Universal central extension of the Lie algebra of exact divergence-free vector fields

The differential δ : Γ ⁢ ( ∧ k T ⁢ M ) → Γ ⁢ ( ∧ k − 1 T ⁢ M ) is given by

Proof

The case k = 1 is the definition of divergence. The case 𝑘 follows from k − 1 using

and L X k ⁢ ι X 1 ∧ ⋯ ∧ X k − 1 ⁢ μ = div ⁢ ( X k ) ⁢ ι X 1 ∧ ⋯ ∧ X k − 1 ⁢ μ − ∑ j = 1 k − 1 ι X 1 ∧ ⋯ ∧ [ X j , X k ] ∧ ⋯ ∧ X k − 1 ⁢ μ . ∎

If α = ι X 1 ∧ X 2 ⁢ μ , then X α = δ ⁢ ( X 1 ∧ X 2 ) is given by

The Leibniz bracket on Γ ⁢ ( ∧ 2 T ⁢ M ) is given by

or, equivalently, by

Proof

If α = ( X 1 ∧ X 2 ) ♭ and β = ( Y 1 ∧ Y 2 ) ♭ , then [ α , β ] = L X α ⁢ β = L X α ⁢ ι Y 2 ⁢ ι Y 1 ⁢ μ . Using L X α ⁢ ι Y j = ι [ X α , Y j ] + ι Y j ⁢ L X α for j ∈ { 1 , 2 } and L X α ⁢ μ = 0 , we find

Substituting (A.1) then yields the required result. ∎

Let us consider a precompact cube U × V ⊂ R k × R n for k ≥ 2 and n ≥ 1 . Given a form α ∈ Ω k , 0 ⁢ ( U × V ) such that

• supp ⁡ ( α ) ⊂ U × V is compact,

• ∫ U α = 0 as an element of C ∞ ⁢ ( V ) ,

Proof

First, we consider a slightly smaller cube U ′′ × V ′ with supp ⁡ ( α ) ⊂ U ′′ × V ′ such that U ′′ ⊂ U ′ ⊂ U and V ′ ⊂ V are relatively compact sets. At the same time, we see R k as S k − N , where 𝑁 is the north pole of the 𝑘-sphere.

• The form 𝛼 extends by zero to a form α ̃ ∈ Ω k , 0 ⁢ ( S k × V ) . The class of α ̃ is trivial since ∫ S k α ̃ = ∫ U α = 0 . (Here we implicitly use H ∙ , 0 ⁢ ( S k × V ) = C ∞ ⁢ ( V ) as follows from the above Künneth theorem.) Let β ∈ Ω k − 1 , 0 ⁢ ( S k × V ) be a primitive of α ̃ .

• Let β ̃ = β | S k ∖ U ′′ ̄ × V . We have d M ⁢ β = α ̃ | S k ∖ U ′′ ̄ × V = 0 . Since

  H k − 1 , 0 ⁢ ( S k ∖ U ′′ ̄ × V ) = H k − 1 ⁢ ( S k ∖ U ′′ ̄ ) ⊗ C ∞ ⁢ ( V ) = 0 ,

  the form β ̃ has a d M -potential γ ∈ Ω k − 2 , 0 ⁢ ( S k ∖ U ′′ ̄ × V ) .

• Let ρ 1 ∈ C c ∞ ⁢ ( S k ∖ U ′′ ̄ ) have compact support and be constantly 1 on S k ∖ U ′ . Similarly, let ρ 2 ∈ C c ∞ ⁢ ( V ) have compact support and be constantly 1 on a neighbourhood of V ′ . We set δ = ρ 2 ⋅ ( β − d M ⁢ ( ρ 1 ⁢ γ ) ) .

Theorem C.1 (Peetre [29])

Let E , F be vector bundles over 𝑀 and let

be a support-decreasing linear map. Then there exists a discrete set Λ ⊂ M such that P | M ∖ Λ is continuous. Moreover, the restriction of 𝑃 to M ∖ Λ is a differential operator of locally finite order: for any p ∈ M ∖ Λ , there exist a chart ( U , x ) and frame { e i } of E U and finitely many nonzero distributions T i σ ⃗ ∈ Γ c ⁢ ( F | U ) ′ such that

for all s = ∑ s i ⁢ e i ∈ Γ c ⁢ ( E | U ) .

Proof

Let 𝒱 be a locally finite cover of 𝑀 such that 𝐸 and 𝐹 trivialise over every V ∈ V . Let e i and f j be the corresponding C ∞ ⁢ ( V ) -bases of Γ ⁢ ( E | V ) and Γ ⁢ ( F | V ) , and write s = ∑ i s i ⁢ e i and t = ∑ j t j ⁢ e j for sections of E | V and F | V , respectively. Then

is a support-decreasing linear map P i ⁢ j : C c ∞ ⁢ ( V ) → C c ∞ ⁢ ( V ) ′ . By the original result [29], there exists a discrete set Λ i ⁢ j V ⊆ V such that P i ⁢ j is continuous on V ∖ Λ i ⁢ j V , and every p ∈ V ∖ Λ i ⁢ j V admits a neighbourhood U i ⁢ j such that

for all f , g ∈ C c ∞ ⁢ ( U i ⁢ j ∖ Λ i ⁢ j V ) . Then 𝑃 is continuous on M ∖ Λ for the discrete set Λ = ⋃ Λ i ⁢ j V . For p ∈ M ∖ Λ , we can find a coordinate neighbourhood U = ⋂ U i ⁢ j such that, with

the result follows with T i σ ⃗ ⁢ ( t ) = ∑ j T i ⁢ j σ ⃗ ⁢ ( t j ) for t ∈ Γ c ⁢ ( F | U ) . ∎

We call a support-decreasing linear map P : Γ c ⁢ ( E ) → Γ c ⁢ ( F ) ′ a differential operator (of locally finite order) if Λ = ∅ , i.e. if 𝑃 is continuous.

Let P : Γ c ⁢ ( E ) → Γ c ⁢ ( F ) ′ be a differential operator. Then 𝑃 induces a continuous morphism of sheaves E E → D F ′ , where E E ⁢ ( U ) = Γ ⁢ ( E | U ) and D F ′ ⁢ ( U ) = Γ c ⁢ ( F | U ) ′ . In particular, it induces a continuous map Γ ⁢ ( E ) → Γ c ⁢ ( F ) ′ .

Proof

This is stated in [29]; however, we will provide here a short explanation. First of all, for any U ⊂ M , 𝑃 can be restricted to a continuous operator P U : Γ c ⁢ ( E | U ) → Γ c ⁢ ( F | U ) ′ by the natural extension Γ c ⁢ ( E | U ) → Γ c ⁢ ( E ) and restriction Γ c ⁢ ( F ) ′ → Γ c ⁢ ( F | U ) ′ . The resulting operator is still support-decreasing, so we only need to show extendibility from Γ c ⁢ ( E | U ) to Γ ⁢ ( E | U ) .

Given s ∈ Γ ⁢ ( E | U ) and compact K ⊆ U , any f K ∈ C c ∞ ⁢ ( U ) with f K | K = 1 yields the same continuous linear functional P U ⁢ ( f K ⁢ s ) ∈ Γ K ⁢ ( F | U ) ′ , where Γ K denotes sections with support in 𝐾. This defines an element in the continuous linear dual of the locally convex injective limit Γ c ⁢ ( F | U ) = lim → ⁡ Γ K ⁢ ( F | U ) , denoted again by P U ⁢ ( s ) . We get a support-decreasing operator P U : Γ ⁢ ( E | U ) → Γ c ⁢ ( F | U ) ′ that extends the originally given one. It is continuous since, for every compact K ⊆ M , the composition of P U with the projection Γ c ⁢ ( F | U ) ′ → Γ K ⁢ ( F | U ) ′ is continuous. ∎

Let U ⊂ R n be connected and open, and let E → U be a vector bundle. For k ≥ 1 , let D : Ω c k ⁢ ( U ) → E = Γ c ⁢ ( E ) ′ be a differential operator (of locally finite order) with D ⁢ d = 0 . Then there exists a differential operator Q : Ω c k + 1 ⁢ ( U ) → E such that D = Q ⁢ d ,

Proof

First, note that 𝐷 uniquely extends from Ω c k ⁢ ( U ) to Ω k ⁢ ( U ) by Lemma C.3. We will work with this extension and note that it vanishes on all exact forms d ⁢ Ω k − 1 ⁢ ( U ) , since Ω c k − 1 ⁢ ( U ) is dense in Ω k − 1 ⁢ ( U ) .

Let Ω ℓ k ⁢ ( U ) denote the space of differential 𝑘-forms with polynomial coefficients of degree ℓ and Ω ≤ ℓ k ⁢ ( U ) the space of differential 𝑘-forms with polynomial coefficients of degree at most ℓ. We construct inductively differential operators D ℓ : Ω k ⁢ ( U ) → E of order ℓ and Q ℓ : Ω k + 1 ⁢ ( U ) → E of order ℓ − 1 such that

1. D ℓ = Q ℓ ⁢ d ,

2. D − D ℓ vanishes on Ω ≤ ℓ k ⁢ ( U ) .

Let E = ∑ μ = 1 n x μ ⁢ ∂ μ be the Euler vector field. Given a differential operator D ℓ of order ℓ which satisfies (1) and (2), we define the following differential operators of order ℓ + 1 :

Here [ A ] ≤ ℓ denotes the part of the differential operator 𝐴 which is of degree at most ℓ.^[1] This notion is not coordinate-invariant, but that does not cause any problem for the proof, since we work in a fixed coordinate system. By construction and induction hypothesis, property (1), i.e. D ℓ + 1 = Q ℓ + 1 ⁢ d , is satisfied.

Let us verify property (2). If α ∈ Ω ≤ ℓ + 1 k ⁢ ( U ) , then d ⁢ α ∈ Ω ≤ ℓ k + 1 ⁢ ( U ) and

This means that [ ( D − D ℓ ) ⁢ ι E ] ≤ ℓ ⁢ d ⁢ α is equal to ( D − D ℓ ) ⁢ ι E ⁢ d ⁢ α . Thus

where in the last equality we use the fact that ( D − D ℓ ) ⁢ d = D ⁢ d − D ℓ ⁢ d = 0 . Now property (2) holds

• for α ∈ Ω ℓ + 1 k ⁢ ( U ) because L E ⁢ α = ( k + ℓ + 1 ) ⁢ α ,

• and for α ∈ Ω ≤ ℓ k ⁢ ( U ) because L E ⁢ α ∈ Ω ≤ ℓ k ⁢ ( U ) and D − D ℓ vanishes on the subspace Ω ≤ ℓ k ⁢ ( U ) .

On any precompact open set V ⋐ U , the order of 𝐷 is bounded by some ℓ. There, we have D | V = D ℓ | V , since a differential operator of order ℓ is completely determined by what it does on polynomials of degree at most 𝑙. Consequently, we have Q ℓ + i | V = Q ℓ | V for all i ≥ 0 . This already implies that Q = lim ℓ → ∞ Q ℓ | Ω c k + 1 ⁢ ( U ) is a support-decreasing operator Ω c k + 1 ⁢ ( U ) → E . Moreover, 𝑄 is continuous, since it is locally continuous, i.e. it is a differential operator. ∎