## 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.)

Zbl 0634.00019