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Weinstein, Alan

The modular automorphism group of a Poisson manifold. (English) Zbl 0902.58013

J. Geom. Phys. 23, No. 3-4, 379-394 (1997).
Author’s abstract: “The modular flow of a Poisson manifold is a one-parameter group of automorphisms determined by the choice of a smooth density on the manifold. When the density is changed, the generator of the group changes by a Hamiltonian vector field, so one has a one-parameter group of “outer automorphisms” intrinsically attached to any Poisson manifold. The group is trivial if and only if the manifold admits a measure which is invariant under all Hamiltonian flows.
The notion of modular flow in Poisson geometry is a classical limit of the notion of modular automorphism group in the theory of von Neumann algebras. In addition, the modular flow of a Poisson manifold is related to modular cohomology classes for associated Lie algebroids and symplectic groupoids. These objects have recently turned out to be important in Poincaré duality theory for Lie algebroids”.
Reviewer: J.Koiller (Rio de Janeiro)

Cited in 5 Reviews
Cited in 96 Documents

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B99 Lie algebras and Lie superalgebras

Keywords:

modular automorphism group; modular flow; Poisson manifold
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References:

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