Lie conformal algebra cohomology and the variational complex

Recall [K] that a Lie conformal algebra over a field F is an F[∂]-module A, endowed with a λ-bracket, that is an F-linear map A ⊗ A → F such that the skew-symmetry

and the Jacobi identity

hold for any a, b, c ∈ A. It is assumed in (0.2) that ∂ is moved to the left.

A module over a Lie conformal algebra A is an F[∂]-module M , endowed with a λ-action, that is an F-linear map A ⊗ M → F[λ] ⊗ M , denoted by a ⊗ b → a λ b, such that sesquilinearity (0.1) holds for a ∈ A, b ∈ M and Jacobi identity (0.3) holds for a, b ∈ A, c ∈ M .

A cohomology theory for Lie conformal algebras was developed in [BKV]. Given a Lie conformal algebra A and an A-module M , one first defines the basic cohomology complex Γ • (A, M ) = k∈Z + Γ k , where Γ k consists of F-linear maps γ : A ⊗k → F[λ 1 , . . . , λ k ] ⊗ M , satisfying certain sesquilinearity and skew-symmetry properties, and endows this complex with a differential δ : Γ k → Γ k+1 , such that δ 2 = 0. This complex is isomorphic to the Lie algebra cohomology complex for the annihilation Lie algebra g -of A with coefficients in the g –module M [BKV,Theorem 6.1].

Next, one endows Γ • (A, M ) with a structure of a F[∂]-module, such that ∂ commutes with δ, which allows one to define the reduced cohomology complex Γ • (A, M ) = Γ • (A, M )/∂ Γ • (A, M ), and this is the Lie conformal algebra cohomology complex, introduced in [BKV].

Our first contribution to this theory is a more explicit construction of the reduced cohomology complex. Namely, we introduce a new cohomology complex C • (A, M ) = ⊕ k∈Z + C k , where C 0 = M/∂M , C 1 = Hom F[∂] (A, M ), and for k ≥ 2, C k consists of poly λ-brackets, namely of F-linear maps c :

satisfying certain sesquilinearity and skewsymmetry conditions, and we endow C • (A, M ) with a square zero differential d. We construct embeddings of complexes:

where C• (A, M ) consists of cocycles which vanish if one of the arguments is a torsion element of A. In fact, Ck = C k , unless k = 1. We show that Γ • (A, M ) = C• (A, M ), provided that, as an F[∂]-module, A is isomorphic to a direct sum of its torsion and a free F[∂]-module (which is always the case if A is a finitely generated F[∂]-module). Our opinion is that the slightly larger complex C • (A, M ) is a more correct Lie conformal algebra cohomology complex than the complex Γ • (A, M ) of [BKV]. This is illustrated by our Theorem 3.1(c), which says that the F[∂]-split abelian extensions of A by M are parameterized by H 2 (A, M ) for the complex C • (A, M ). This holds for the cohomology theory of [BKV] only if A is a free F[∂]-module.

Following [BKV], we also consider the superspace of basic chains Γ • (A, M ) and its subspace of reduced chains Γ • (A, M ) (they are not complexes in general). Corresponding to the embeddings of complexes (0.4), we introduce the vector superspaces of chains C • (A, M ) and C• (A, M ), and the maps:

We develop the theory further in the important for the calculus of variations case, when the A-module M is endowed with a commutative associative product, such that ∂ and a λ for all a ∈ A are derivations of this product. In this case one can endow the superspace Γ • (A, M ) with a commutative associative product [BKV]. Furthermore, we introduce a Lie algebra bracket on the space g := Π Γ 1 (A, M ) (Π, as usual, stands for reversing of the parity). Let g = ηg ⊕ g ⊕ F∂ η be a Z-graded Lie superalgebra extension of g, where η is an odd indeterminate, η 2 = 0. We endow Γ • (A, M ) with a structure of a g-complex, which is a Z-grading preserving Lie superalgebra homomorphism ϕ : g → End F Γ • (A, M ), such that ϕ(∂ η ) = δ. We also show that ϕ( g) lies in the subalgebra of derivations of the superalgebra Γ • (A, M ). For each X ∈ g we thus have the Lie derivative L X = ϕ(X) and the contraction operator ι X = ϕ(ηX), satisfying all usual relations, in particular, the Cartan formula L X = ι X δ + δι X .

Denoting by g ∂ the centralizer of ∂ in g, we obtain the induced structure of a g ∂ -complex for Γ • (A, M ), which we, furthermore, extend to the larger complex C • (A, M ). Namely, we introduce a canonical Lie algebra bracket on all spaces of 1-chains with reversed parity (see (0.5)), so that all the maps ΠC 1 ։ Π C1 → ΠΓ 1 ֒→ Π Γ 1 are Lie algebra homomorphisms, and the embeddings (0.4) are morphisms of complexes, endowed with a corresponding Lie algebra structure.

What does it all have to do with the calculus of variations? In order to explain this, introduce the notion of an algebra of differentiable functions (in ℓ variables). This is a differential algebra, i.e., a unital commutative associative algebra V with a derivation ∂, endowed with commuting derivations ∂ ∂u (n) i , i ∈ I = {1, . . . , ℓ}, n ∈ Z + , such that only a finite number of ∂f ∂u (n) i are non-ze

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