The openness condition for a coadjoint orbit projection of the semidirect product Lie group M((n, p), ℝ) ⋊ GL (n, ℝ)

Abstract

Let G(n, p) be the semidirect product Lie group of the vector space K : = M((n,p),ℝ) of n × p real matrices and the Lie group L : = GL(n,ℝ) of n × n real invertible matrices. Moreover, we denote by g(n, p) the Lie algebra of G(n, p) whose the dual vector space is g*(p, n). In this paper, we study the projection of a coadjoint orbit of G(n, p) from g*(p, n) to K*. The main purpose is to give necessary and sufficient conditions for the openness of a coadjoint orbit projection. In this research, we applied the study literature method by studying the openness of a coadjoint orbits. As the main result, we proved the openness condition for coadjoint orbits projections in K*. For the future research, the openness of coadjoint orbits of G(n, p) still needs to be investigated more.