Note that in the second example, the standard inner product onR^3 provides an
SO(3)invariant identification of the Lie algebra with its dual, and as a result
the adjoint and co-adjoint actions are much the same. In the case ofH 3 , there
is no invariant inner product onh 3 , so the adjoint and co-adjoint actions are
rather different, explaining the different actions ofH 3 onMandMdescribed
earlier.
15.6 For further reading
For a general discussion of vector fields onRn, see [48]. See [2], [8] and [13] for
more on Hamiltonian vector fields and the moment map. For more on the duals
of Lie algebras and co-adjoint orbits, see [9] and [51].