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A sufficient condition for reduction by stages: Proof and applications. (English. Russian original) Zbl 1160.37022

Sb. Math. 199, No. 5, 663-671 (2008); translation from Mat. Sb. 199, No. 5, 35-44 (2008).
The problem of symplectic reduction is stated as follows: given a symplectic action of a Lie group on a symplectic manifold having a moment map, take the quotient of a level set of the moment map by the action of a suitable subgroup to form a new symplectic manifold. The problem of implementing reduction by stages (MMOPR-reduction) was put forward [J. E. Marsden, G. Misiołek, J.-P. Ortega, M. Perlmutter, T. S. Ratiu, Hamiltonian reduction by stages. Lecture Notes in Mathematics 1913. Berlin: Springer. (2007; Zbl 1129.37001)] in the following form: starting with a big group \(M\) with normal subgroup \(N\) and trying to reduce first by \(N\) and then by some quotient group \(M_{\nu}/N_{\nu}, M_{\nu}\subset M, N_{\nu}\subset N\), would give the right framework for the theory of reduction by stages. It was showed that while the corresponding question for Poisson reduction is quite simple, the symplectic case is not so easy. The authors modify the MMOPR-reduction by stages to solve this problem in another way. They replace the quotient group \(M_{\nu}/N_{\nu}\) by the group \(M_{\nu}\), considering the action of \(M_{\nu}\) on the first reduced space bearing in the mind that the action of the subgroup \(N_{\nu}\subset M_{\nu}\) on this space is trivial. This action of \(M_{\nu}\) has a natural equivariant moment map and admits the usual MW-reduction procedure. The resulting manifold (the second reduced space) is the same as in the MMOPR-reduction by stages theorem basing on this fact is shorter and clearer. Weaker assumptions are used to carry out the reduction by stages procedure. In particular the normal subgroup \(N\subset M\) is not necessarily closed and the first reduced space is not necessarily connected.

MSC:

37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
53D20 Momentum maps; symplectic reduction

Citations:

Zbl 1129.37001
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