On the symplectic reductions. (Q1848456)

The main results of this paper concern symplectic reduction methods for symplectic G-spaces and for Poisson actions of Poisson Lie groups on symplectic manifolds, without using momentum mappings. More precisely the authors prove that if we have a symplectic action of a Lie group \(G\) on a symplectic manifold \(M\), for each regular point \(x\in M\) of rank \(t\), there is a connected \((n-t)\)-dimensional submanifold \(N_x\) of \(M\) through \(x\) such that the tangent space of \(N_x\) in each point is the symplectic orthogonal complement space of the tangent of the \(G\)-orbit at this point. If one assumes that the subgroup \(K\) of \(G\) that preserves \(N_x\) acts freely and properly, then there is a unique symplectic form on the quotient space \(K/N_x\) such that this space is the symplectic reduced phase space of \(M\). A similar result is obtained for Poisson actions of a connected Poisson Lie group on a symplectic manifold.