Neeb, Karl-Hermann Lie algebra extensions and higher order cocycles. (English) Zbl 1105.53064 J. Geom. Symmetry Phys. 5, 48-74 (2006). Summary: In this note we present an abstract approach, based on Lie algebra cohomology, to the Lie algebra extensions associated to symplectic manifolds. We associate to any Lie algebra cocycle of degree at least two an abelian extension by some space \(\mathfrak a\) and central extensions of subalgebras analogous to the Lie algebras of symplectic, respectively, Hamiltonian vector fields. We even obtain a Poisson bracket on \(\mathfrak a\) compatible with the Hamiltonian Lie subalgebra. We then describe how this general approach provides a unified treatment of cocycles defined by closed differential forms on Lie algebras of vector fields on possibly infinite dimensional manifolds. Cited in 1 ReviewCited in 3 Documents MSC: 17B56 Cohomology of Lie (super)algebras 17B63 Poisson algebras 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B66 Lie algebras of vector fields and related (super) algebras Keywords:Lie algebra cohomology; symplectic manifolds; Poisson bracket × Cite Format Result Cite Review PDF