Adjoint orbits of sl(2, R) and their geometry

Abstract

Let SL(2,R) be the special linear group and sl(2, R) its Lie algebra. We study geometric properties associated to the adjoint orbits. In particular, we show that just three possibilities arise: either the adjoint orbit is a one-sheeted hyperboloid, or a two-sheeted hyperboloid, or else a cone. In addition, we introduce a speci?c potential and study the corresponding gradient vector ?eld and its dynamics when we restrict to the adjoint orbit. We also describe the symplectic structure on these adjoint orbits coming from the well known Kirillov-Kostant-Souriau symplectic form on coadjoint orbits.