nLab Heisenberg Lie n-algebra

Idea

The notion of Heisenberg Lie $n$-algebra is the generalization to n-plectic geometry of the notion of Heisenberg Lie algebra in symplectic geometry.

The Heisenberg Lie $n$-algebra integrates to the Heisenberg n-group.

Definition

We discuss a generalization of the notion of Heisenberg Lie algebra from ordinary symplectic geometry to a notion of Heisenberg Lie n-algebra in higher geometric quantization of n-plectic geometry.

The following definition is naturally motivated from the fact that:

1. The ordinary Heisenberg Lie algebra is the sub-Lie algebra of the

Poisson bracket Lie algebra, the one underlying the corresponding Poisson algebra (see below) on the constant and linear functions.

1. The generalization of Poisson brackets to Poisson Lie n-algebras in n-plectic geometry for all $n$ is established (see there).

In view of this, the following definition takes the Heisenberg Lie $n$-algebra to be the sub-Lie $n$-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.

Definition

Let $n \in \mathbb{N}$, let $(V, \omega)$ be an n-plectic vector space.

The corresponding $n$-plectic manifold is the n-plectic manifold $(V, \mathbf{\omega})$, with $V$ now the canonical smooth manifold structure on the given vector space, and with

$$\mathbf{\omega} \in \Omega^{n+1}(V)$$

the differential form obtained by left (right) translating $\omega$ along $V$.

Explicitly, for all vector fields $\{v_i \in \Gamma(T V)\}_{i = 1}^n$ and all points $x \in V$ we set

$$\mathbf{\omega}_x(v_1, \cdots, v_n) := \omega(v_1(x), \cdots, v_n(x)) \,.$$

Here on the right – and in all of the following – we are using that every tangent space $T_x V$ of $V$ is naturally identified with $V$ itself

$$T_x V \stackrel{\simeq}{\to} V \,.$$

Definition

Let $n \in \mathbb{N}$, let $(V, \omega)$ be an n-plectic vector space and let $(V, \mathbf{\omega})$ be the corresponding n-plectic manifold.

The Heisenberg Lie $n$-algebra $Heis(V,\omega)$ is the sub-Lie n-algebra of the Poisson Lie n-algebra $\mathcal{P}(V, \omega)$ on those differential forms which are either linear or constant (with respect to left/right translation on $V$).

All one has to observe is:

Proposition

This is indeed a sub-Lie $n$-algebra.

Proof

We need to check that the linear and constant forms are closed under the L-infinity algebra brackets of $\mathcal{P}(V, \omega)$.

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 $\mathbf{\omega}$ with the Hamiltonian vector fields of linear or constant forms. Since $\mathbf{\omega}$ 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 $\mathbf{\omega}$ gives a constant form.

Related concepts

• the universal enveloping E-n algebra of a Heisenberg Lie n-algebra deserves to be called a Weyl 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 $\mathbb{G}$-principal ∞-connection)

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

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\infty$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega \mathbb{G})$-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 topological part of the Heisenberg Lie 2-algebra of the string sigma-model called the WZW model has been discussed (not under this name, though) in

• John Baez, Chris Rogers, Categorified Symplectic Geometry and the String Lie 2-Algebra. arXiv:0901.4721.

and shown to be the string Lie 2-algebra.

General 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)