COMPACT SYMMETRIC SPACES, TRIANGULAR FACTORIZATION, AND CAYLEY COORDINATES

作     者：Habermas, Derek 

作者机构：SUNY Coll Potsdam Dept Math Potsdam NY 13676 USA 

出 版 物：《PACIFIC JOURNAL OF MATHEMATICS》 (Pac. J. Math.)

年 卷 期：2011年第253卷第1期

页      面：57-73页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：compact symmetric space triangular factorization ldu factorization Bruhat decomposition Cayley map Cayley coordinates symplectic leaves compute computation concrete classical connected component Cartan embedding antidiagonal antitranspose 

摘      要：Let U/K represent a connected, compact symmetric space, where theta is an involution of U that fixes K, phi : U/K - U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection of phi(U/K) with the Bruhat decomposition of G corresponding to a theta-stable triangular, or LDU, factorization of the Lie algebra of G. When g is an element of phi(U/K) is generic, the corresponding factorization g = ld(g)u is unique, where l is an element of N-, d(g) is an element of H, and u is an element of N+. We present an explicit formula for d in Cayley coordinates, compute it in several types of symmetric spaces, and use it to identify representatives of the connected components of the generic part of phi(U/K). This formula calculates a moment map for a torus action on the highest dimensional symplectic leaves of the Evens-Lu Poisson structure on U/K.