On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations

We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of
$\text{sl}{(2,\mathbb{R})}$
. The reduction process is a bi-Hamiltonian reduction that can be canonically performed on every bi-Hamiltonian manifold.