The following definition is naturally motivated from the fact that:
In view of this, the following definition takes the Heisenberg Lie -algebra to be the sub-Lie -algebra of the Poisson Lie n-algebra on the linear and constant differential forms.
First we need the following definition, which is elementary, but nevertheless worth making explicit once.
Let , let be an n-plectic vector space.
the differential form obtained by left (right) translating along .
Explicitly, for all vector fields and all points we set
Here on the right – and in all of the following – we are using that every tangent space of is naturally identified with itself
All one has to observe is:
This is indeed a sub-Lie -algebra.
We need to check that the linear and constant forms are closed under the L-infinity algebra brackets of .
The only non-trivial such brackets are the unary one, and the ones on elements all of degree 0.
The unary bracket is given by the de Rham differential. Since this sends a linear form to a constant form and a constant form to 0, our sub-complex is closed under this.
Similarly, the brackets on elements all in degree 0 is given by contraction of with the Hamiltonian vector fields of linear or constant forms. Since is a constant form, and since the de Rham differential of a linear or constant form is constant (or even 0), these Hamiltonian vector fields are necessarily constant. Hence their contraction with gives a constant form.
|cohesive ∞-groups:||Heisenberg ∞-group||quantomorphism ∞-group||∞-bisections of higher Courant groupoid||∞-bisections of higher Atiyah groupoid|
|L-∞ algebras:||Heisenberg L-∞ algebra||Poisson L-∞ algebra||Courant L-∞ algebra||twisted vector fields|
higher and integrated Kostant-Souriau extensions:
|geometry||structure||unextended structure||extension by||quantum extension|
|higher prequantum geometry||cohesive ∞-group||Hamiltonian symplectomorphism ∞-group||moduli ∞-stack of -flat ∞-connections on||quantomorphism ∞-group|
|1||symplectic geometry||Lie algebra||Hamiltonian vector fields||real numbers||Hamiltonians under Poisson bracket|
|1||Lie group||Hamiltonian symplectomorphism group||circle group||quantomorphism group|
|2||2-plectic geometry||Lie 2-algebra||Hamiltonian vector fields||line Lie 2-algebra||Poisson Lie 2-algebra|
|2||Lie 2-group||Hamiltonian 2-plectomorphisms||circle 2-group||quantomorphism 2-group|
|n-plectic geometry||Lie n-algebra||Hamiltonian vector fields||line Lie n-algebra||Poisson Lie n-algebra|
|smooth n-group||Hamiltonian n-plectomorphisms||circle n-group||quantomorphism n-group|
(extension are listed for sufficiently connected )
and shown to be the string Lie 2-algebra.
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)