Reduction of Poisson algebras at nonzero momentum values

A reduction of a Poisson manifold using the ideal I(J) generated by the momentum map was introduced by Sniatycki and Weinstein (1983). This reduction has been extended to nonzero momentum values mu by two methods: by shifting to zero momentum on a larger space, the product with the coadjoint orbit; and by the method of Wilbour sind Kimura (1991, 1993)using the modified ideal I(J - mu). It is shown that these two methods produce isomorphic reduced algebras under the assumptions that the symmetry group is connected and that the stabilizer group of mu also is connected. If the latter assumption fails, the shifted reduced algebra is isomorphic to a (possibly proper) subalgebra of the Wilbour-Kimura algebra.