  This paper is a rigorous study of the dual pair structure of the ideal fluid
and the dual pair structure for the $n$-dimensional Camassa-Holm (EPDiff)
equation, including the proofs of the necessary transitivity results. In the
case of the ideal fluid, we show that a careful definition of the momentum maps
leads naturally to central extensions of diffeomorphism groups such as the
group of quantomorphisms and the Ismagilov central extension.
