The relation between local and global dual pairs
نویسندگان
چکیده
In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of the extra conditions. 1 Regular dual pairs The set-up we consider is the following. Let (M,ω) be a symplectic manifold (we assume our manifolds to be paracompact), (P1, {·, ·}1) and (P2, {·, ·}2) two Poisson manifolds, and π1 : M → P1 and π2 : M → P2 two surjective submersive Poisson maps. (In Remark 1.7 we describe the effect of replacing the condition “submersion” by “open”.) Let Fj denote the algebra of the pull–backs of smooth functions on Pj , that is, Fj = π ∗ j (C ∞(Pj)). Since π1 and π2 are Poisson it follows that F1 and F2 are Poisson subalgebras of C(M). If U ⊂ M is open, we write Fj(U) for the algebra Fj(U) = π ∗ j (C ∞(πj(U))). This is a Poisson subalgebra of C(U). For a subset A ⊂ C(U) we write A for the centralizer of A with respect to the Poisson structure on (U,ω|U ), that is, A := {f ∈ C(U) | {f, g} = 0 for all g ∈ A}, where {·, ·} is the restriction of the Poisson bracket to U . Department of Mathematics. UMIST. PO Box 88, Manchester M60 1QD, United Kingdom. [email protected]. Département de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16 route de Gray, 25030 Besançon cédex, France. [email protected]. Centre Bernoulli, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland. [email protected].
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