Dual vector fields ii: calculating the Jacobian
Given a Lie algebra with a chosen basis, the change of coordinates relating coordinates of the first and second kinds near the identity of the corresponding local group yields some remarkable vector fields and dual vector fields. One family of vector fields is dual to a representation of the Lie algebra acting on a Fock-type space. To this representation an abelian family of dual vector fields is associated. The exponential of these commuting operators acting on an appropriate vacuum yields the same result as does the local group element generated by the nonabelian Lie algebra. Another family of dual vector fields gives a representation of the Lie algebra, yet acting on an appropriate vacuum, yields the same result as an abelian Lie algebra. An essential component of these constructions is the Jacobian of the change of coordinates. Here we present a formula for this Jacobian using the pi-matrices that play a fundamental rôle in our approach to representations of Lie algebras.