The Lie n-algebra that generalizes the Poisson bracket from symplectic geometry to n-plectic geometry: the Poisson bracket -algebra of local observables in higher prequantum geometry.
More discussion is here at n-plectic geometry.
Applied to the symplectic current (in the sense of covariant phase space theory, de Donder-Weyl field theory) this is the higher current algebra (see there) of conserved currents of a prequantum field theory.
Throughout, Let be a smooth manifold, let a natural number and a closed differential (n+1)-form on . The pair we may regard as a pre-n-plectic manifold.
We define two L-∞ algebras defined from this data and discuss their equivalence. Either of the two or any further one equivalent to the two is the Poisson bracket Lie -albebra of . The first definition is due to (Rogers 10), the second due to (FRS 13b). Here in notation we follow (FRS 13b).
(FRS 13b, def. 2.1.3)
The L-∞ algebra has as underlying chain complex the truncated and modified de Rham complex
with the Hamiltonian pairs, def. 1, in degree 0 and with the 0-forms (smooth functions) in degree , and its non-vanishing -brackets are as follows:
(FRS 13b, prop. 3.1.2)
Let be any Cech-Deligne-cocycle relative to an open cover of , which gives a prequantum n-bundle for . The L-∞ algebra is the dg-Lie algebra (regarded as an -algebra) whose underlying chain complex is
with differential given by (where refers to total complex of the Cech-de Rham double complex).
The non-vanishing dg-Lie bracket on this complex are defined to be
(FRS 13b, def./prop. 4.2.1)
There is an equivalence in the homotopy theory of L-∞ algebras
between the -algebras of def. 2 and def. 3 (in particular def. 3 does not depend on the choice of ) whose underlying chain map satisfies
(FRS 13b, theorem 4.2.2)
The extension theorem
Given a pre n-plectic manifold , then the Poisson bracket Lie -algebra from above is an extension of the Lie algebra of Hamiltonian vector fields , def. 1 by the cocycle infinity-groupoid for ordinary cohomology with real number coefficients in that there is a homotopy fiber sequence in the homotopy theory of L-infinity algebras of the form
where the cocycle , when realized on the model of def. 2, is degreewise given by by contraction with .
This is FRS13b, theorem 3.3.1.
As a corollary this means that the 0-truncation is a Lie algebra extension by de Rham cohomology, fitting into a short exact sequence of Lie algebras
slice-automorphism ∞-groups in higher prequantum geometry
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for -principal ∞-connection)
(extension are listed for sufficiently connected )
The Poisson bracket -algebra was introduced in
Discussion in the broader context of higher differential geometry and higher prequantum geometry is in
Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)
Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)