2020/04/25 by Francisco Rubilar, Rubilar, Francisco, Leonardo Schultz +1
Mathematics · #53A05 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.DG #math.RT #msc:53A05
paper · pdf · doi:10.48550/arxiv.2004.12180
4 figures. To appear in Pro Mathematica
arxiv created 2020/04/25 · openalex publication_date 2020/04/25 · arxiv updated 2020/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let SL(n,ℝ) be the special linear group and \mathfraksl(n,ℝ) its Lie algebra. We study geometric properties associated to the adjoint orbits in the simplest non-trivial case, namely, those of \mathfraksl(2,ℝ). 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 specific potential and study the corresponding gradient vector field and its dynamics when we restricted to the adjoint orbit. We conclude by describing the symplectic structure on these adjoint orbits coming from the well known Kirillov-Kostant-Souriau symplectic form on coadjoint orbits.