Courant, Ted; Weinstein, Alan 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. Cited in 8 ReviewsCited in 56 Documents 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.) Keywords:Poisson structure; presymplectic structure; Integrability conditions for Dirac structures Citations:Zbl 0634.00019 × Cite Format Result Cite Review PDF