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0742.58016
Arms,
Judith M.; Cushman,
Richard H.; Gotay,
Mark J.
A universal reduction procedure for Hamiltonian group actions. (English)
[CA] The geometry of Hamiltonian systems, Proc. Workshop, Berkeley/CA (USA)
1989, Math. Sci. Res. Inst. Publ. 22, 33-51 (1991).
[For the entire collection see Zbl
0733.00016.]\par This is an important contribution to the theory of
reduction through a Hamiltonian group action (of a Lie group $G$) on a symplectic
manifold $(M,\omega)$, which generalizes the well-known Marsden-Weinstein
procedure to the case of singularities of the momentum mapping $J$. It is
first observed that for each $\mu\in J(M)$, there is a homeomorphism of
the reduced space $M\sb \mu=J\sp{-1}(\mu)/G\sb \mu$ onto $\Pi(J\sp{- 1}(\mu))$,
where $\Pi$ is the projection $M\to M/G$. The space $M/G$ being naturally
equipped with the structure of a Poisson variety, it is proved that each
$M\sb \mu$ inherits such a structure from $M/G$. For singular values $\mu$,
the reduced Poisson algebra may generally contain nontrivial Casimirs and
$M\sb \mu$ need not be a finite union of symplectic leaves of $M/G$. It
is further shown that this degeneracy of the reduced Poisson algebra does
not occur when the action of $G$ is proper, and that the singularities of
$J\sp{-1}(\mu)$ then are quadratic. If in addition the Marsden-Weinstein
reduction applies, it is shown to coincide with the universal reduction
discussed here. The final sections of the paper deal with the interesting
special case of a compact Lie group and a fully worked out illustration
of the theory with the spherical pendulum.
[ W.Sarlet
(Gent) ]
Keywords: Hamiltonian group actions; Poisson structure; reduction