nLab
Heisenberg Lie n-algebra

Context

Symplectic geometry

Geometric quantization

Contents

Idea

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

The Heisenberg Lie nn-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.

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

In view of this, the following definition takes the Heisenberg Lie nn-algebra to be the sub-Lie nn-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 nn \in \mathbb{N}, let (V,ω)(V, \omega) be an n-plectic vector space.

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

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

the differential form obtained by left (right) translating ω\omega along VV.

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

ω x(v 1,,v n):=ω(v 1(x),,v n(x)). \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 xVT_x V of VV is naturally identified with VV itself

T xVV. T_x V \stackrel{\simeq}{\to} V \,.
Definition

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

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

All one has to observe is:

Proposition

This is indeed a sub-Lie nn-algebra.

Proof

We need to check that the linear and constant forms are closed under the L-infinity algebra brackets of 𝒫(V,ω)\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 ω\mathbf{\omega} with the Hamiltonian vector fields of linear or constant forms. Since ω\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 ω\mathbf{\omega} gives a constant form.

slice-automorphism ∞-groups in higher prequantum geometry

cohesive ∞-groups:Heisenberg ∞-group\hookrightarrowquantomorphism ∞-group\hookrightarrow∞-bisections of higher Courant groupoid\hookrightarrow∞-bisections of higher Atiyah groupoid
L-∞ algebras:Heisenberg L-∞ algebra\hookrightarrowPoisson L-∞ algebra\hookrightarrowCourant L-∞ algebra\hookrightarrowtwisted vector fields

higher and integrated Kostant-Souriau extensions

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

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

and shown to be the string Lie 2-algebra.

Revised on February 20, 2013 21:43:25 by Urs Schreiber (80.81.16.253)