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Left-invariant distributions and metric Hamiltonians on ${\rm SL}(n,{\mathbb R})$ induced by its Killing form

Abraham Bobadilla Osses, Mauricio Godoy Molina

Abstract

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 ${\mathfrak{sl}}(n,{\mathbb R})$. Diagonalizing the Gram matrix $Gr$ associated with this scalar product we find a basis of ${\mathfrak{sl}}(n,{\mathbb R})$ 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 $n=2$.

Left-invariant distributions and metric Hamiltonians on ${\rm SL}(n,{\mathbb R})$ induced by its Killing form

Abstract

From the classical theory of Lie algebras, it is well-known that the bilinear form defines a non-degenerate scalar product on the simple Lie algebra . Diagonalizing the Gram matrix associated with this scalar product we find a basis of of eigenvectors of which produces a family of bracket generating distributions on . Consequently, the bilinear form 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 .

Paper Structure

This paper contains 9 sections, 9 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Concerning ${\rm SL}(n,{\mathbb R})$
  4. Non-degenerate metrics on bracket generating distributions
  5. General framework
  6. Hamiltonian formalism
  7. Coordinate description of the metric Hamiltonian
  8. Commuting Hamiltonians
  9. Solutions of \ref{['eq:Hamsys']} for $n=2$

Key Result

Proposition 3.1

The Gram matrix $Gr$ of the bilinear form $B$ in the ordered basis ${\mathscr{B}}$ is the $(n^2-1)\times(n^2-1)$ block diagonal matrix in which the $2\times 2$ block $J=$ appears $\dfrac{n(n-1)}{2}$ times.

Theorems & Definitions (18)

  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • proof
  • Example 3.6
  • ...and 8 more