Extensions of the Poisson bracket to differential forms and multi-vector fields. (English) Zbl 0748.58008

Let \((M\{.,.\})\) be a Poisson manifold, where the Poisson bracket \(\{.,.\}\) is defined on the algebra of smooth functions on \(M\). This theory as well as its extensions are presented by A. Weinstein in J. Differ. Geom. 18, 523-557 (1983; Zbl 0524.58011).
Let \(\Lambda^*(M)\) denote the differential exterior algebra of \(M\). The authors extend the notion of Poisson bracket to the space \(\Lambda^*(M)/d\Lambda^*(M)\), of so-called co-exact forms on \(M\). For that extended bracket notion the authors give the basic constructions and formulae of the standard Hamiltonian formalism. By duality the Poisson bracket is extended also to the space of co-exact multi-vector fields. They define the graded Lie algebra homomorphisms connecting these extended brackets.


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58A15 Exterior differential systems (Cartan theory)
17B70 Graded Lie (super)algebras


Zbl 0524.58011
Full Text: DOI


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