nLab
Poisson bracket Lie n-algebra

Idea

The Lie n-algebra that generalizes the Poisson bracket from symplectic geometry to n-plectic geometry: the Poisson bracket $L_{\mathrm{\infty }}$-algebra of local observables in higher prequantum geometry.

The detailed discussion is currently still here at n-plectic geometry.

Related concepts

• Poisson n-algebra

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:	Heisenberg ∞-group	$\hookrightarrow$	quantomorphism ∞-group	$\hookrightarrow$	∞-bisections of higher Courant groupoid	$\hookrightarrow$	∞-bisections of higher Atiyah groupoid
L-∞ algebras:	Heisenberg L-∞ algebra	$\hookrightarrow$	Poisson L-∞ algebra	$\hookrightarrow$	Courant L-∞ algebra	$\hookrightarrow$	twisted vector fields

higher and integrated Kostant-Souriau extensions

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $𝔾$-principal ∞-connection)

$$(\Omega 𝔾)\mathrm{FlatConn}(X)\to \mathrm{QuantMorph}(X,\nabla )\to \mathrm{HamSympl}(X,\nabla )$$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\mathrm{\infty }$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega 𝔾)$-flat ∞-connections on $X$	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
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)

References

The Poisson bracket $L_{\mathrm{\infty }}$-algebras were proposed in

• Chris Rogers, $L_{\mathrm{\infty }}$ algebras from multisymplectic geometry , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).

A comprehensive account is in

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)

An general account in higher geometry is in

• Higher geometric prequantum theory