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

TL;DR

This paper explores left-invariant distributions and metric Hamiltonians on SL(n,R) derived from the Killing form, analyzing their geometric structures, Poisson relations, and explicit solutions for the case n=2.

Contribution

It constructs and studies geometric structures on SL(n,R) induced by the Killing form, including Hamiltonian systems and their properties, with explicit solutions for n=2.

Findings

Identified eigenvector basis for the Gram matrix of the Killing form.

Established sub-pseudo-Riemannian structures on distributions.

Provided explicit solutions for Hamiltonian systems when n=2.

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 $n=2$.