The Diffiety Institute Preprint Series

Preprint DIPS–1/98 January 26, 1998

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The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Cohomology background in geometry of PDE

by

Joseph KRASIL’SHCHIK

Cohomology background in geometry of PDE

Joseph KRASIL’SHCHIK

Abstract. Using techniques of Frölicher – Nijenhuis brackets, we associate to any formally integrable equation E a cohomology theory (C-complex) H_{C}^{∗}(E) related to deformations of the equation structure on the infinite prolongation E^{∞}. A subgroup in H_{C}^{1}(E) is identified with recursion operators acting on the Lie algebra sym E of symmetries. On the other hand, another subgroup of H_{C}^{∗}(E) can be understood as the algebra of supersymmetries of the “superization” of the equation E. This pass to superequations makes it possible to obtain a well-defined action of recursion operators in nonlocal setting. Relations to Poisson structures on E^{∞} are briefly discussed.

This text is based on my lecture at the Moscow Conference, as well as on series of papers [Kr1, Kr2, Kr3, KrKe1] published in recent years. Unlike Jim Stasheff ([Sta]), I was able to use only one letter of my native language here.

In [Vin1] A. Vinogradov showed that Lagrangian formalism with constraints, as well as the theory of conservation laws, is, in essence, a cohomological theory expressed in terms of the C-spectral sequence. My main purpose here is to show that the theory of symmetries for partial differential equations (including the theory of recursion operators for these symmetries, [Olv]) is of cohomological nature as well. The corresponding cohomologies are determined by the Frölicher – Nijenhuis bracket due to flatness of the Cartan connection and are called C-cohomologies. Moreover, the term E_{1} of the C-spectral sequence and C-cohomologies are in a sense mutually dual. Using this duality, one can introduce the concept of Poisson structure on a differential equation as a cochain map of a special type, [Kr3, Kr4]. On the other hand, both cohomology theories can be considered as horizontal cohomologies with appropriate coefficients [Kr4, Ver] which gives a unified way to estimate them [Ver].

Section 1 is a brief introduction to the theory of Frölicher – Nijenhuis brackets in a general algebraic framework. In Section 2, I construct the complex associated to a flat connection in a commutative algebra and introduce the corresponding cohomologies. Section 3 contains specialization of these construction in the case of the algebra of smooth functions on the infinite prolongation of a formally integrable equation endowed with the Cartan connection. In Section 4, it is shown that to obtain a self-contained, theory one needs to extend the initial setting with graded variables naturally associated to any differential equation and to take into consideration nonlocal picture. Section 5 contains a brief outline of some problems related to the main topic.

Contents

1. Bracket formalism in commutative algebras 2 2. Connections and cohomologies 4 3. Nonlinear differential equations:

C-cohomologies, deformations and recursion operators 8 4. Nonlocal and superextensions 13 5. Concluding remarks 16 References 16

1991 Mathematics Subject Classification. Primary 58G05; Secondary 58G37, 58H10, 35Q53, 16W55.

This work was partially supported by the Russian Foundation for Base Research Grant N 97-01-00462 and by the INTAS Grant N 96-0793.

1. Bracket formalism in commutative algebras

Let 𝔽 be a field of characteristic zero and A be a commutative unitary 𝔽-algebra. We shall use below notations and definitions from differential calculus over commutative algebras (see for details [Vin2, KLV, Kr5]). Recall the basics:

• D(P) is the module of P -valued derivations A → P, where P is an A-module;

• D_{i}(P) is the module of P -valued derivations skew-symmetric i-derivations. In particular, D_{1}(P) = D(P);

• Λ^{i}(A) is the module of differential i-forms of the algebra A;

• d : Λ^{i}(A) → Λ^{i+1}(A) is the de Rham differential.

If A is the algebra of smooth functions on a smooth manifold M, we use the notation D(M) and Λ^{i}(M) for D(A) and Λ^{i}(A) respectively. The modules Λ^{i}(A) are representative objects for the functors D_{i} : P ⇒ D_{i}(P), i.e., D_{i}(P) = hom_{A}(Λ^{i}(A), P). The isomorphism D(P) = hom_{A}(Λ^{1}(A), P) can be expressed in more exact terms: for any derivation X : A → P there exists a uniquely defined homomorphism f_{X} : Λ^{1}(A) → P satisfying X = f_{X} ∘ d. Denote by ⟨Z, ω⟩ ∈ P the value of the derivation Z ∈ D_{i}(P) at ω ∈ Λ^{i}(A).

Both Λ^{∗}(A) = ⊕_{i⩾0}Λ^{i}(A) and D_{∗}(A) = ⊕_{i⩾0}D_{i}(A) are endowed with the structures of commutative superalgebras with respect to the wedge product operation ∧ : Λ^{i}(A)⊗Λ^{j}(A) → Λ^{i+j}(A), D_{i}(A)⊗D_{j}(A) → D_{i+j}(A), the de Rham differential d : Λ^{∗}(A) → Λ^{∗}(A) becoming a derivation. Note also that D_{∗}(P) = ⊕_{i⩾0}D_{i}(P) is a D_{∗}(A)-module.

Using the paring ⟨·, ·⟩ and the wedge product, we define the interior product (or contraction) i_{X} ω ∈ Λ^{j−i}(A) of X ∈ D_{i}(A) and ω ∈ Λ^{j}(A), i ⩽ j, by setting

⟨Y, i_{X} ω⟩ = ⟨Y ∧ X, ω⟩ (1.1)

where Y is an arbitrary element of D_{j−i}(A). We formally set i_{X} ω = 0 for i > j. We now define the Lie derivative of ω along X as

L_{X} ω = (i_{X} ∘ d −(−1)^{X} d ∘ i_{X})ω = [i_{X}, d]ω (1.2)

where (as everywhere below) (−1)^{a} substitutes (−1)^{deg a} for any element a of a “superobject” (i.e., algebra or module), while [·, ·] denotes the supercommutator.

Consider now the graded module D(Λ^{∗}(A)) of Λ^{∗}(A)-valued derivations A → Λ^{∗}(A) corresponding to form-valued vector fields (or, which is the same, vectorvalued differential forms) on a smooth manifold. Note that the graded structure in D(Λ^{∗}(A)) is determined by the splitting D(Λ^{∗}(A)) = ⊕_{i⩾0}D(Λ^{i}(A)) and thus elements of grading i are derivations X such that im X ⊂ Λ^{i}(A). We shall need three algebraic structures associated to D(Λ^{∗}(A)). First note that D(Λ^{∗}(A)) is a graded Λ^{∗}(A)-module: for any X ∈ D(Λ^{∗}(A)), ω ∈ Λ^{∗}(A) and a ∈ A we set (ω ∧ X)a = ω ∧ X(a). Second, we can define interior product i_{X} ω ∈ Λ^{i+j−1}(A) of X ∈ D(Λ^{i}(A)) and ω ∈ Λ^{j}(A) using the same definition (1.1). Finally, we can contract elements of D(Λ^{∗}(A)) with each other by setting

(i_{X} Y)a = i_{X}(Y a), X, Y ∈ D(Λ^{∗}(A)), a ∈ A.

Two properties of contractions are essential in the sequel.

Proposition 1.1. Let X, Y ∈ D(Λ^{∗}(A)) and ω, θ ∈ Λ^{∗}(A). Then

i_{X}(ω ∧ θ) = i_{X}(ω) ∧ θ + (−1)^{ω(X−1)}ω ∧ i_{X}(θ), (1.3)

i_{X}(ω ∧ Y) = i_{X}(ω) ∧ Y + (−1)^{ω(X−1)}ω ∧ i_{X}(Y), (1.4)

[i_{X}, i_{Y}] = i_{[[X,Y]]^{rn}}, (1.5)

where

[[X, Y]]^{rn} = i_{X}(Y) − (−1)^{(X−1)(Y −1)} i_{Y} (X). (1.6)

Proof. Equalities (1.3) and (1.4) are proved by direct use of definitions. To prove (1.5), it suffices to use expression (1.6).

Definition 1.1. The element [[X, Y]]^{rn} defined by (1.6) is called the Richardson – Nijenhuis bracket of elements X and Y.

Directly from Proposition 1.1 we obtain the following

Proposition 1.2. For any X, Y, Z ∈ D(Λ^{∗}(A)) and ω ∈ Λ^{∗}(A) one has

[[X, Y]]^{rn} + (−1)^{(X+1)(Y +1)}[[Y, X]]^{rn} = 0, (1.7)

∮ (−1)^{(Y +1)(X+Z)}[[[[X, Y]]^{rn}, Z]]^{rn} = 0, (1.8)

[[X, ω ∧ Y]]^{rn} = i_{X}(ω) ∧ Y + (−1)^{(X+1)ω}ω ∧ [[X, Y]]^{rn}. (1.9)

Here and below the symbol ∮ denotes the sum of cyclic permutations.

Note that Proposition 1.2 means that D(Λ^{∗}(A))^{↓} is a Gerstenhaber algebra with respect to the Richardson –Nijenhuis bracket [K-S]. Here the superscript ↓ denotes the shift of grading by 1.

Similar to (1.2) define the Lie derivative of ω ∈ Λ^{∗}(A) along X ∈ D(Λ^{∗}(A)) by

L_{X} ω = (i_{X} ∘ d +(−1)^{X} d ∘ i_{X})ω = [i_{X}, d]ω (1.10)

(change of sign is due to the fact that deg(i_{X}) = deg(X) − 1). From the properties of i_{X} and d we obtain

Proposition 1.3. For any X ∈ D(Λ^{∗}(A)) and ω, θ ∈ Λ^{∗}(A) one has

L_{X}(ω ∧ θ) = L_{X}(ω) ∧ θ + (−1)^{Xω}ω ∧ L_{X}(θ), (1.11)

L_{ω∧X} = ω ∧ L_{X} +(−1)^{ω+X} d(ω) ∧ i_{X}, (1.12)

[L_{X}, d] = 0. (1.13)

Our main concern now is to analyze the commutator [L_{X}, L_{Y}] of two Lie derivatives. It may be done efficiently for the following class of algebras.

Definition 1.2. We say that a commutative algebra A is smooth, if Λ^{1}(A) is a projective module of finite type.

Proposition 1.4. Let A be a smooth algebra. Then for any elements X, Y ∈ D(Λ^{∗}(A)) there exists a uniquely determined element [[X, Y]]^{fn} such that

[L_{X}, L_{Y}] = L_{[[X,Y]]^{fn}}. (1.14)

Proof. To prove existence, note that for smooth algebras one has

D_{i}(P) = hom_{A}(Λ^{i}(A), P) = P ⊗_{A} hom_{A}(Λ^{i}(A), A) = P ⊗_{A} D_{i}(A)

for any A-module P and integer i ⩾ 0. Using this identification represent elements X, Y ∈ D(Λ^{∗}(A)) in the form

X = ω ⊗ X′, Y = θ ⊗ Y′, ω, θ ∈ Λ^{∗}(A), X′, Y′ ∈ D(A).

Then it is easily checked that the element

Z = ω ∧ θ ⊗ [X′, Y′] + ω ∧ L_{X′} θ ⊗ Y + (−1)^{ω} d ω ∧ i_{X′} θ ⊗ Y′

− (−1)^{ωθ}θ ∧ L_{Y′} ω ⊗ X′ − (−1)^{(ω+1)θ} d θ ∧ i_{Y′} ω ⊗ X′ (1.15)

= ω ∧ θ ⊗ [X′, Y′] + L_{X} θ ⊗ Y′ − (−1)^{ωθ} L_{Y} ω ⊗ X′

satisfies (1.14).

Uniqueness follows from the fact that L_{X}(a) = X(a) for a ∈ A.

Definition 1.3. The element [[X, Y]]^{fn} defined by formula (1.14) is called the Frölicher – Nijenhuis bracket of elements X and Y.

The basic properties of this bracket are summarized in the following

Proposition 1.5. Let A be a smooth algebra, X, Y, Z ∈ D(Λ^{∗}(A)) and ω ∈ Λ^{∗}(A). Then

[[X, Y]]^{fn} + (−1)^{XY} [[Y, X]]^{fn} = 0, (1.16)

∮ (−1)^{Y (X+Z)}[[X, [[Y, Z]]^{fn}]]^{fn} = 0, (1.17)

i_{[[X,Y]]^{fn}} = [L_{X}, i_{Y}] + (−1)^{X(Y +1)} L_{i_{Y} X}, (1.18)

(1.19) i_{Z} [[X, Y]]^{fn} = [[i_{Z} X, Y]]^{fn} + (−1)^{X(Z+1)}[[X, i_{Z} Y]]^{fn}

+(−1)^{X} i_{[[Z,X]]^{fn}} Y − (−1)^{(X+1)Y} i_{[[Z,Y]]^{fn}} X,

[[X, ω ∧ Y]]^{fn} = L_{X} ω ∧ Y − (−1)^{(X+1)(Y +ω)} d ω ∧ i_{Y} X + (−1)^{Xω}ω ∧ [[X, Y]]^{fn}. (1.20)

Note that the first two equalities in the previous proposition mean that the module D(Λ^{∗}(A)) is a Lie superalgebra with respect to the Frölicher – Nijenhuis bracket.

Remark 1.1. The above exposed algebraic scheme obtains a geometrical realization, if one takes C^{∞}(M) for the algebra A, M being a smooth finite-dimensional manifold.^{1} The algebra A = C^{∞}(M) is smooth in this case. However, in geometrical theory of differential equations we have to work with infinite-dimensional manifolds of the form N = \underleftarrow{lim}_{\{π_{k+1,k}\}}N_{k}, where π_{k+1,k} : N_{k+1} → N_{k} are surjections of finite-dimensional smooth manifolds. The corresponding algebraic object is a filtered algebra A = ⋃_{k∈ℤ} A_{k}, A_{i} ⊂ A_{k+1}, where all A_{k} are subalgebras in A. A self-contained differential calculus over A is constructed, if one considers the category of all filtered A-modules with filtered homomorphisms for morphisms between them. Then all functors of differential calculus in this category become filtered, as well as their representative objects.

In particular, the A-modules Λ^{i}(A) are filtered by A_{k}-modules Λ^{i}(A_{k}). We say that the algebra A is finitely smooth, if Λ^{1}(A_{k}) is a projective A_{k}-module of finite type for any k ∈ ℤ. For finitely smooth algebras, elements of D(P) may be represented as formal infinite sums ∑_{k} p_{k} ⊗ X_{k}, such that any finite sum S_{n} = ∑_{k⩽n} p_{k} ⊗X_{k} is a derivation A_{n} → P_{n+s} for some fixed s ∈ ℤ. Any derivation X is completely determined by the system {S_{n}} and Proposition 1.5 obviously remains valid.

1To obtain exact correspondence between algebraic and geometrical pictures, one needs to restrict oneself with the subcategory of all A-modules consisting of the so-called geometrical modules.

See the details in [KLV].

2. Connections and cohomologies

We now introduce the second object of our interest. Let A be an 𝔽-algebra and B be an algebra over A. We shall assume that the corresponding homomorphism φ : A → B is an embedding. Let P be a B-module; then it is an A-module as well and we can consider the B-module D(A, P) of P -valued derivations A → P.

Definition 2.1. Let ∇^{•} : D(A, •) ⇒ D(•) be a natural transformations of functors D(A, •) : A ⇒ D(A, P), D(•) : P ⇒ D(•) in the category of B-modules, i.e., a system of homomorphisms ∇^{P} : D(A, P) → D(P) such that the diagram

D(A, P) ⟶^{∇^{P}} D(P)
D(A,f)↓  ↓^{D(f)}
D(A, Q) ⟶^{∇^{Q}} D(Q)

is commutative for any B-homomorphism f : P → Q. We say that ∇^{•} is a connection in the triad (A, B, φ), if ∇^{P} (X) |_{A} = X for any X ∈ D(A, P).

Here and below we use the notation Y |_{A} = Y ∘ φ for any Y ∈ D(P).

Remark 2.1. When A = C^{∞}(M), B = C^{∞}(E), φ = π^{∗}, where M, E are smooth manifolds and π : E → M is a smooth fiber bundle, Definition 2.1 reduces to the ordinary definition of a connection in the bundle π. In fact, if we have a connection ∇^{•} in the sense of Definition 2.1, then the correspondence

D(A) ↪ D(A, B) ⟶^{∇^{B}} D(B)

allows one to lift any vector field on M up to a π-protectible field on E. Conversely, if ∇ is such a correspondence, then we can construct a natural transformation ∇^{•} of the functors D(A, •) and D(•) due to the fact that for smooth manifolds one has D(A, P) = P ⊗_{A} D(A) and D(P) = P ⊗_{B} D(P), P being an arbitrary B-module. We use the notation ∇ = ∇^{B} in the sequel.

Definition 2.2. Let ∇^{•} be a connection in (A, B, φ) and X, Y ∈ D(A, B) be two derivations. The curvature form of the connection ∇^{•} on the pair X, Y is defined by

R_{∇}(X, Y) = [∇(X), ∇(Y)] − ∇(∇(X) ∘ Y − ∇(Y) ∘ X). (2.1)

Note that (2.1) makes sense, since ∇(X) ∘ Y − ∇(Y) ∘ X is a B-valued derivation of A.

Consider now the de Rham differential d = d_{B} : B → Λ^{1}(B). Then the composition d_{B} ∘φ : A → B is a derivation. Consequently, we may consider the derivation ∇(d_{B} ∘φ) ∈ D(Λ^{1}(B)).

Definition 2.3. The element U_{∇} ∈ D(Λ^{1}(B)) defined by

U_{∇} = ∇(d_{B} ∘φ) − d_{B} (2.2)

is called the connection form of ∇.

Directly from definition we obtain the following

Lemma 2.1. The equality

i_{X}(U_{∇}) = X − ∇(X |_{A}) (2.3)

holds for any X ∈ D(B).

Using this result, we now prove Proposition 2.2. If B is a smooth algebra, then

i_{Y} i_{X} [[U_{∇}, U_{∇}]]^{fn} = 2R_{∇}(X |_{A} , Y |_{A}) (2.4)

for any X, Y ∈ D(B).

Proof. First note that deg U_{∇} = 1. Then using (1.19) and (1.16) we obtain

i_{X} [[U_{∇}, U_{∇}]]^{fn} = [[i_{X} U_{∇}, U_{∇}]]^{fn} + [[U_{∇}, i_{X} U_{∇}]]^{fn} − i_{[[X,U_{∇}]]^{fn}} U_{∇} − i_{[[X,U_{∇}]]^{fn}} U_{∇}
= 2([[i_{X} U_{∇}, U_{∇}]]^{fn} − i_{[[X,U_{∇}]]^{fn}} U_{∇}).

Applying i_{Y} to the last expression and using (1.17) and (1.19), we get now

i_{Y} i_{X} [[U_{∇}, U_{∇}]]^{fn} = 2([[i_{X} U_{∇}, i_{Y} U_{∇}]]^{fn} − i_{[[X,Y]]^{fn}} U_{∇}).

But [[V, W]]^{fn} = [V, W] for any V, W ∈ D(Λ^{0}(A)) = D(A). Hence, by (2.3), we have

i_{Y} i_{X} [[U_{∇}, U_{∇}]]^{fn} = 2([X − ∇(X |_{A}), Y − ∇(Y |_{A})] − ([X, Y] − ∇([X, Y] |_{A}))).

It only remains to note now that ∇(X |_{A}) |_{A} = X |_{A} and [X, Y] |_{A} = X ∘ Y |_{A} − Y ∘ X |_{A}.

Definition 2.4. A connection ∇ in (A, B, φ) is called flat, if R_{∇} = 0.

Thus for flat connections we have

[[U_{∇}, U_{∇}]]^{fn} = 0. (2.5)

Let U ∈ D(Λ^{1}(B)) be an element satisfying (2.5). Then from the graded Jacobi identity (1.17) we obtain 2[[U, [[U, X]]^{fn}]]^{fn} = [[[[U, U]]^{fn}, X]]^{fn} = 0 for any X ∈ D(Λ^{∗}(A)). Consequently, the operator ∂_{U} = [[U, ·]]^{fn} : D(Λ^{i}(B)) → D(Λ^{i+1}(B)) defined by ∂_{U}(X) = [[U, X]]^{fn} satisfies ∂_{U} ∘ ∂_{U} = 0.

Consider now the case U = U_{∇}, where ∇ is a flat connection.

Definition 2.5. An element X ∈ D(Λ^{∗}(B)) is called vertical, if X(a) = 0 for any a ∈ A. Denote the B-submodule of such elements by D^{v}(Λ^{∗}(B)).

Lemma 2.3. Let ∇ be a connection in (A, B, φ). Then (i) an element X ∈ D(Λ^{∗}(B)) is vertical if and only if i_{X} U_{∇} = X; (ii) the connection form U_{∇} is vertical, U_{∇} ∈ D^{v}(Λ^{1}(B)); (iii) the map ∂_{U_{∇}} preserves verticality, ∂_{U_{∇}}(D^{v}(Λ^{i}(B))) ⊂ D^{v}(Λ^{i+1}(B)).

Proof. To prove (i), use Lemma 2.5: from (2.3) it follows that i_{X} U_{∇} = X if and only if ∇(X |_{A}) = 0. But ∇(X |_{A}) |_{A} = X |_{A}. The second statements follows from the same lemma and from first one:

i_{U_{∇}} = U_{∇} − ∇(U_{∇} |_{A}) = U_{∇} − ∇((U_{∇} − ∇(U_{∇} |_{A}) |_{A} ) = U_{∇}.

Finally, (iii) is a consequence of (1.19).

Definition 2.6. Denote the restriction ∂_{U_{∇}} |_{D^{v}(Λ^{∗}(A))} by ∂_{∇} and call the complex

0 → D^{v}(A) ⟶^{∂_{∇}} D^{v}(Λ^{1}(A)) → … → D^{v}(Λ^{i}(A)) ⟶^{∂_{∇}} D^{v}(Λ^{i+1}(A)) → … (2.6)

the U-complex of the triple (A, B, φ). The corresponding cohomology groups are denoted by H_{∇}^{i}(B; A, φ), H_{∇}^{∗}(B; A, φ) = ⨁_{i⩾0} H_{∇}^{i}(B; A, φ) and are called the Ucohomologies of the triple (A, B, φ).

Introduce the notation

d_{∇}^{v} = L_{U_{∇}} : Λ^{i}(B) → Λ^{i+1}(B). (2.7)

Proposition 2.4. Let ∇ be a flat connection in the triple (A, B, φ) and B be a smooth (or finitely smooth) algebra. Then for any X, Y ∈ D^{v}(Λ^{∗}(A)) and ω ∈ Λ^{∗}(A) one has

∂_{∇}[[X, Y]]^{fn} = [[∂_{∇}X, Y]]^{fn} + (−1)^{X}[[X, ∂_{∇}Y]]^{fn}, (2.8)

[i_{X}, ∂_{∇}] = (−1)^{X} i_{∂_{∇}X}, (2.9)

∂_{∇}(ω ∧ X) = (d_{∇}^{v} − d)(ω) ∧ X + (−1)^{ω}ω ∧ ∂_{∇}X, (2.10)

[d_{∇}^{v}, i_{X}] = i_{∂_{∇}X} +(−1)^{X} L_{X} . (2.11)

Proof. Equality (2.8) is a direct consequence of (1.17). Equality (2.9) follows from (1.19). Equality (2.10) follows from (1.20) and (2.3). Finally, (2.11) is obtained from (1.18).

Corollary 2.5. The cohomology module H_{∇}^{∗}(B; A, φ) inherits the graded Lie algebra structure with respect to [[·, ·]]^{fn}, as well as the contraction operation.

Proof. Note that D^{v}(Λ^{∗}(A)) is closed with respect to the Frölicher – Nijenhuis bracket: to prove this fact, it suffices to apply (1.19). Then the first statement follows from (2.8). The second one is a consequence of (2.9).

Remark 2.2. We preserve the same notations for the inherited structures. Note, in particular, that H_{∇}^{0}(B; A, φ) is a Lie algebra with respect to the Frölicher – Nijenhuis bracket (which reduces to the ordinary Lie bracket in this case). Moreover, H_{∇}^{1}(B; A, φ) is an associative algebra with respect to the inherited contraction, while the action

R_{Ω} : X ↦ i_{X} Ω, X ∈ H_{∇}^{0}(B; A, φ), Ω ∈ H_{∇}^{1}(B; A, φ)

is a representation of this algebra as endomorphisms of H_{∇}^{0}(B; A, φ).

Consider now the map d_{∇}^{v} : Λ^{∗}(B) → Λ^{∗}(B) defined by (2.7) and define d_{∇}^{h} = d_{B} − d_{∇}^{v}.

Proposition 2.6. Let B be a (finitely) smooth algebra and ∇ be a smooth connection in the triple (B; A, φ). Then

(i) The pair (d_{∇}^{h}, d_{∇}^{v}) forms a bicomplex, i.e.

(2.12) d_{∇}^{v} ∘ d_{∇}^{v} = 0, d_{∇}^{h} ∘ d_{∇}^{h} = 0, d_{∇}^{h} ∘ d_{∇}^{v} + d_{∇}^{v} ∘ d_{∇}^{h} = 0.

(ii) The differential d_{∇}^{h} possesses the following properties

[d_{∇}^{h}, i_{X}] = − i_{∂_{∇}X}, (2.13)

∂_{∇}(ω ∧ X) = − d_{∇}^{h}(ω) ∧ X + (−1)^{ω}ω ∧ ∂_{∇}X, (2.14)

where ω ∈ Λ^{∗}(B), X ∈ D^{v}(Λ^{∗}(B)).

Proof. (i) Since deg d_{∇}^{v} = 1, we have

2 d_{∇}^{v} ∘ d_{∇}^{v} = [d_{∇}^{v}, d_{∇}^{v}] = [L_{U_{∇}}, L_{U_{∇}}] = L_{[[U_{∇},U_{n}a]]^{fn}} = 0.

Since d_{∇}^{v} = L_{U_{∇}}, the identity [d_{B}, d_{∇}^{v}] = 0 holds (see (1.13)), and it finishes the proof of the first part.

(ii) To prove (2.13), note that [d_{∇}^{h}, i_{X}] = [d_{B} − d_{∇}^{h}, i_{X}] = (−1)^{X} L_{X} −[d_{∇}^{v}, i_{X}], and (2.13) holds due to (2.11). Finally, (2.14) is just the other form of (2.10).

Definition 2.7. Let ∇ be a connection in (A, Bφ).

(i) The bicomplex (B, d_{∇}^{h}, d_{∇}^{v}) is called the variational bicomplex associated to the connection ∇.

(ii) Corresponding spectral sequence is called ∇-spectral sequence of the triple (A, Bφ).

Obviously, the ∇-spectral sequence converges to the de Rham cohomology of B.

To finish this section, note the following. Since the module Λ^{1}(B) is generated by the image of the operator d_{B} : B → Λ^{1}(B) while the graded algebra Λ^{∗}(B) is generated by Λ^{1}(B), we have the direct sum decomposition

Λ^{∗}(B) = ⨁_{i⩾0} ⨁_{p+q=i} Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B),

where

Λ_{v}^{p}(B) = \underbrace{Λ_{v}^{1}(B) ∧ … ∧ Λ_{v}^{1}(B)}_{p times}, Λ_{h}^{q}(B) = \underbrace{Λ_{h}^{1}(B) ∧ … ∧ Λ_{h}^{1}(B)}_{q times},

while Λ_{v}^{1}(B) ⊂ Λ^{1}(B), Λ_{h}^{1}(B) ⊂ Λ^{1}(B) are spanned in Λ^{1}(B) by the images of d_{∇}^{v} and d_{∇}^{h} respectively. Obviously,

d_{∇}^{h} (Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B)) ⊂ Λ_{v}^{p}(B) ⊗ Λ_{h}^{q+1}(B),
d_{∇}^{v} (Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B)) ⊂ Λ_{v}^{p+1}(B) ⊗ Λ_{h}^{q}(B).

Denote by D^{p,q}(B) the module D^{v}(Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B)). Then, obviously, D^{v}(B) = ⨁_{i⩾0} ⨁_{p+q=i} D^{p,q}(B), while from (2.13) and (2.14) we obtain

∂_{∇}(D^{p,q}(B)) ⊂ D^{p,q+1}(B).

Consequently, the module H_{∇}^{∗}(B; A, φ) is split as

H_{∇}^{∗}(B; A, φ) = ⨁_{i⩾0} ⨁_{p+q=i} H_{∇}^{p,q}(B; A, φ) (2.15)

with obvious meaning of the notation H_{∇}^{p,q}(B; A, φ).

3. Nonlinear differential equations: C-cohomologies, deformations and recursion operators

Now we apply the above exposed algebraic results to the case of infinitely prolonged differential equation. In what follows, we use the introductory material of A. Vinogradov’s paper published in this volume.

Let π : E → M be a locally trivial vector^{2} bundle, dim M = n, dim π = m, and E ⊂ J^{k}(π) be a k-th order differential equation in the bundle π. Consider the prolongations E^{l} ⊂ J^{k+l}(π) and the maps π_{k+l+1,k+l} : E^{l+1} → E^{l}.

Definition 3.1. Equation E ⊂ J^{k}(π) is called formally integrable if all prolongations E^{l} are smooth manifolds and the maps π_{k+l+1,k+l} : E^{l+1} → E^{l} are surjections and affine fiber bundles.

In what follows, we assume that all equations under consideration are formally integrable.

Let E^{∞} ⊂ J^{∞}(π) be the infinite prolongation of E with the natural projections π_{∞,k+l} : E^{∞} → E^{l}, π_{∞} : E^{∞} → M. We use the following notations below:

• F(E) denotes the filtered algebra of smooth (local) functions on E^{∞}, i.e., F(E) = ⋃_{l⩾0} F_{l}(E), where F_{l}(E) = C^{∞}(E^{l}),

• F(E, ξ) = ⋃_{l⩾0} F_{l}(E, ξ) is the filtered F(E)-module of sections of the pull-back π_{∞}^{∗}ξ, where ξ : F → M is a vector bundle,

• for any filtered F(E)-module P, D_{i}(P) denotes the module of P -valued filtered i-derivations of F(E),

• in particular, D(E) = D_{1}(E) is identified with the module of vector fields on E^{∞}. We also denote by D^{v}(P) the submodule of π_{∞}-vertical derivations, • Λ^{∗}(E) = ⨁_{i⩾0} Λ^{i}(E) is the graded filtered algebra of local differential forms

2In fact, all necessary construction can be defined in the case of an arbitrary locally trivial bundle, but it adds to technical difficulties only and does not affect the general conceptual scheme.

on E^{∞}, i.e., Λ^{i}(E) = ⋃_{l⩾0} Λ^{i}(E^{l}).

For the “empty” equation E^{∞} = J^{∞}(π) we write F(π), D(π), Λ^{i}(π), etc.

Let ξ : F → M, ξ′ : F′ → M be two vector bundles and Δ : Γ(ξ) → Γ(ξ′) be a C^{∞}(M)-linear differential operator taking sections of ξ to sections of ξ′. Let f be a (local) section of π and j_{∞}(f) ∈ J^{∞}(π) be its infinite jet. Set

j_{∞}(f)^{∗}(CΔ(φ)) = Δ(j_{∞}(f)^{∗}φ), (3.1)

where φ ∈ F(π, ξ).

Proposition 3.1 (see [KLV]). Equality (3.1) uniquely defines an F(π)-linear differential operator CΔ : F(π, ξ) → F(π, ξ′) possessing the following properties:

(i) The correspondence Δ ↦ CΔ is C^{∞}(M)-linear. (ii) If ξ′′ : F′′ → M is another vector bundle and Δ′ : Γ(ξ′) → Γ(ξ′′) is a linear

differential operator, then

C(Δ′ ∘ Δ) = C(Δ′) ∘ C(Δ). (3.2)

(iii) If ϵ : E^{∞} ↪ J^{∞}(π) is an infinite prolongation of a formally integrable equation, then any operator of the form CΔ admits restriction onto E^{∞}, i.e., there exists an operator C̃Δ such that the diagram

F(π, ξ) ⟶^{CΔ} F(π, ξ′)
ϵ^{∗}↓  ↓_{ϵ^{∗}}
F(E, ξ) \underrightarrow{C̃Δ} F(E, ξ′)

is commutative.

Note also that (3.1) means that operators CΔ admit restrictions onto graphs of infinite jets.

Definition 3.2. Operators of the form C̃Δ : F(E, ξ) → F(E, ξ′) are called Cdifferential (or total) operators. The F(E)-module of such operators is denoted by C Diff(ξ, ξ′).

In particular, if X ∈ D(M) is a vector field on the manifold M, then CX is a vector field on J^{∞}(π) admitting restriction onto E^{∞}. Such fields are called total derivatives. From the definition and Proposition 3.1 it follows that the correspondence X ↦ CX is a connection in the bundle π_{∞} : E^{∞} → M.

Definition 3.3. The correspondence X ↦ CX is called the Cartan connection on E.

Proposition 3.2. The Cartan connection is flat for any formally integrable equation.

Proof. It is a direct consequence of (3.2).

Thus, to any formally integrable equation E ⊂ J^{k}(π) we can associate the complex (D(Λ^{i}(E)), ∂_{C}) and the cohomology theory determined by the Cartan connection. We denote the corresponding cohomology modules by H_{C}^{∗}(E) = ⨁_{i⩾0} H_{C}^{i}(E). In the case of “empty” equation, we use the notation H_{C}^{∗}(π) = ⨁_{i⩾0} H_{C}^{i}(π).

Definition 3.4. Let E ⊂ J^{k}(π) be a formally integrable equation and C be the Cartan connection in the bundle π_{∞} : E^{∞} → M. Then

(i) The connection form U_{C} = U_{C}(E) ∈ D^{v}(Λ^{1}(E)) is called the structural element of the equation E.

(ii) The modules H_{C}^{i}(E) are called C-cohomologies of E.

The following result contains an interpretation of the first two modules of Ccohomologies.

Theorem 3.3. For any formally integrable equation E ⊂ J^{k}(π), one has

(i) The module H_{C}^{0}(E) as a Lie algebra is isomorphic to the algebra sym E of higher symmetries of the equation E.

(ii) The module H_{C}^{1}(E) coincides with the equivalence classes of nontrivial vertical deformations of the structural element.

Proof. To prove (i), take a vertical vector field Y ∈ D^{v}(E) and an arbitrary field Z ∈ D(E). Then, due to (1.19), one has

i_{Z} ∂_{C}Y = i_{Z} [[U_{C}, Y]]^{fn} = [i_{Z} U_{C}, Y] − i_{[Z,Y]} U_{C}
= [Z^{v}, Y] − [Z^{v} + (Z − Z^{v}), Y]^{v} = [Z − Z^{v}, Y]^{v},

where Z^{v} = i_{Z} U_{C}. Hence, ∂_{C}Y = 0 if and only if [Z−Z^{v}, Y]^{v} = 0 for any Z ∈ D(E). But the last equality holds if and only if [CX, Y] = 0 for any X ∈ D(M) which means that

ker (∂_{C} : D^{v}(E) → D^{v}(Λ^{1}(E))) = sym E.

Consider the second statement. Let U(ε) be a deformation of the structural element satisfying U(ε) ∈ D^{v}(Λ^{1}(E)), [[U(ε), U(ε)]]^{fn} = 0 and U(0) = U_{C}. Then U(ε) = U_{C} + U_{1}ε + O(ε^{2}). Consequently,

[[U(ε), U(ε)]]^{fn} = [[U_{C}, U_{C}]]^{fn} + 2[[U_{C}, U_{1}]]^{fn}ε + O(ε^{2}) = 0,

from where it follows that [[U_{C}, U_{1}]]^{fn} = ∂_{C}U_{1} = 0. Hence the linear part of the deformation U(ε) determines an element of H_{C}^{1}(E) and vice versa. On the other hand, let A : E^{∞} → E^{∞} be a diffeomorphism^{3} of E^{∞}. Define the action A^{∗} of A on the elements Ω ∈ D(Λ^{∗}(E)) in such a way that the diagram

Λ^{∗}(E) ⟶^{L_{Ω}} Λ^{∗}(E)
A^{∗}↓  ↓_{A^{∗}}
Λ^{∗}(E) ⟶^{L_{Ω}} Λ^{∗}(E)

is commutative. Then, if A_{t} is a one-parameter group of diffeomorphisms, then, obviously,

\frac{d}{dt}|_{t=0} A_{t,∗}(L_{Ω}) = \frac{d}{dt}|_{t=0} A_{t}^{∗} ∘ L_{Ω} ∘(A_{t}^{∗})^{−1} = [L_{X}, L_{Ω}] = L_{[[X,Ω]]^{fn}} .

Hence, infinitesimal action is given by the Frölicher – Nijenhuis bracket. Taking Ω = U_{C} and X ∈ D^{v}(E), we see that im ∂_{C} consists of infinitesimal deformation arising due to infinitesimal action of diffeomorphisms on the structural element. Such deformations are natural to be called trivial.

Remark 3.1. From the general theory [Ger], we obtain that the module H_{C}^{2}(E) consists of obstructions to prolongation of infinitesimal deformations up to formal ones. In the case under consideration, elements H_{C}^{2}(E) have another nice interpretation discussing later (see Remark 3.14).

We shall now compute the modules H_{C}^{p}(π), p ⩾ 0. To this, recall the splitting Λ^{i}(E) = ⨁_{p+q=i} Λ_{v}^{p}(E) ⊗ Λ_{h}^{q}(E) (see Section 2).

Theorem 3.4. One has H_{C}^{p}(π) = F(π, π) ⊗_{F(π)} Λ_{v}^{p}(E) for any p ⩾ 0.

3Since E^{∞} is, in general, infinite-dimensional, vector fields on E^{∞} do not usually possess oneparameter groups of diffeomorphisms. Thus the arguments below are of a heuristic nature

Proof. Define a filtration in D^{v}(Λ^{∗}(π)) by setting

F^{l}D^{v}(Λ^{p}(π)) = {X ∈ D^{v}(Λ^{p}(π)) | X |_{F_{l−p−1}} = 0}.

Evidently,

F^{l}D^{v}(Λ^{p}(π)) ⊂ F^{l+1}D^{v}(Λ^{p}(π)), ∂_{π}(F^{l}D^{v}(Λ^{p}(π))) ⊂ F^{l}D^{v}(Λ^{p+1}(π)).

Thus we obtain the spectral sequence associated to this filtration. To compute the term E_{0}, choose local coordinates x_{1}, …, x_{n}, u^{1}, …, u^{m} in the bundle π and consider the corresponding special coordinates u_{σ}^{j}, j = 1, …, m, σ = (σ_{1}, …, σ_{n}) in J^{∞}(π). In these coordinates, the structural element is represented as

U_{π} = ∑_{|σ|⩾0} ∑_{j=1}^{m} (d u_{σ}^{j} − ∑_{i=1}^{n} u_{σ+1_{i}} d x_{i}) ⊗ \frac{∂}{∂u_{σ}^{j}} , (3.3)

where σ + 1_{i} = (σ_{1}, …, σ_{i} + 1, …, σ_{n}), while for X = ∑_{σ,j} θ_{σ}^{j} ⊗ ∂/∂u_{σ}^{j}, θ ∈ Λ^{∗}(π), one has

∂_{π}(X) = ∑_{|σ|⩾0} ∑_{j=1}^{m} ∑_{i=1}^{n} d x_{i} ∧ (θ_{σ+1_{i}}^{j} − D_{i}(θ_{σ}^{j})) ⊗ \frac{∂}{∂u_{σ}^{j}} , (3.4)

where

D_{i} = C (\frac{∂}{∂x_{i}} ) = \frac{∂}{∂x_{i}} + ∑_{σ,j} u_{σ+1_{i}}^{j} \frac{∂}{∂u_{σ}^{j}}

is the total derivative along x_{i}.

Obviously, the term E_{0}^{p,−q} = F^{p}D^{v}(Λ^{p−q}(π))/F^{p−1}D^{v}(Λ^{p−q}(π)), p ⩾ 0, 0 ⩽ q ⩽ p, identifies with the tensor product Λ^{p−q}(π) ⊗_{F(π)} Γ(π_{∞,q−1}^{∗}(π_{q,q−1})), where π_{∞,q−1} : J^{∞}(π) → J^{q−1}(π), π_{q,q−1} : J^{q}(π) → J^{q−1}(π) are natural projections. These modules can be locally represented as F(π, π)⊗Λ^{p−q}(π)-valued homogeneous polynomials of order q, while the differential ∂_{0}^{p,−q} : E_{0}^{p,−q} → E_{0}^{p,−q+1} acts as the Spencer δ-differential (or , which is the same, as the Koszul differential). Hence, all homology groups are trivial except for the term E_{0}^{p,0} and one has coker ∂_{0}^{p,0} = F(π, π) ⊗_{F(π)} Λ_{v}^{p}(π). Consequently, only the 0-th line survives in the term E_{1} and this line is of the form

F(π, π) ⟶^{∂_{1}^{0,0}} F(π, π) ⊗_{F(π)} Λ_{v}^{1}(π) → …

… → F(π, π) ⊗_{F(π)} Λ_{v}^{p}(π) ⟶^{∂_{1}^{p,0}} F(π, π) ⊗_{F(π)} Λ_{v}^{p+1}(π) → …

But the image of ∂_{C} contains at least one horizontal component (see the end of Section 2). Therefore, all differentials ∂_{1}^{p,0} vanish.

Let us now establish the correspondence between the last result (describing Ccohomology in terms of Λ_{v}^{∗}(π)) and representation H_{C}^{∗}(π) as classes of derivations F(π) → Λ^{∗}(π). To do this, for any ω = (ω^{1}, …, ω^{m}) ∈ F(π, π) ⊗_{F(π)} Λ_{v}^{∗}(π) set

�_{ω} = ∑_{σ,j} D_{σ}(ω^{j}) ⊗ \frac{∂}{∂u_{σ}^{j}} , (3.5)

where D_{σ} = D_{1}^{σ_{1}} ∘ ⋯ ∘ D_{n}^{σ_{n}} for σ = (σ_{1}, … σ_{n}).

Definition 3.5. The element �_{ω} ∈ D^{v}(Λ^{∗}(π)) defined by (3.5) is called evolution superderivation with the generating section ω ∈ Λ_{v}^{∗}(π).

Proposition 3.5. Definition of �_{ω} is independent of coordinate choice.

Proof. It is easily checked that

�_{ω}(F(π)) ⊂ Λ_{v}^{∗}(π), �_{ω} ∈ ker ∂_{C}.

But derivations possessing these properties are uniquely determined by their restriction onto F_{0}(π) (see [Kr1, KLV] for details).

From this result and from Corollary 2.5 it follows that given two evolution superderivations �_{ω}, �_{θ}, the elements

(i) [[�_{ω} , �_{θ} ]]^{fn}, (ii) i_{�_{ω}} (�_{θ})

are evolution superderivations as well.

In the first case, the corresponding generating section is called the Jacobi superbracket of elements ω = (ω^{1}, …, ω^{m}) and θ = (θ^{1}, …, θ^{m}) and is denoted by {ω, θ}. The components of this bracket are expressed by

{ω, θ}^{j} = L_{�_{ω}} (θ^{j}) − (−1)^{ωθ} L_{�_{θ}} (ω^{j}), j = 1, …, m. (3.6)

Obviously, the module F(π, π) ⊗_{F(π)} Λ_{v}^{∗}(π) is a graded Lie algebra with respect to the Jacobi superbracket.

In the case (ii), the generating section is i_{�_{ω}} (θ). Note now that any element ρ ∈ Λ_{v}^{1}(π) is of the form ρ = ∑_{σ,α} a_{σ,α}ω_{σ}^{α}, where

ω_{σ}^{α} = d_{C}^{v} u_{σ}^{α} = d u_{σ}^{α} − ∑_{i=1}^{n} u_{σ+1_{i}}^{α} d x_{i}.

Hence, if θ ∈ F(π, π) ⊗_{F(π)} Λ_{v}^{1}(π) and θ^{j} = ∑_{σ,α} a_{σ,α}^{j}ω_{σ}^{α}, then

( i_{�_{ω}} (θ))^{j} = ∑_{σ,α} a_{σ,α}^{j}D_{σ}(ω^{α}). (3.7)

In particular, we see that (3.7) establishes an isomorphism between the modules F(π, π)⊗_{F(π)}Λ_{v}^{1}(π) and C Diff(π, π) and defines the action of C-differential operators on elements of Λ_{v}^{∗}(π). This is a really well-defined action because of the fact that i_{CX} ω = 0 for any X ∈ D(M) and ω ∈ Λ_{v}^{∗}(π).

Consider now a differential equation E ⊂ J^{k}(π) and assume that it is determined by a differential operator Δ ∈ F(π, ξ). Denote by ℓ_{E} the restriction of the operator of universal linearization ℓ_{Δ} onto E^{∞}. Let ℓ_{E}^{[p]} be the extension of ℓ_{E} to F(π, π)⊗_{F(π)} Λ_{v}^{p}(E) which is well defined due to the above said. Then the module H_{C}^{p,0}(E) is identified with the set of evolution superderivations �_{ω} whose generating sections ω ∈ F(π, π) ⊗_{F(π)} Λ_{v}^{p}(E) satisfy the equation

ℓ_{E}^{[p]}(ω) = 0 (3.8)

(see [Ver] for the proof and more details). If E satisfies the assumptions of the 2-lines theorem (see [Vin1, Ver]), then H_{C}^{p,1}(E) is identified with the cokernel of ℓ_{E}^{[p−1]} and thus

H_{C}^{i}(E) = ker ℓ_{E}^{i} ⊕ coker ℓ_{E}^{[i−1]}

in this case.

As it was noted in Remark 2.2, H_{C}^{1}(E) is an associative algebra with respect to contraction represented in the algebra of endomorphisms of H_{C}^{0}(E). It is easily seen that the action of the H_{C}^{0,1}(E) is trivial while H_{C}^{1,0}(E) acts on H_{C}^{0}(E) = sym E as C-differential operators (see above).

Definition 3.6. Elements of the module H_{C}^{1,0}(E) are called recursion operators for symmetries of the equation E.

We use the notation R(E) for the algebra of recursion operators.

Remark 3.2. The algebra R(E) is always nonempty, since it contains the structural element U_{E} which is the unit of this algebra. “Usually” this is the only solution of (3.8) for p = 1. This fact apparently contradicts to practical experience (cf. with well-known recursion operators for the KdV and other integrable systems). The reason is that these operators contain nonlocal terms like D^{−1} or of a more complicated form. An adequate framework to deal with such constructions will be described in the next section.

Remark 3.3. Let φ ∈ sym E be a symmetry and R ∈ R(E) be a recursion operator. Then we obtain a series of symmetries φ_{0} = φ, φ_{1} = R(φ), …, φ_{n} = R^{n}(φ), . . .. Using identity (1.19), one can compute the commutators [φ_{m}, φ_{n}] in terms of [[φ, R]]^{fn} ∈ H_{C}^{1,0}(E) and [[R, R]]^{fn} ∈ H_{C}^{2,0}(E). In particular, it can be shown that when both [[φ, R]]^{fn} and [[R, R]]^{fn} vanish, all symmetries φ_{n} mutually commute (see [Kr2]).

For example, if E is an evolution equation, H_{C}^{p,0}(E) = 0 for all p ⩾ 2. Hence, if φ is a symmetry and R is a φ-invariant recursion operator (i.e., such that [[φ, R]]^{fn} = 0), then R generates a commutative series of symmetries. This is exactly the case for the KdV and other evolution integrable equations.

4. Nonlocal and superextensions

Let E ⊂ J^{k}(π) be a differential equation and E^{∞} ⊂ J^{∞}(π) be its infinite prolongation. Consider the algebra Λ_{v}^{∗}(E) and a vector field X ∈ D(M). Define C_{S}X : Λ_{v}^{∗}(E) → Λ_{v}^{∗}(E) by setting

C_{S}X(ω) = L_{CX}(ω), ω ∈ Λ_{v}^{∗}(E).

Then

C_{S}(fX)(ω) = L_{CfX}(ω) = f L_{CX}(ω) + d f ∧ i_{CX} ω = f L_{CX}(ω) = fC_{S}(X)(ω)

since, by definition, i_{CX} ω = 0 for all X ∈ D(M), ω ∈ Λ_{v}^{∗}(E). Obviously one has C_{S}X |_{D(M)} = X, and we obtain a flat connection in the algebra Λ_{v}^{∗}(E).

Definition 4.1. The pair SE = (Λ_{v}^{∗}(E), C_{S}) is called the superization of the equation E.

Remark 4.1. The object SE may be considered as a superdifferential equation. Its even part is the equation E, while the odd one may be described as follows. Let E be given by F_{α}(x_{1}, …, x_{n}, …, u_{σ}^{j}, …) = 0 in a special coordinate system (x_{i}, u_{σ}^{j}). Denote by v^{j} the form d u^{i} − ∑_{i} u_{1_{i}}^{j} d x_{i}. Then the odd component of SE is of the form

ℓ_{F_{α}}v ≡ ∑ \frac{∂F_{α}}{∂u_{σ}^{j}} D_{σ}v^{j} = 0.

Thus, SE governs the behavior of the generators in Λ_{v}^{∗}(E).

The construction of SE is a particular case of a more general concept.

Definition 4.2. Let E be a differential equation and A be a commutative graded F(E)-algebra. Suppose that there exists a flat connection C_{A} in A as in a C^{∞}(M)- algebra. Then (A, C_{A}) is called an extension of the equation E, if C_{A}X |_{F(E)} = CX for any derivation X : C^{∞}(M) → A. Two extensions, (A, C_{A}) and (B, C_{B}), are said to be equivalent, if there exists an F(E)-isomorphism f : A → B such that the diagram

A ⟶^{f} B
C_{A}(X)↓  ↓^{C_{B}(f∘X)}
A ⟶^{f} B

is commutative for any derivation X : C^{∞}(M) → A.

For any extension (A, C_{A}), one can repeat all the constructions of Section 2 and to define symmetries, C-cohomologies, etc. Let us describe the symmetries of the superization SE.

Lemma 4.1. Let E be a formally integrable equation. Then a derivation X ∈ D^{v}(Λ^{∗}(E)) lies in H_{C}^{∗,0}(E) if and only if [[CY , X]]^{fn} = 0 for any Y ∈ D(M).

Proof. Let X ∈ D^{v}(Λ^{∗,0}(E)). Then ∂_{C}X ∈ D^{v}(Λ^{∗,1}(E)) and consequently ∂_{C}X = 0 if and only if i_{CY} ∂_{C}X = 0 for all Y ∈ D(M). Use (1.19) now:

i_{CY} ∂_{C}X = i_{CY} [[U_{C}, X]]^{fn} = [[i_{CY} U_{C}, X]]^{fn}−[[U_{C}, i_{CY} X]]^{fn}−i_{[[CY ,U_{C}]]^{fn}} X−i_{[[CY ,X]]^{fn}} U_{C}.

But all terms in right-hand side vanish except for the last one, which equals to −[[CY , X]]^{fn} due to verticality of [[CY , X]]^{fn}.

Theorem 4.2. For any formally integrable equation one has

sym SE = H_{C}^{∗,0}(E) ⊕ H_{C}^{∗,0}(E).

Proof. Let X : Λ^{∗,0}(E) → Λ^{∗,0}(E) be a vertical derivation. Any such a derivation can be uniquely represented in the form

X = L_{X_{0}}^{C} + i_{X_{1}}, X_{0}, X_{1} ∈ D^{v}(Λ^{∗,0}(E)),

where L_{X_{0}}^{C} = [i_{X_{0}}, d_{C}^{v}]. Then, using Proposition 1.5, we obtain

[C_{S}Y, X] = [L_{CY} , L_{X_{0}}^{C} + i_{X_{1}}] = [L_{CY} , [i_{X_{0}}, d_{C}^{v}]] + [L_{CY} , i_{X_{1}}]
= [[L_{CY} , i_{X_{0}}], d_{C}^{v}] + [i_{X_{0}}, [L_{CY} , d_{C}^{v}]] + i_{[[CY ,X_{1}]]^{fn}} − L_{i_{X_{1}} CY} .

where Y ∈ D(M). But the second and fourth summands vanish while the first one equals to L_{[[CY ,X_{0}]]^{fn}}. Hence [[CY , X_{0}]]^{fn} = 0, [[CY , X_{1}]]^{fn} = 0, which gives the result by Lemma 4.1.

Thus we see that there are two embeddings of H_{C}^{∗,0}(E) into sym SE: the first one, L^{C}, given by the Lie derivative and another one, i, defined by the interior product. Note that if R ∈ R(E) is a recursion operator and φ ∈ sym E is a symmetry, then [i(φ), L^{C}(R)] = L^{C}(Rφ) and thus the action of recursion operators on symmetries is expressed in terms of the Lie bracket in sym SE. Note also that the algebra sym SE is always nonempty: it contains two elements, L^{C}(U_{E}) and i(U_{E}), at least.

Consider a particular and a very important case of extensions. Let τ : Ẽ → E^{∞} be a fiber bundle and F_{τ} (E) denote the function algebra C^{∞}(Ẽ).

Definition 4.3 (cf. [KrVin]). We say that τ : Ẽ → E^{∞} is a covering over E, if there exists an extension structure (F_{τ} (E), C_{τ}) in F_{τ} (E). A symmetry of (F_{τ} (E), C_{τ}) is called a nonlocal symmetry of the type τ of the equation E, or τ-symmetry.

More general, let (A, C_{A}) be an extension of E and (B, C_{B}) be an extension of (A, C_{A}). We say that (B, C_{B}) is a covering over (A, C_{A}), if they are of the same grading.

Geometrically, it means that a flat connection C_{τ} is given in the bundle π_{∞} ∘ τ : Ẽ → M satisfying C_{τ} X(f) = CX(f) for any f ∈ F(E).

Example 4.1. Let E be given by an operator Δ : Γ(π) → Γ(π^{1}). The infinite prolongation Δ^{∞} determines the map φ_{Δ} : J^{∞}(π) → J^{∞}(ξ^{1}). Then, under unrestrictive regularity conditions, the image φ_{Δ}J^{∞}(π) is of the form E_{1}^{∞} for some equation E_{1} ⊂ J^{k_{1}}(π^{1}). This equation E_{1} consists of compatibility conditions for E and is generated by the Nöther identities of E. If E_{1} is a proper equation, i.e., E_{1}^{∞} does not coincide with J^{∞}(π^{1}), we can repeat the construction, etc. Thus we obtain the sequence of coverings

E^{∞} ⊂ J^{∞}(π) ⟶^{φ_{Δ}} E_{1}^{∞} ⊂ J^{∞}(π^{1}) ⟶^{φ_{Δ_{1}}} … ⟶^{φ_{Δ_{s−1}}} E_{s}^{∞} ⊂ J^{∞}(π^{s}) ⟶^{φ_{Δs}} …

which may be called the nonlinear compatibility sequence of E (cf. [Ver]) and was discovered by A. Vinogradov some 10 years ago, [Vin3].

Consider on J^{∞}(π^{1}) the evolution derivation �_{δ} whose generating section δ is the diagonal in F(π^{1}, π^{1}) (i.e., j_{∞}(φ)^{∗}(δ) = φ for any φ ∈ Γ(π^{1})). The composition �_{δ}^{Δ} = φ_{Δ}^{∗} ∘ �_{δ} is a F(π)-valued derivation of F(π^{1}).^{4} Then the contraction i_{�_{δ}^{Δ}} coincides with the Koszul – Tate differential while elements of Λ_{v}^{1}(E_{1}) are identified with anti-fields.

Example 4.2. Let E be an equation and (A, C_{A}) be its extension. Consider a horizontal 1-form ω ∈ Λ_{h}^{1}(A) and the algebra A_{ω} = A(u_{ω}), where the variable u_{ω} carries the same grading as the one inherited by ω from A. For any X ∈ D(M), define C_{A_{ω}}u_{ω} = i_{CX} ω. Then the pair (A_{ω}, C_{A_{ω}}) is the extension of E and a covering over (A, C_{A}).

If two horizontal forms, ω and ω′, are d_{C}^{h}-homologous (say, ω′ = ω+d_{C}^{h} a, a ∈ A), then the coverings (A_{ω}, C_{ω}) and (A_{ω′}, C_{ω′}) are equivalent and the corresponding isomorphism is given by u_{ω} ↦ u_{ω′} − a. Thus we established a correspondence between the group H_{h}^{0,1}(A) and the set of classes of equivalent coverings over A. Note that the d_{C}^{h}-cohomology class of ω is “killed” in (A_{ω}, C_{ω}), since, by definition,

i_{CX} d_{C}^{h} u_{ω} = L_{CX} u_{ω} = i_{CX} ω

for any X ∈ D(M).

Let now ω_{1}, …, ω_{α}, … be an ℝ-basis in H_{h}^{0,1}(A). Then by a similar construction, we can consider the extension (K^{1}A, C^{1}), where K^{1}A = A(u_{ω_{1}}, …, u_{ω_{α}}, …) and the connection C_{A}^{1} is given as above. Evidently the equivalence class of K^{1}A is independent of basis choice. All elements of H_{h}^{0,1}(A) will be killed in K^{1}A, but new ones may arise and we shall kill them by constructing K^{2}A = K^{1}(K^{1}A), etc. Thus we obtain the series of extensions

(A, C_{A}) ⊂ (K^{1}A, C_{A}^{1}) ⊂ ⋯ ⊂ (K^{s}A, C_{A}^{s}) ⊂ (K^{s+1}A, C_{A}^{s+1}) ⊂ …

whose direct limit is denoted by (K^{∗}A, C^{∗}) and is called the universal Abelian covering (or extension) of A (cf. [Kh]). By construction, H_{h}^{0,1}(K^{∗}A) = 0.

Let again E be an equation (or a superequation) and τ : Ẽ → E^{∞} be a covering. The any C-differential operator Δ on E^{∞} can be lifted up to an operator Δ̃ on Ẽ. In particular, we can consider the operator ℓ̃_{E} and solutions of the equations

ℓ̃_{E}φ = 0, φ ∈ Γ((π_{∞} ∘ τ)^{∗}, π). (4.1)

Solutions of (4.1) are called shadows of τ-symmetries. We say that a shadow φ is recoverable, if there exists a τ-symmetry S such that S |_{F(E)} = �_{φ} |_{F(E)} . Of course, not all shadows are recoverable, but the universal Abelian covering is a happy case.

Denote by 𝔨^{s} : 𝔎_{E}^{s} → E^{∞} the covering corresponding to the extension K^{s}F(E) and by 𝔨^{∗} : 𝔎_{E}^{∗} → E^{∞} the covering corresponding to K^{∗}F(E)

Theorem 4.3 ([Kh]). Any 𝔨^{∗}-shadow is recoverable in the covering 𝔨^{∗}.

Though the proof in [Kh] was given for nongraded equations, it remains to be literary valid for the super case as well. Note now that by construction any 𝔨^{s}- shadow is also a 𝔨^{∗}-shadow. Hence we have a tautological consequence of the above theorem.

4We consider F(π) as an F(π^{1})-algebra due to φ_{Δ}^{∗}.

Corollary 4.4. Any 𝔨^{s}-shadow is recoverable in the covering 𝔨^{∗}.

Using this result, we prove the following theorem stating existence of well-defined action of nonlocal recursion operators in 𝔨^{∗}.

Theorem 4.5 (action of recursion operators in K^{∗}). Let E be a formally integrable equation and SE be its superization. Consider the universal Abelian extension K^{∗}SE and a 𝔨^{∗}-shadow ω of grading 1. Then, if S = S_{0} is a symmetry of 𝔎_{E}^{∗}, then there exists a series of symmetries in 𝔎_{E}^{∗} defined by S_{i+1} = i_{S_{i}} ω.

Proof. First, following Theorem 4.3 let us recover the shadow ω up to a symmetry S_{ω} (of grading 1) in K^{∗}SE. Let S be a symmetry of K^{∗}E. Then the restriction i_{S} |_{SE} is a shadow in K^{∗}E (it is a consequence of Remark 4.1) and thus can be lifted up to a symmetry S^{−} of K^{∗}SE. Since S^{−} is of grading −1, the commutator [S^{−}, S_{ω}] is of grading 0 and thus can be restricted onto K^{∗}E ⊂ K^{∗}SE which gives the result desired.

Example 4.3. Let u_{t} = u_{xxx} + uu_{x} be the KdV equation. Choose the functions t, x, u_{0}, …, u_{k}, …, where u_{k} corresponds to ∂^{k}u/∂x^{k}, for the internal coordinates in E^{∞}. Denote by ω_{k} the form d_{C}^{v} u_{k}. The form ω = u_{0} d x + (u_{2} + 1/2u_{0}^{2}) d t is a d_{C}^{h}-closed form with nontrivial cohomology class. Let u_{−1} be the new variable in the corresponding covering. Then ρ = ω_{2} + 2/3u_{0}ω_{0} + 1/3u_{1}ω_{−1} is a shadow in 𝔨^{∗}. The corresponding recursion operator is D_{x}^{2} +2/3u_{0} +1/3u_{1}ω_{−1} and coincides with the classical one, [Olv]. Other, less trivial examples one can find in [KrKe2].

5. Concluding remarks

I want to finish this paper with mentioning some topics not discussed above — partially because of lack of space, but mainly because they need to be more clarified.

1. C-cohomologies and Poisson structures. As it was mentioned in the Introduction, the term E_{1} of Vinogradov’s C-spectral sequence and C-complex introduced here are in a sense mutually dual: while C-cohomologies contain information on (super)symmetries of equation E, the (n − 1)-st line in E_{1}(E) consists of (super)conservation laws of the same equation. Moreover, elements of H_{C}^{∗}(E) act on E_{1}(E) by the Lie derivative and if a cochain map P : E_{0}(E) → D^{v}(Λ^{∗}(E)) is given, we can define the bracket on E_{1}(E),

⟨[ω], [ω′]⟩_{P} = [L_{P(ω)} ω′], ω, ω′ ∈ E_{0}(E),

where [ ·] denotes the cohomology class. In [Kr3, Kr4] it was shown that in the case when E of evolution equations and when P is a C-differential operator, ⟨·, ·⟩_{P} is a Poisson bracket, if P (i) satisfies two conditions (skew-symmetry and the Jacobi identity) deduced in [AsVin] for the case J^{∞}(π) and (ii) is a solution of the operator equation

ℓ_{E} ∘ P + P^{∗} ∘ ℓ_{E}^{∗} = 0,

where Δ^{∗} denotes the operator formally adjoint to Δ. It seems that the same is true when E satisfies assumptions of the 2-lines theorem.

2. Generalized connections and higher differential forms. The definition of a connection used here (see Definition 2.1) may be fruitfully used in different situation generating corresponding cohomology theories. For example, taking a pair of algebras A ⊂ B, one can consider natural transformations of the functors Diff_{∗}(A, •) and Diff_{∗}(•). Then, literary repeating the scheme of Section 2, we shall obtain a differential in the module Diff_{∗}^{v}(Λ^{σ(∗)}), where Λ^{σ(∗)} denotes the module of higher differential forms of the algebra B, [VeVi]. It is almost obvious that its 0-cohomology coincides with the module of secondary differential operators (when B = F(E), see [GVY] for the definition). It would be quite interesting to consider arising invariants in more details.

3. Relation to the Koszul – Tate complex. Finally, I would like to note that Example 4.1 puts the Koszul – Tate complex in a natural geometric framework which opens promising perspectives for further constructions.

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The Diffiety Institute and Moscow Institute for Municipal Economy Current address: 1st Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia

E-mail address: josephk@glasnet.ru, josephk@mail.ecfor.rssi.ru