# Compact symmetric spaces, triangular factorization, and Cayley coordinates - Mathematics > Representation Theory

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Abstract: Let U-K represent a connected, compact symmetric space, where theta is aninvolution of U that fixes K, phi: U-K to U is the geodesic Cartan embedding,and G is the complexification of U. We investigate the intersection of phiU-Kwith the Bruhat decomposition of G corresponding to a theta-stable triangular,or LDU, factorization of the Lie algebra of G. When g in phiU-K is generic,the corresponding factorization g=ldgu is unique, where l in N^-, dg in H,and u in N^+. In this paper we present an explicit formula for d in Cayleycoordinates, compute it in several types of symmetric spaces, and use it toidentify representatives of the connected components of the generic part ofphiU-K. This formula calculates a moment map for a torus action on thehighest dimensional symplectic leaves of the Evens-Lu Poisson structure on U-K.

Author: ** Derek Habermas**

Source: https://arxiv.org/