-compact Lie Group, withK its maximal compact subgroup, such that K contains a one-dimensional center C, then the coset space G/K is an Hermitian symmetric non-compact space. SL(2,R)/U(1) is the simplest example of such a space. It is only when G/K is an Hermitian symmetric space that there exists unitary discrete representations of G. We will here study string theories defi...