2024/10/20 by Abraham Bobadilla Osses, Osses, Abraham Bobadilla, Mauricio Godoy Molina +1
Mathematics · #15B30 #17B20 #53C17 #53C50 #Advanced Algebra and Geometry #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Rings and Algebras (math.RA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2410.15478
openalex publication_date 2024/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
From the classical theory of Lie algebras, it is well-known that the bilinear form B(X,Y)=\rm tr(XY) defines a non-degenerate scalar product on the simple Lie algebra \mathfraksl(n,\mathbb R). Diagonalizing the Gram matrix Gr associated with this scalar product we find a basis of \mathfraksl(n,\mathbb R) of eigenvectors of Gr which produces a family of bracket generating distributions on \rm SL(n,\mathbb R). Consequently, the bilinear form B defines sub-pseudo-Riemannian structures on these distributions. Each of these geometric structures naturally carries a metric quadratic Hamiltonian. In the present paper, we construct in detail these manifolds, study Poisson-commutation relations between different Hamiltonians, and present some explicit solutions of the corresponding Hamiltonian system for n=2.