Manchester Geometry Seminar 1999/2000

Formally, a Poisson structure on a manifold M is a Lie algebra structure on the real vector space of smooth functions on M satisfying, additionally, the Leibniz rule with respect to the ordinary multiplication.

Poisson geometry has three principal sources:

1. It is a more convenient context in which to study many aspects of symplectic geometry.
2. It is the theory dual to Lie algebra theory and, more generally, Lie algebroid theory.
3. It is a (semi-)classical limit of quantum theory.

Every symplectic structure (i.e., a non-degenerate closed 2-form) gives rise to a Poisson bracket. Conversely, a Poisson manifold is foliated by symplectic leaves. Poisson manifolds are natural framework for Hamiltonian mechanics. Many constructions in symplectic geometry destroy the symplectic structure but preserve the Poisson bracket. Put another way, the only morphisms in the symplectic category are the symplectomorphisms, which are diffeomorphisms, but the obvious concept of Poisson map allows a much greater flexibility and power. The dual to every Lie algebra g has a natural Poisson structure. In this example the bracket of two linear functions is linear and this characterizes duals of Lie algebras amongst all Poisson structures. Here the symplectic leaves are precisely the coadjoint orbits of any Lie group which integrates g. There is a lot of evidence that Lie actually thought more in terms of these duals than in terms of what we call a Lie algebra. A Poisson bracket is the first term in the power series of a deformation quantization. Also, Poisson-Lie groups are very definitely a good approximation to quantum groups. (Both objects, in particular, arise in integrable systems.)

A "Poisson reduction" is a procedure which allows to decrease the order of a Hamiltonian system with symmetries by constructing a new Poisson bracket on a manifold of smaller dimension. (Elementary example is obtaining the Fubini-Studi form on CPⁿ from the canonical bracket on Cⁿ⁺¹.) In the talk we describe a single framework which unifies classical Marsden-Weinstein reduction with various approaches which have been made in recent years to reduction for Poisson group actions. In particular we show how the description of Poisson homogeneous spaces due to Drinfeld and developed by Lu arises naturally from a consideration of double structures.

Basic notions of Poisson geometry and double structures will be included. Look for more details here (PDF file)...