THE EXPLICIT FORMULAS OF PARAMETRIZATION OF COADJOINT ORBITS OF THE HEISENBERG LIE GROUP

Abstract

This research focuses on the Heisenberg Lie group. The aim is to determine the coadjoint orbits and their parametrizations. The method used in this research involves constructing the parametrization of coadjoint orbit for Heisenberg Lie group corresponding to the Heisenberg Lie algebra of dimension 2n+1. Furthermore, the obtained results are specialized to the cases of n=1, 2, and 3 which  correspond to the Heisenberg Lie algebras of dimensions 3, 5, and 7. The main results are the explicit formulas of coadjoint orbits  for the Heisenberg Lie group H_1, H_2, and H_3 which are expressed by the equations (〖Ad〗^* H_1 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^',γ^'∈R}, (〖Ad〗^* H_2 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^'∈R^2,γ^'∈R}, and (〖Ad〗^* H_3 ) l_(α,β,γ)={l_(α^',β^',γ^' ):α^',β^'∈R^3,γ^'∈R}.  In addition, their associated parametrizations are  given by the explicit formulas ψ(γZ^*,u)=∑_(i=1)^n▒(u_i X_i^*+u_(n+i) Y_i^* ) +γZ^* for n=1, 2, and 3. As a further study, various types of Lie groups can be explored to determine coadjoint orbits and their parametrization. Two Lie groups that are interesting to investigate further regarding their coadjoint orbits and parametrization are the diamond and Jacobi groups.