DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 381–391

A SUPERSYMMETRY APPROACH TO POISSON STRUCTURES OVER DIFFERENTIAL EQUATIONS

I. S. KRASIL’SHCHIK

Abstract. An exposition of Poisson structures theory over nonlinear partial differential equations is given. The approach is based on consideration of d_{h}-invariant Hamiltonian formalism in the superalgebra Λ^{∗}(E^{∞}), d_{h} being the horizontal differential.

Relations between supersymmetries and Poisson structures are established. A local description of Poisson structures for the two cases is given: E^{∞} = J^{∞}(π) and E being a system of evolution equations.

1. Basic algebraic structures related to nonlinear partial differential equations.

When dealing with so-called integrable systems, one always meets with the following concepts:

- symmetries,

- conservation laws,

- Hamiltonian maps (Poisson structures),

- recursion operators. In fact, all these concepts can be defined^{1} for any nonlinear partial differential equation and only the problem of existence depend on a particular type of the equation at hand. Moreover, the basic constructions are of a purely algebraic (cohomological, to be more exact) nature and analytical reasons become important at the final step — in obtaining computational formulae.

Relations between basic PDE structures and algebraic constructions can be expressed by the following scheme:

1991 Mathematics Subject Classification. 35A30, 58F05, 58G05, 16W55, 81T60.

Key words and phrases. Nonlinear partial differential equations, supersymmetry, Poisson structures, Hamiltonian formalism.

The research described in this publication was made possible in part by Grant N ^{\underline{∘}} N2F000 from the International Science Foundation and by Grant N ^{\underline{∘}} N2F300 from the International Science Foundation and Russian Government.

This paper is in final form and no version of it will be submitted for publication elsewhere. 1and efficiently computed

Conservation Hamiltonian maps Symmetries
laws  and recursion

operators

Initial  Differential  C-differential  Vector-valued
setting  forms on E^{∞},  operators  forms on E^{∞},
de Rham  U-differential

differential

Defining  Cartan  Schouten  Nijenhuis
structure  ideal  bracket  bracket

Cohomology  Term E_{1}  Homotopy classes U-cohomology

of C-spectral sequence

Analytical  ℓ_{E}^{∗}φ = 0  Δ ∘ ℓ_{E}^{∗} + ℓ_{E} ∘ Δ = 0  ℓ_{E}ω = 0

formulae

References [10, 12, 13, 15] [1, 6, 7, 11]  [3, 4]

Consider these constructions in more details.

2. A general setting [8].

Let M be a smooth manifold, dim M = n, π : E → M be a locally trivial vector bundle and π_{k} : J^{k}(π) → M be the bundle of it k-jets. We consider a formally integrable equation E ⊂ J^{k}(π) and its prolongations:

E^{s} ⊂ J^{k+s}(π), E^{s+1} ⊂ J^{k+s+1}(π), …, E^{∞} ⊂ J^{∞}(π).

For any formally integrable equation E the Cartan connection

C : D(M) → D(E^{∞}), C : X ↦ CX

is defined by

j_{∞}^{∗}(f)(CX(φ)) = X(j_{∞}^{∗}(f)φ), φ ∈ C^{∞}(E^{∞}), f ∈ Γ(π),

where for any manifold N by D(N) denotes the Lie algebra of vector fields on N is denoted, Γ(π) is the module of sections for π and j_{k}(f) denotes the k-jet of the section f. The Cartan connection is flat, i.e.

C[X, Y] = [CX, CY], X, Y ∈ D(M)

and in local coordinate presentation coincides with the total derivatives. The Cartan connection determines two mutually dual splittings

D(E^{∞}) = CD(E^{∞}) ⊕ D^{v}(E^{∞})

and

Λ^{1}(E^{∞}) = C^{1}Λ(E^{∞}) ⊕ Λ_{h}^{1}(E^{∞}), (1)

where CD(E^{∞}) is spanned in D(E^{∞}) by the fields of the form CX, X ∈ D(M), D^{v}(E^{∞}) consists of π_{∞}-vertical fields, C^{1}Λ(E^{∞}) are 1-forms vanishing on CD(E^{∞}) and Λ_{h}^{1}(E^{∞}) are horizontal 1-forms on E^{∞}. From (1) it follows that

Λ^{i}(E^{∞}) = ∑_{p+q=i} C^{p}Λ(E^{∞}) ∧ Λ_{h}^{q}(E^{∞}), (2)

where

C^{p}Λ(E^{∞}) = \underbrace{C^{1}Λ(E^{∞}) ∧ … ∧ C^{1}Λ(E^{∞})}_{p times},

while Λ_{h}^{q}(E^{∞}) is the module of horizontal q-forms.

3. The C-spectral sequence [10, 12, 13, 15].

Consider the ideal CΛ^{∗}(E^{∞}) generated in Λ^{∗}(E^{∞}) by C^{1}Λ(E^{∞}). Then the filtration in Λ^{∗}(E^{∞}) by the powers of CΛ^{∗}(E^{∞}) is in agreement with the action of the de Rham differential d. Corresponding spectral sequence (so-called Vinogradov’s C-spectral sequence) converges to de Rham cohomology of M and possesses the following properties^{2}:

Theorem 1. Let E ⊂ J^{k}(π) be an equation. Then locally one has:

(i) The term E_{0} = E_{0}(E) is of the form E_{0}^{p,q} = C^{p}Λ(E^{∞}) ∧ Λ_{h}^{q}(E^{∞}) while d_{0}^{p,q} coincides with the horizontal differential d_{h} with coefficients in C^{p}Λ(E^{∞}).

(ii) The group E_{1}^{0,n−1} coincides with the group of conservation laws of the equation E.

(iii) The group E_{1}^{1,n−1} consists of generating functions of conservation laws. (iv) In the case E^{∞} = J^{∞}(π), only the groups E_{1}^{0,0} and E_{1}^{p,n} are nontrivial in

the term E_{1}, p ≥ 0.

(v) If E is a ”good” (roughly speaking, not highly overdetermined) equation, only the groups E_{1}^{0,0}, E_{1}^{p,n−1} and E_{1}^{p,n} are nontrivial in the term E_{1}, p ≥ 0. In this case the differential d_{1}^{0,n−1} : E_{1}^{p,n−1} → E_{1}^{p,n} is an embedding.

(vi) Generating functions of conservation laws satisfy the equation

ℓ_{E}^{∗}φ = 0, (3)

where ℓ_{E}^{∗} is the formally adjoint to the universal linearization operator of the equation E.

2To simplify exposition, we consider the case H^{∗}(M, ℝ) = ℝ.

4. The U-cohomology [3, 4].

This cohomology theory arises via the following scheme. Let k be a zero characteristic field and A be a commutative unitary k-algebra. Consider tensor modules Λ^{i}(A) ⊗_{A} D(A), where D(A) is the Lie algebra of derivations A → A and define a contraction

(ω ⊗ X)⌋θ =^{def} ω ∧ (X⌋θ), ω, θ ∈ Λ^{∗}(A), X ∈ D(A).

Then, using the standard scheme [2] one defines the operation of “Lievization” L_{Ω} : Λ^{∗}(A) → Λ^{∗}(A), Ω ∈ Λ^{i}(A) ⊗_{A} D(A), and it can be shown that the commutator [L_{Ω}, L_{Θ}] is again of the form L_{Ξ} for some Ξ ∈ Λ^{∗}(A) ⊗_{A} D(A). The element Ξ =^{def} [[Ω, Θ]]^{F N} is uniquely defined and is called the Frölicher-Nijenhuis bracket of Ω and Θ. The module Λ^{∗}(A) ⊗ D(A) is a super Lie algebra with respect to [[•, •]]^{F N} and for any element Ω ∈ Λ^{1}(A) ⊗ D(A) the map ∂_{Ω} =^{def} [[Ω, •]]^{F N} satisfies ∂_{Ω} ∘ ∂_{Ω} = 0.

Let U_{C} ∈ Λ^{1}(E^{∞})⊗D^{v}(E^{∞}) be the connection form of the Cartan connection on E. Then flatness of C is equivalent to [[U_{C}, U_{C}]]^{F N} = 0. Corresponding differential can be restricted onto Λ^{∗}(E^{∞})⊗D^{v}(E^{∞}) and this restriction is denoted by ∂_{C}. The groups H_{C}^{∗}(E^{∞}) =^{def} ker ∂_{C}/ im ∂_{C} are called U-cohomology groups of the equation E and enjoy the following properties:

Theorem 2. Let E ⊂ J^{k}(π) be a formally integrable equation. Then (i) Consider splitting (2). Then

∂_{C}(C^{p}(E^{∞}) ∧ Λ_{h}^{q}(E^{∞}) ⊗ D^{v}(E^{∞})) ⊂ C^{p}(E^{∞}) ∧ Λ_{h}^{q+1}(E^{∞}) ⊗ D^{v}(E^{∞}).

(ii) The differential

∂_{C}^{p,q} : C^{p}(E^{∞}) ∧ Λ_{h}^{q}(E^{∞}) ⊗ D^{v}(E^{∞}) → C^{p}(E^{∞}) ∧ Λ_{h}^{q+1}(E^{∞}) ⊗ D^{v}(E^{∞})

coincides with the horizontal differential with the coefficients in C^{p}(E^{∞})⊗D^{v}(E^{∞}). Consequently, one has the splitting H_{C}^{i}(E^{∞}) = ∑_{p+q=i} H_{C}^{p,q}(E^{∞}).

(iii) The group H_{C}^{∗}(E^{∞}) is a module over the term E_{1} of the C-spectral sequence and inherits both the contraction operation and the Frölicher-Nijenhuis bracket from Λ^{∗}(E^{∞}) ⊗ D^{v}(E^{∞}).

(iv) For any formally integrable equation E one has

H_{C}^{p,0} = ker ℓ_{E}^{[p]},

where ℓ_{E}^{[p]} : C^{p}Λ(E^{∞}) → C^{p}Λ(E^{∞}) is a natural extension of the universal linearization onto C^{p}Λ(E^{∞}). The correspondence between elements of ker ℓ_{E}^{[p]} is established by the formula

Ω ↦ � _{Ω} =^{def} ∑ D_{σ}(Ω)\frac{∂}{∂u_{σ}^{α}} ,

where u_{σ}^{α} in J^{∞}(π) are local coordinates corresponding to local coordinates (x_{1}, …,

x_{n}, u^{1}, …, u^{m}) in J^{0}(π) and D_{σ} =^{def} D_{x_{1}}^{σ_{1}} ∘ ⋯ ∘ D_{x_{n}}^{σ_{n}} for σ = (σ_{1}, …, σ_{n}) and D_{x_{i}}

being the total derivative along x_{i}.

(v) In particular, H_{C}^{0,0} = ker ℓ_{E} coincides with the Lie algebra sym E of higher symmetries for E.

(vi) The group H_{C}^{1,0} is an associative algebra with respect to contraction operation ⌋ and U_{C} is its unit.

(vii) The action H_{C}^{0,0}⌋H_{C}^{1,0} ⊂ H_{C}^{0,0} is a representation of this algebra and the elements of the algebra H_{C}^{1,0} = ker ℓ_{E}^{[1]} can be identified with recursion operators for symmetries of E. To be more precise, if ω ∈ C^{1}Λ(E^{∞}) is a solution of the equation

ℓ_{E}^{[1]}ω = 0 (4)

and φ ∈ sym E is a symmetry, then the action of corresponding recursion operator is expressed by

R_{ω}φ =^{def} � _{φ}⌋ω.

5. Super Poisson structures [5] and δ-invariant theory.

Let again A be a commutative unitary k-algebra and A = ∑_{α≥0} A_{α}, A_{0} = A, be a graded commutative A-algebra, i.e.

ab = (−1)^{ab}ba

for any a, b ∈ A (here and below, to simplify the notations, we write (−1)^{θ} instead of (−1)^{deg θ}. For any i ≥ 0 the module D_{i}(A) of i-derivations is defined together with the evaluation map

D_{i}(A) ⊗k A → D_{i−1}(A), Δ ↦ Δ(a) Δ ∈ D_{i}(A), a ∈ A.

For any two elements Δ ∈ D_{i}(A), Δ′ ∈ D_{j}(A) we define by induction their Schouten bracket [[Δ, Δ′]]^{S} ∈ D_{i+j−1}(A) by setting [[Δ, Δ′]]^{S} = [Δ, Δ′], when Δ, Δ′ ∈ D_{1}(A), and

[[Δ, Δ′]]^{S}a = [[Δ, Δ′a]]^{S} + (−1)^{aΔ′+j}[[Δa, Δ′]]^{S}, a ∈ A.

Then D_{∗}(A) = ∑_{i≥0} D_{i}(A) is a bigraded Lie algebra with respect to [[•, •]]^{S}.

We say that a bivector P ∈ D_{2}(A) is a super Poisson structure on A, if (a) P is an even element and (b)

[[P, P]]^{S} = 0. (5)

We also say that two Poisson structures P, P′ commute, if [[P, P′]]^{S} = 0. To any Poisson structure P we relate corresponding Poisson map f_{P} : Λ^{1}(A) → D_{1}(A) defined by f_{P} (da) = P a, d being the de Rham differential.

For any a, b ∈ A one defines the Poisson bracket corresponding to P by {a, b}_{P} =^{def} P (a, b). The basic property (5) of P implies that this bracket satisfies the super Jacobi identity.

Assume now that the algebra A is endowed with an odd differential δ : A → A, δ ∘ δ = 0. Then δ-invariant calculus over A can be constructed. Namely, we consider differential operators Δ : A → A satisfying [δ, Δ] = 0 and classes of such operators modulo operators of the form Δ = [δ, ∇]. Denote the set of these classes by Diff(A; δ). Then Diff(A; δ) acts on the homology group H(A; δ) = ker δ/ im δ and in fact it is this action which really matters in the theory.

The set Diff(A; δ) inherits the associative algebra structure with respect to composition of differential operators. One can extend the action of δ onto Diff(A; δ) by setting δ([Δ]) = [δ ∘ Δ], where Δ : A → A is a differential operator and [·] denotes its class. Obviously, δ ∘ δ = 0 again which makes it possible to define modules Diff^{i}(A; δ) =^{def} Diff(A, Diff^{i−1}(A; δ); δ). Thus, the whole conceptual scheme of differential calculus in commutative algebras [14] can be carried over into the δ-invariant setting. In particular, one can define modules of δ-invariant polyderivations D_{i}(A; δ) as well as of symmetric Diff_{i}^{sym}(A; δ) and skew-symmetric Diff_{i}^{alt}(A; δ) polydifferential operators respectively and define the evaluation maps

D_{i}(A; δ) ⊗ H(A; δ) → D_{i−1}(A; δ),
Diff_{i}^{sym}(A; δ) ⊗ H(A; δ) → Diff_{i−1}^{sym}(A; δ),
Diff_{i}^{alt}(A; δ) ⊗ H(A; δ) → Diff_{i−1}^{alt}(A; δ).

The construction of super Schouten bracket is obviously δ-invariant and consequently one obtains the brackets

[[D_{i}(A; δ), D_{j}(A; δ)]]^{S} ⊂ D_{i+j}(A; δ),
[[ Diff_{i}^{alt}(A; δ), Diff_{j}^{alt}(A; δ)]]^{S} ⊂ Diff_{i+j}^{alt}(A; δ).

A δ-invariant Poisson structure in A is defined as an even element P ∈ D_{2}(A; δ) satisfying [[P, P]]^{S} = 0 . In a similar manner one can define higher Poisson structures as elements of Diff_{2}^{alt}( CA; δ) complying with the same conditions.

6. Realization for nonlinear differential equations.

Let now E be a formally integrable equation and take the algebra Λ^{∗}(E^{∞}) = ∑_{p,q} C^{p}Λ(E^{∞})∧Λ_{h}^{q}Λ(E^{∞}) for the A and the differential d_{0} : C^{p}Λ(E^{∞})∧Λ_{h}^{q}(E^{∞}) → C^{p}Λ(E^{∞})∧Λ_{h}^{q+1}(E^{∞}) of the of C-spectral sequence term E_{0} for δ. Use the notation

D_{i}(Λ^{∗}(E^{∞}); d_{0}) =^{def} D_{i}(E).

Let us first describe the group D_{1}(E). To do this, we use the following wellknown property of graded derivations of the algebra Λ^{∗}(N), N being a smooth manifold: Any X ∈ D_{1}(Λ^{∗}(N)) is uniquely represented as the sum of a Lie derivative and a contraction:

X = L_{X_{0}} + i_{X_{1}}, (6)

for some uniquely determined X_{0}, X_{1} ∈ Λ^{∗}(N) ⊗ D(N). Thus we have the maps

Δ_{0} : D(Λ^{∗}(N)) → Λ^{∗}(N) ⊗ D(N), Δ_{0}(X) = X_{0},
Δ_{1} : D(Λ^{∗}(N)) → Λ^{∗}(N) ⊗ D(N), Δ_{1}(X) = X_{1},

the first one being a homomorphism and the second one being a first order differential operator.

Using decomposition (6) and basic properties of the Frölicher-Nijenhuis bracket [4], one can see that

[X, Y]_{0} = [[X_{0}, Y_{0}]]^{F N} + X_{1}⌋Y_{0} − (−1)^{XY} Y_{1}⌋X_{0},

[X, Y]_{1} = [[X_{1}, Y_{1}]]^{RN} + [[X_{0}, Y_{1}]]^{F N} − (−1)^{XY} [[Y_{0}, X_{1}]]^{F N}, (7)

for any X, Y ∈ D(Λ^{∗}(N)). Here [[•, •]]^{RN} is the so-called Richardson-Nijenhuis bracket defined as

[[Ω, Θ]]^{RN} =^{def} Ω⌋Θ − (−1)^{(Ω+1)(Θ+1)}Θ⌋Ω,

for any Ω, Θ ∈ Λ^{∗}(N) ⊗ D(N).

The following description of D_{1}(E) is valid:

Theorem 3. For any formally integrable equation E one has an isomorphism

D_{1}(E) = H_{C}^{∗}(E^{∞}) ⊕ H_{C}^{∗}(E^{∞}).

In particular, the action of elements X = (X_{0}, X_{1}) ∈ H_{C}^{∗,0}(E^{∞}) ⊕ H_{C}^{∗,0}(E^{∞}) on D_{0}(E) = E_{1}^{∗,∗} is as follows:

([Ω_{0}], [Ω_{1}])[ω] = [L_{� _{Ω_{0}}} ω + i_{� _{Ω_{1}}} ω],

where [·] denotes the cohomology class.

We use the notation S(E) = H_{C}^{∗,0}(E^{∞}) ⊕ H_{C}^{∗,0}(E^{∞}) and call elements of S(E) supersymmetries of the equation E. Note that the commutator of superderivations induces a bracket in S(E) which combines elements of the Jacobi-Richardson-Frölicher-Nijenhuis brackets, namely, if X = (� _{Ω_{0}}, � _{Ω_{1}}) and Y = (� _{Θ_{0}}, � _{Θ_{1}}), then [X , Y] = (� _{Ξ_{0}}, � _{Ξ_{1}}) with

Ξ_{0} = {Ω_{0}, Θ_{0}} + � _{Ω_{0}}⌋Ω_{1} − (−1)^{Ω_{0}(Θ_{1}+1)}� _{Ω_{0}}⌋Θ_{1},
Ξ_{1} = � _{Ω_{1}}⌋Θ_{1} − (−1)^{(Ω_{1}+1)(Θ_{1}+1)}� _{Θ_{1}}⌋Ω_{1} +
{Ω_{0}, Θ_{1}} − (−1)^{Θ_{0}(Ω_{1}+1)}{Θ_{0}, Ω_{1}},

where {•, •} denotes the super Jacobi bracket.

We proceed now with description of D_{2}(E). Note that having an element X ∈ D_{2}(Λ^{∗}(E^{∞})) we get the evaluation map f_{X} : Λ^{∗}(E^{∞}) → D_{1}(E^{∞}) determined by f_{X}(ω) = Xω. Taking compositions f_{X} ∘ Δ_{i}, i = 0, 1, we obtain the operators

X_{0} : Λ^{∗}(E^{∞}) → Λ^{∗}(E^{∞}) ⊗ D(E^{∞}),
X_{1} : Λ^{∗}(E^{∞}) → Λ^{∗}(E^{∞}) ⊗ D(E^{∞}).

It can be shown [6] that when one considers d_{0}-homotopy classes, then description of these classes reduces to description of classes of second order differential operators Δ : Λ^{∗}(E^{∞}) → Λ^{∗}(E^{∞}) ⊗ D^{v}(E^{∞}) satisfying

Δ ∘ d_{0} + ∂_{C} ∘ Δ = 0. (8)

To be more precise, any element of D_{2}(E) is determined by a pair (Δ_{0}, Δ_{1}) of differential operators satisfying (8), such that Δ_{0} is a derivation and Δ_{1} is a second order differential operator whose symbol is determined by Δ_{0} (the details see in [6]. Due to the two-lines theorem [15] one obtains that for ”good” equations such operators act trivially on corresponding cohomologies.

This observation forces one to generalize the concept of a Poisson structure for differential equations. We do it in the next section.

7. Poisson structures and superization.

Definition 1. Let E ⊂ J^{k}(π) be a formally integrable equation.

(i) Say that a pair P = (Δ_{0}, Δ_{1}) of C-differential operators

Δ_{0}, Δ_{1} : Λ^{∗}(E^{∞}) → Λ^{∗}(E^{∞}) ⊗ D^{v}(E^{∞})

is Poissonnian, if

(a) both Δ_{0} and Δ_{1} satisfy equation (8), Δ_{0} is an even operator and deg Δ_{1} = deg Δ_{0} + 1;

(b) The bracket {•, •}_{P} defined by

{ω, θ}_{P} =^{def} L_{Δ_{0}ω}θ + i_{Δ_{1}ω}θ

is skew-symmetric and satisfies the graded Jacobi identity.

(ii) A Poisson structure on E related to the pair P = (Δ_{0}, Δ_{1}) is the corresponding cohomology map

P : E_{1}^{∗,∗} → H_{C}^{∗}(E^{∞}) ⊕ H_{C}^{∗}(E^{∞}).

(iii) A Poisson structure is said to be of a normal type, if

P(E_{1}^{∗,p_{0}}) ⊂ H_{C}^{∗,0}(E^{∞}) ⊕ H_{C}^{∗,0}(E^{∞}),

where p_{0} is the number of the first nontrivial line in E_{1}(E).

Thus, Poisson structures of a normal type send conservation laws of the equation E (as well as other elements of the (n − 1)-st line in E_{1}) to supersymmetries of this equation.

To get a characterization of the above defined Poisson structures we need the concept of superization of the equation E (see [3]).

Namely, extend the algebra C^{∞}(E^{∞}) by adding Cartan forms to it, i.e. elements of the modules C^{p}Λ(E^{∞}), p > 0, as variables of the degree p. Then the Cartan connection extends onto C^{∗}Λ(E^{∞}) = ∑_{p≥0} C^{p}Λ(E^{∞}). In fact

L_{C(fX)}ω = L_{fCX}ω = fL_{CX}ω + dω ∧ CX⌋ω = fL_{CX}ω.

Hence, one can introduce an object whose function algebra is generated over C^{∞}(E^{∞}) by new odd variables ω ∈ C^{1}Λ(E^{∞}). This object can be interpreted as a super differential equation. Denote it by S(E). Consequently, C^{∞}(SE) = ∑_{p≥0} C^{p})Λ(E^{∞}). Then all basic concepts, such as symmetries, conservation laws, etc., can be generalized onto this object. In particular, one has Proposition 4. For any formally integrable equation E the following facts hold:

(i) Symmetries of superization coincide with supersymmetries of the initial equation, i.e.

sym (SE) = S(E).

(ii) The system (E_{0}^{p,q}(E), d_{0}) is equivalent to the horizontal complex of the superized equation S(E). In particular, the group of conservation laws for the superized equation coincides with E_{1}^{∗,n−1}(E).

(iii) The system (C^{p}Λ(E^{∞})∧Λ_{h}^{q}(E^{∞})⊗D^{v}(E^{∞}), ∂_{C}) for the equation E is equivalent to the 0-th column of a similar system for superization.

The identifications above allow one to prove the following result [6]:

Theorem 5. Let E be the ”empty” equation, i.e. E^{∞} = J^{∞}(π). Then locally one has

(i) All nontrivial Poisson structures for E are of a normal type.

(ii) To any Poisson structure P on E there corresponds a super C-differential operator Δ : E_{1}^{1,n}(SE) → sym (SE) by which P is determines uniquely.

(iii) Corresponding Poisson bracket {•, •}_{Δ} if skew-symmetric if and only if

Δ + Δ^{∗} = 0, (9)

where Δ^{∗} is formally adjoint to Δ.

(iv) The bracket {•, •}_{Δ} satisfies the graded Jacobi identity if and only if

[� _{Δφ}, Δ] = ℓ_{Δφ} ∘ Δ + Δ ∘ ℓ_{Δφ}^{∗}, (10)

where � _{ψ} is the evolutionary super vector field corresponding to ψ.

(v) Two Poisson structures P and P′ commute if and only if

(11) [� _{Δφ}, Δ′] + [� _{Δ′φ}, Δ] = ℓ_{Δφ} ∘ Δ′ + ℓ_{Δ′φ} ∘ Δ + Δ ∘ ℓ_{Δ′φ}^{∗} + Δ′ ∘ ℓ_{Δφ}^{∗}.

The proof of this theorem uses the theory of formally adjoint operator presented for the graded case in [9] and is a straightforward generalizations of the results obtained in [1, 11].

Previous result easily generalizes to the case of a determined system of evolution equations [7] and we obtain the following

Theorem 6. Let E be a determined system of evolution equations presented as

u_{t}^{j} = Ψ^{j}(x, …, u_{σ}^{α}, …),

where j, α = 1, …, m, |σ| ≤ k, x = (x_{1}, …, x_{n}) and

u_{σ}^{j} = \frac{∂^{|σ|}u^{j}}{∂x_{1}^{σ_{1}} … ∂x_{n}^{σ_{n}}} .

Then Poisson structures of a normal type on E are determined by operators Δ satisfying Theorem 5 and such that

ℓ_{E} ∘ Δ + Δ ∘ ℓ_{E}^{∗} = 0, (12) where ℓ_{E} is the universal linearization operator for the equation E and ℓ_{E}^{∗} is its formally adjoint.

8. Example.

Consider the KdV equation E presented as

u_{t} = uu_{x} + u_{xxx}.

Then solving (12) in this case one obtains two independent solutions

Δ_{1} = D_{x},

where D_{x} is the total derivative with respect to x, and

Δ_{2} = D_{x}^{3} + \frac{2}{3}uD_{x} + \frac{1}{3}u_{x},

which coincide with well-known Hamiltonian maps for the KdV.

9. Acknowledgments.

I am grateful to the Erwin Schrödinger International Institute for Mathematical Physics (Wien, Austria), where a part this work was finished. I am also grateful to Professor Demeter Krupka for support of my participation in the Conference.

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