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Beyond Poisson structures. (English) Zbl 0698.58020

Actions hamiltoniennes de groupes. Troisième théorème de Lie, Sémin. Sud-Rhodan. Géom. VIII, Lyon/France 1986, Trav. Cours 27, 39-49 (1988).
[For the entire collection see Zbl 0634.00019.]
The notions of Poisson, symplectic, and presymplectic structures on a manifold are generalized. A Poisson structure on a manifold X can be viewed as a skew endomorphism \(T^*X\to TX\) satisfying certain integrability properties. A presymplectic structure is an endomorphism \(TX\to T^*X\), and a symplectic structure is an invertible presymplectic structure. The graphs of these endomorphisms are sub-bundles \(L\subset TX\oplus T^*X\), which are maximally isotropic. A Dirac structure is defined to be such a sub-bundle L. Thus at some points the manifold may be Poisson, at some presymplectic, and at some neither. Integrability conditions for Dirac structures are given. A bracket is defined for local sections of \(L\to X\). When L is integrable the bracket satisfies Jacobi’s identity.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B65 Infinite-dimensional Lie (super)algebras
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Citations:

Zbl 0634.00019