in the following way

i∆ ω ≡ ∆ ω = 0, if s > r,

i∆ ω = ∆(ω), if s = r, due to the de¬nition of Λr ,

ia ω = aω, if a ∈ A = D0 (A),

1

This distinction between ¬rst n gradings and additional (n + 1)-st one will be pre-

served both for graded forms and graded polyderivations throughout the whole chapter.

1. GRADED CALCULUS 247

and for r > s set by induction

i∆ (da § ω) = i∆(a) (ω) + (’1)∆·a+s da § i∆ (ω). (6.2)

Proposition 6.4. Let A be an n-graded commutative algebra.

(i) For any ∆ ∈ Ds (A) de¬nition (6.2) determines an (n + 1)-graded

di¬erential operator

i∆ : Λ — ’ Λ —

of the order s.

(ii) In particular, if ∆ ∈ D1 (A), then i∆ is a graded derivation of Λ— :

ω ∈ Λr , θ ∈ Λ— .

i∆ (ω § θ) = i∆ (ω) § θ + (’1)∆·ω+r ω § i∆ θ,

Now we consider tensor products of the form Λr —A Ds (A) and generalize

contraction and wedge product operations as follows

(ω — ∆) § (θ — ∆) = (’1)∆·θ (ω § θ) — (∆ § ),

iω—∆ (θ — ) = ω § i∆ (θ) — ,

where ω, θ ∈ Λ— , ∆, ∈ D— (A). Let us de¬ne the Richardson“Nijenhuis

bracket in Λ— — Ds (A) by setting

[[„¦, ˜]]rn = i„¦ (˜) ’ (’1)(ω+∆)·(˜+ )+(q’s)(r’s)

i˜ („¦), (6.3)

s

where „¦ = ω — ∆ ∈ Λr — Ds (A), ˜ = θ — ∈ Λq — Ds (A). In what follows,

we con¬ne ourselves with the case s = 1 and introduce an (n + 1)-graded

structure into Λ— — D1 (A) by setting

gr(ω — X) = (gr(ω) + gr(X), r), (6.4)

where gr(ω) and gr(X) are initial n-gradings of the elements ω ∈ Λr , X ∈

D1 (A). We also denote by „¦ and „¦1 the ¬rst n and (n + 1)-st gradings of

„¦ respectively in the powers of (’1).

Proposition 6.5. Let A be an n-graded commutative algebra. Then:

(i) For any two elements „¦, ˜ ∈ Λ— — D1 (A) one has

[i„¦ , i˜ ] = i[[„¦,˜]]rn .

1

Hence, the Richardson“Nijenhuis bracket [[·, ·]]rn = [[·, ·]]rn determines

1

— — D (A) the structure of (n + 1)-graded Lie algebra with respect

in Λ 1

to the grading in which (n+1)-st component is shifted by 1 with respect

to (6.4), i.e.,

(ii) [[„¦, ˜]]rn + (’1)„¦·˜+(„¦1 +1)(˜1 +1) [[˜, „¦]]rn = 0,

(iii) (’1)˜·(„¦+Ξ)+(˜1 +1)(„¦1 +Ξ1 ) [[[[„¦, ˜]]rn , Ξ]]rn = 0, where, as before,

denotes the sum of cyclic permutations.

(iv) Moreover, if ρ ∈ Λ— , then

[[„¦, ρ § θ]]rn = („¦ ρ) § ˜ + (’1)„¦·ρ § [[„¦, ˜]]rn .

(v) In conclusion, the composition of two contractions is expressed by

+ (’1)„¦1 i„¦§˜ .

i„¦ —¦ i ˜ = i „¦ ˜

248 6. SUPER AND GRADED THEORIES

1.4. De Rham complex and Lie derivatives. The de Rham di¬er-

ential d : Λr ’ Λr+1 is de¬ned as follows. For r = 0 it coincides with the

derivation d : A ’ Λ1 introduced in Proposition 6.2. For any adb ∈ Λ1 ,

a, b ∈ A, we set

d(adb) = da § db

and for a decomposable form ω = θ §ρ ∈ Λr , θ ∈ Λr , ρ ∈ Λr , r > 1, r , r <

r, set

dω = dθ § ρ + (’1)θ1 θ § dρ.

By de¬nition, d : Λ— ’ Λ— is a derivation of grading (0, 1) and, obviously,

d —¦ d = 0.

Thus, one gets a complex

d

0 ’ A ’ Λ1 ’ · · · ’ Λr ’ dΛr+1 ’ · · · ,

’

which is called the de Rham complex of A.

Let X ∈ D1 (A) be a derivation. A Lie derivative LX : Λ— ’ Λ— is

de¬ned as

LX = [iX , d] = iX —¦ d + d —¦ iX . (6.5)

Thus for any ω ∈ Λ— one has

LX ω = X dω + d(X ω).

The basic properties of LX are described by

Proposition 6.6. For any commutative n-graded algebra A one has

(i) If ω, θ ∈ Λ— , then

LX (ω § θ) = LX ω § θ + (’1)X·ω ω § LX θ,

i.e., LX really is a derivation of grading (gr(X), 0).

(ii) [LX , d] = LX —¦ d ’ d —¦ LX = 0.

(iii) For any a ∈ A and ω ∈ Λ— one has

LaX (ω) = aLX ω + da § iX (ω).

(iv) [LX , iY ] = [iX , LY ] = i[X,Y ] .

(v) [LX , LY ] = L[X,Y ] .

Now we extend the classical de¬nition of Lie derivative onto the elements

of Λ— — D1 (A) and for any „¦ ∈ Λ— — D1 (A) de¬ne

L„¦ = [i„¦ , d] = i„¦ —¦ d + (’1)„¦1 d —¦ i„¦ .

If „¦ = ω — X, then one has

Lω—X = ω § LX + (’1)ω1 dω § iX .

Proposition 6.7. For any n-graded commutative algebra A the follow-

ing statements are valid :

1. GRADED CALCULUS 249

(i) For any „¦ ∈ Λ— — D1 (A) one has

L„¦ (ρ § θ) = L„¦ (ρ) § θ + (’1)„¦·ρ+„¦1 ·ρ1 ρ § L„¦ θ, ρ, θ ∈ Λ— ,

i.e., L„¦ is a derivation of Λ— whose grading coincides with that of „¦.

(ii) [L„¦ , d] = L„¦ —¦ d ’ (’1)„¦1 d —¦ L„¦ = 0.

(iii) Lρ§„¦ = ρ § L„¦ + (’1)ρ1 +„¦1 dρ § i„¦ , ρ ∈ Λ— .

To formulate properties of L„¦ similar to (iv) and (v) of Proposition 6.6,

one needs a new notion.

1.5. Graded Fr¨licher“Nijenhuis bracket. We shall now study the

o

commutator of two Lie derivatives.

Proposition 6.8. Let, as before, A be an n-graded commutative alge-

bra.

(i) For any two elements „¦, ˜ ∈ Λ— — D1 (A), the commutator of

corresponding Lie derivatives [L„¦ , L˜ ] is of the form LΞ for some

Ξ ∈ Λ— — D1 (A).

(ii) The correspondence L : Λ— — D1 (A) ’ D1 (Λ— ), „¦ ’ L„¦ , is injec-

tive and hence Ξ in (i) is de¬ned uniquely. It is called the (graded)

Fr¨licher“Nijenhuis bracket of the elements „¦, ˜ and is denoted by

o

Ξ = [[„¦, ˜]]fn . Thus, by de¬nition, one has

[L„¦ , L˜ ] = L[[„¦,˜]]fn .

(iii) If „¦ and ˜ are of the form

ω, θ ∈ Λ— , X, Y ∈ D1 (A),

„¦ = ω — X, ˜ = θ — Y,

then

[[„¦, ˜]]fn = (’1)X·θ ω § θ — [X, Y ] + ω § LX θ — Y

+ (’1)„¦1 dω § (X θ) — Y

’ (’1)„¦·˜+„¦1 ·˜1 θ § LY ω — X

’ (’1)„¦·˜+(„¦1 +1)·˜1 dθ § (Y ω) — X

= (’1)X·θ ω § θ — [X, Y ] + L„¦ (θ) — Y

’ (’1)„¦·˜+„¦1 ·˜1 L˜ (ω) — X. (6.6)

(iv) If „¦ = X, ˜ = Y ∈ D1 (A) = Λ0 — D1 (A), then the graded Fr¨licher“

o

Nijenhuis bracket of „¦ and ˜ coincides with the graded commutator

of vector ¬elds:

[[X, Y ]]fn = [X, Y ].

The main properties of the Fr¨licher“Nijenhuis bracket are described by

o

Proposition 6.9. For any „¦, ˜, Ξ ∈ Λ— — D1 (A) and ρ ∈ Λ— one has

(i)

[[„¦, ˜]]fn + (’1)„¦·˜+„¦1 ·˜1 [[˜, „¦]]fn = 0. (6.7)

250 6. SUPER AND GRADED THEORIES

(ii)

(’1)(„¦+Ξ)·˜+(„¦1 +Ξ1 )·˜1 [[„¦, [[˜, Ξ]]fn ]]fn = 0, (6.8)

i.e., [[·, ·]]fn de¬nes a graded Lie algebra structure in Λ— — D1 (A).

(iii)

[[„¦, ρ § ˜]]fn = L„¦ (ρ) § ˜ ’ (’1)„¦·(˜+ρ)+(„¦1 +1)·(˜1 +ρ1 ) dρ § i˜ „¦

+ (’1)„¦·ρ+„¦1 ·ρ1 · ρ § [[„¦, ˜]]fn . (6.9)

(iv)

[L„¦ , i˜ ] + (’1)„¦·˜+„¦1 ·(˜1 +1) L˜ = i[[„¦,˜]]fn . (6.10)

„¦

(v)

iΞ [[„¦, ˜]]fn = [[iΞ „¦, ˜]]fn + (’1)„¦·Ξ+„¦1 ·(Ξ1 +1) [[„¦, iΞ ˜]]fn

+ (’1)„¦1 i[[Ξ,„¦]]fn ˜ ’ (’1)„¦·˜+(„¦1 +1)·˜1 i[[Ξ,˜]]fn „¦. (6.11)

Remark 6.5. Similar to the commutative case, identity (6.11) can be

taken for the inductive de¬nition of the graded Fr¨licher“Nijenhuis bracket.

o

Let now U be an element of Λ1 — D1 (A) and let us de¬ne the operator

‚U = [[U, ·]]fn : Λr — D1 (A) ’ Λr+1 — D1 (A). (6.12)

Then from the de¬nitions it follows that

‚U (U ) = [[U, U ]]fn = (1 + (’1)U ·U )LU —¦ LU (6.13)

and from (6.7) and (6.8) one has

(1 + (’1)U ·U )‚U (‚U „¦) + (’1)U ·U [[„¦, [[U, U ]]fn ]]fn = 0

for any „¦ ∈ Λ— — D1 (A).

We are interested in the case when (6.12) is a complex, i.e., ‚U —¦ ‚U = 0,

and give the following

Definition 6.1. An element U ∈ Λ1 — D1 (A) is said to be integrable, if

(i) [[U, U ]]fn = 0 and

(ii) (’1)U ·U equals 1.

From the above said it follows that for an integrable element U one has

‚U —¦ ‚U = 0, and we can introduce the corresponding cohomologies by

ker(‚U : Λr — D1 (A) ’ Λr+1 — D1 (A))

r

HU (A)

= .

im(‚U : Λr’1 — D1 (A) ’ Λr — D1 (A))

The main properties of ‚U are described by

Proposition 6.10. Let U ∈ Λ1 — D1 (A) be an integrable element and

„¦, ˜ ∈ Λ— — D1 (A), ρ ∈ Λ— . Then

(i) ‚U (ρ § „¦) = LU (ρ) § „¦ ’ (’1)U ·(„¦+ρ) dρ § i„¦ U + (’1)U ·ρ+ρ1 ρ § ‚U „¦.

2. GRADED EXTENSIONS 251

(ii) [LU , i„¦ ] = i‚U „¦ + (’1)U ·„¦+„¦1 L„¦ U .

(iii) [i„¦ , ‚U ]˜ + (’1)U ·˜ i[[„¦,˜]]fn U = [[i„¦ U , ˜]]fn + (’1)U ·„¦+„¦1 i‚U „¦ ˜.

(iv) ‚U [[„¦, ˜]]fn = [[‚U „¦, ˜]]fn + (’1)U ·„¦+„¦1 [[„¦, ‚U ˜]]fn .

From the last equality it follows that the Fr¨licher“Nijenhuis bracket is

o

— r

inherited by the module HU (A) = r≥0 HU (A) and thus the latter forms

an (n + 1)-graded Lie algebra with respect to this bracket.

2. Graded extensions

In this section, we adapt the cohomological theory of recursion opera-

tors constructed in Chapter 5 (see also [55, 58]) to the case of graded (in

particular, super) di¬erential equations. Our ¬rst step is an appropriate

de¬nition of graded equations (cf. [87] and the literature cited there). In

what follows, we still assume all the modules to be projective and of ¬nite

type over the main algebra A0 = C ∞ (M ) or to be ¬ltered by such modules

in a natural way.

2.1. General construction. Let R be a commutative ring with a unit

and A’1 ‚ A0 be two unitary associative commutative Zn -graded R-alge-

bras. Let D = D0 ‚ D(A’1 , A0 ) be an A0 -submodule in the module

D(A’1 , A0 ) = {‚ ∈ homR (A’1 , A0 ) | ‚(aa )

= ‚a · a + (’1)a·‚ a · ‚a , a, a ∈ A’1 }.

Let us de¬ne a Zn -graded A0 -algebra A1 by the generators

a ∈ A0 , ‚ ∈ D0 , gr[‚, a] = gr(‚) + gr(a),

[‚, a],

with the relations

[‚, a0 ] = ‚a0 ,

[‚, a + a ] = [‚, a] + [‚, a ],

[a ‚ + a ‚ , a] = a [‚ , a] + [‚ , a],

[‚, aa ] = [‚, a] · a + (’1)‚·a a · [‚, a ],

where a0 ∈ A’1 , a, a , a ∈ A, ‚, ‚ , ‚ ∈ D0 .

For any ‚ ∈ D0 we can de¬ne a derivation ‚ (1) ∈ D(A0 , A1 ) by setting

‚ (1) (a) = [‚, a], a ∈ A1 .

Obviously, ‚ (1) a = ‚a for a ∈ A0 . Denoting by D1 the A1 -submodule in

D(A0 , A1 ) generated by the elements of the form ‚ (1) , one gets the triple

{A0 , A1 , D1 }, A0 ‚ A1 , D1 ‚ D(A0 , A1 ),

which allows one to construct {A1 , A2 , D2 }, etc. and to get two in¬nite

sequences of embeddings

A’1 ’ A0 ’ · · · ’ Ai ’ Ai+1 ’ · · ·

252 6. SUPER AND GRADED THEORIES

and

D0 ’ D1 ’ · · · ’ Di ’ Di+1 ’ · · · ,

where Ai+1 = (Ai )1 , Di+1 = (Di )1 ‚ D(Ai’1 , Ai ), and Di ’ Di+1 is a

morphism of Ai+1 -modules.

Let us set

D∞ = inj lim Di .

A∞ = inj lim Ai ,

i’∞ i’∞

Then D∞ ‚ D(A∞ ) and any element ‚ ∈ D0 determines a derivation

D(‚) = ‚ (∞) ∈ D(A∞ ). The correspondence D : D0 ’ D(A∞ ) possesses

the following properties

D(X)(a) = X(a) for a ∈ A’1 ,

D(aX) = aD(X) for a ∈ A0 .

Moreover, by de¬nition one has

[D(X), D(Y )](a) = D(X)(Y (a)) ’ (’1)X·Y D(Y )(X(a)),

a ∈ A’1 , X, Y ∈ D0 .

2.2. Connections. Similar to Chapter 5, we introduce the notion of a

connection in the graded setting.

Let A and B be two n-graded algebras, A ‚ B. Consider modules the

of derivations D(A, B) and D(B) and a B-linear mapping

: D(A, B) ’ D(B).

The mapping is called a connection for the pair (A, B), or an (A, B)-

connection, if

(X)|A = X.

From the de¬nition it follows that is of degree 0 and that for any

derivations X, Y ∈ D(A, B) the element

(X) —¦ Y ’ (’1)X·Y (Y ) —¦ X

again lies in D(A, B). Thus one can de¬ne the element

( (X) —¦ Y ’ (’1)X·Y

R (X, Y ) = [ (X), (Y )] ’ (Y ) —¦ X)

which is called the curvature of the connection and possesses the following

properties

R (X, Y ) + (’1)X·Y R (Y, X) = 0, X, Y ∈ D(A, B),

a ∈ B,

R (aX, Y ) = aR (X, Y ),

R (X, bY ) = (’1)X·b bR (X, Y ), b ∈ B.

is called ¬‚at, if R (X, Y ) = 0 for all X, Y ∈ D(A, B).

A connection

Evidently, when the grading is trivial, the above introduced notions

coincide with the ones from Chapter 5.

2. GRADED EXTENSIONS 253

2.3. Graded extensions of di¬erential equations. Let now M be

a smooth manifold and π : E ’ M be a smooth locally trivial ¬bre bundle

over M . Let E ‚ J k (π) be a k-th order di¬erential equation represented as

a submanifold in the manifold of k-jets for the bundle π. We assume E to

be formally integrable and consider its in¬nite prolongation E i ‚ J ∞ (π).

Let F(E) be the algebra of smooth functions on E ∞ and CD(E) ‚

D(E) = D(F(E)) be the Lie algebra generated by total derivatives CX,

X ∈ D(M ), C : D(M ) ’ D(E) being the Cartan connection on E ∞ (see

Chapter 2).

Let F be an n-graded commutative algebra such that F0 = F(E). De-

note by CD0 (E) the F-submodule in D(F(E), F) generated by CD(E) and

consider the triple (F(E), F, CD0 (E)) as a starting point for the construc-

tion from Subsection 2.1. Then we shall get a pair (F∞ , CD∞ (E)), where

def

CD∞ (E) = (CD0 (E))∞ . We call the pair (F∞ , CD∞ (E)) a free di¬erential

F-extension of the equation E.

The algebra F∞ is ¬ltered by its graded subalgebras Fi , i = ’1, 0, 1, . . . ,

and we consider its ¬ltered graded CD∞ (E)-stable ideal I. Any vector ¬eld

(derivation) X ∈ CD∞ (E) determines a derivation XI ∈ D(FI ), where

FI = F/I. Let CDI (E) be an FI -submodule generated by such deriva-

tions. Obviously, it is closed with respect to the Lie bracket. We call the

pair (FI , CDI (E)) a graded extension of the equation E, if I © F(E) = 0,

where F(E) is considered as a subalgebra in F∞ .

Let F’∞ = C ∞ (M ). In an appropriate algebraic setting, the Cartan

connection C : D(F’∞ ) ’ D(F(E)) can be uniquely extended up to a con-

nection

CI : D(F’∞ , FI ) ’ CDI (E) ‚ D(FI ).

In what follows we call graded extensions which admit such a connection

C-natural. From the ¬‚atness of the Cartan connection and from the de¬ni-

tion of the algebra CD∞ (E) (see Subsection 2.1) it follows that CI is a ¬‚at

connection as well, i.e.,

RCI (X, Y ) = 0,

where X, Y ∈ D(F’∞ , FI ), for any C-natural graded extension

(FI , CDI (E)).

2.4. The structural element and C-cohomologies. Let us consider

a C-natural graded extension (FI , CDI (E)) and de¬ne a homomorphism UI ∈

homFI (D(FI ), D(FI )) by

UI (X) = X ’ CI (X’∞ ), X ∈ D(FI ), X’∞ = X|F’∞ . (6.14)

The element UI is called the structural element of the graded extension

(FI , CDI (E)).

Due to the assumptions formulated above, UI is an element of the module

D1 (Λ— (FI )), where FI is ¬nitely smooth (see Chapter 4) graded algebra, and

consequently can be treated in the same way as in the nongraded situation.

254 6. SUPER AND GRADED THEORIES

Theorem 6.11. For any C-natural graded extension (FI (E), CDI (E)),

the equation E being formally integrable, its structural element is integrable:

[[UI , UI ]]fn = 0.

Proof. Let X, Y ∈ D(FI ) and consider the bracket [[UI , UI ]]fn as an

element of the module homFI (DI (E) § DI (E), DI (E)). Then applying (6.11)

twice, one can see that

[[UI , UI ]]fn (X, Y ) = µ (’1)U ·Y [UI (X), UI (Y )] ’ (’1)U ·Y UI ([UI (X), Y ])

2

’ UI ([X, UI (Y )]) + UI ([X, Y ]) , (6.15)

where µ = (’1)X·Y (1 + (’1)U ·U ). Expression (6.15) can be called the graded

Nijenhuis torsion (cf. [49]).

From (6.14) if follows that the grading of UI is 0, and thus (6.15) trans-

forms to

[[UI , UI ]]fn (X, Y ) = (’1)X·Y · 2 [UI (X), UI (Y )] ’ UI [UI (X), Y ]

2

’ UI [X, UI (Y )] + UI [X, Y ] . (6.16)

Now, using de¬nition (6.14) of UI , one gets from (6.16):

[[UI , UI ]]fn (X, Y )

= (’1)X·Y · 2 [CI (X’∞ ), CI (Y’∞ )] ’ CI ([CI (X’∞ ), Y ]’∞ )

’ CI ([X, CI (Y’∞ ]’∞ ) + CI ((CI ([X, Y ]’∞ ))’∞ .

But for any vector ¬elds X, Y ∈ D(FI ) one has

(CI (X’∞ ))’∞ = X’∞ .

and

[X, Y ]’∞ = X —¦ Y’∞ ’ (’1)X·Y Y —¦ X∞ .

Hence,

[[UI , UI ]]fn (X, Y ) = (’1)X·Y · 2 [CI (X’∞ ), CI (Y’∞ )]

’ CI (CI (X’∞ ) —¦ Y’∞ ’ (’1)X·Y CI (Y’∞ ) —¦ X’∞ )

= (’1)X·Y 2RCI (X, Y ) = 0.

Hence, with any C-natural graded E-equation, in an appropriate alge-

braic setting, one can associate a complex

0 ’ D(FI ) ’ Λ1 (FI ) — D(FI ) ’ · · ·

‚

· · · ’ Λr (FI ) — D(FI ) ’I Λr+1 (FI ) — D(FI ) ’ · · · ,

’ (6.17)

2. GRADED EXTENSIONS 255

where ‚I („¦) = [[UI , „¦]]fn , „¦ ∈ Λr (FI ) — D(FI ), with corresponding cohomol-

ogy modules.

Like in Chapters 4 and 5, we con¬ne ourselves with a subtheory of this

cohomological theory.

2.5. Vertical subtheory.

Definition 6.2. An element „¦ ∈ Λ— (FI ) — D(FI ) is called vertical, if

L„¦ (•) = 0 for any • ∈ F’∞ ‚ FI = Λ0 (FI ).

Denote by D v (FI ) the set of all vertical vector ¬elds from D(FI ) =

Λ0 (FI ) — D(FI ).

Proposition 6.12. Let (FI , CDI (E)) be a C-natural graded extension of

an equation E. Then

(i) The set of vertical elements in Λr (FI ) — D(FI ) coincides with the

module Λr (FI ) — Dv (FI ).

(ii) The module Λ— (FI ) — Dv (FI ) is closed with respect to the Fr¨licher“

o

Nijenhuis bracket as well as with respect to the contraction operation:

[[Λr (FI ) — Dv (FI ), Λs (FI ) — Dv (FI )]]fn ‚Λr+s (FI ) — Dv (FI ),

Λr (FI ) — Dv (FI ) Λs (FI ) — Dv (FI ) ‚Λr+s’1 (FI ) — Dv (FI ).

(iii) An element „¦ ∈ Λ— (FI ) — D(FI ) lies in Λ— (F) — D v (FI ) if and only

if

i„¦ (UI ) = „¦.

(iv) The structural element is vertical : UI ∈ Λ1 (FI ) — Dv (FI ).

From the last proposition it follows that complex (6.17) can be restricted

up to

0 ’ D v (FI ) ’ Λ1 (FI ) — Dv (FI ) ’ · · ·

‚

· · · ’ Λr (FI ) — Dv (FI ) ’I Λr+1 (FI ) — Dv (FI ) ’ · · ·

’ (6.18)

Cohomologies

ker(‚I : Λr (FI ) — Dv (FI ) ’ Λr+1 (FI ) — Dv (FI ))

r

HI (E) =

im(‚I : Λr’1 (FI ) — Dv (FI ) ’ Λr (FI ) — Dv (FI ))

are called C-cohomologies of a graded extension. The basic properties of the

di¬erential ‚I in (6.18) are corollaries of Propositions 6.9 and 6.12:

Proposition 6.13. Let (FI (E), CDI (E)) be a C-natural graded extension

of the equation E and denote by LI the operator LUI . Then for any „¦, ˜ ∈

Λ— (FI ) — Dv (FI ) and ρ ∈ Λ— (FI ) one has

(i) ‚I (ρ § „¦) = (LI (ρ) ’ dρ) § „¦ + (’1)ρ1 · ρ § ‚I „¦,

(ii) [LI , i„¦ ] = i‚I „¦ + (’1)„¦1 L„¦ ,

(iii) [i„¦ , ‚I ]˜ = (’1)„¦1 (‚I „¦) ˜,

(iv) ‚I [[„¦, ˜]]fn = [[‚I „¦, ˜]]fn + (’1)„¦1 [[„¦, ‚I ˜]]fn .

256 6. SUPER AND GRADED THEORIES

Let dh = d ’ LI : Λ— (FI ) ’ Λ— (FI ). From (6.13) and Proposition 6.6 (ii)

it follows that dh —¦dh = 0. Similar to the nongraded case, we call dh the hori-

zontal di¬erential of the extension (FI , CDI (E)) and denote its cohomologies

—

by Hh (E; I).

Corollary 6.14. For any C-natural graded extension one has

— —

r

(i) The module HI (E) = r≥0 HI (E) is a graded Hh (E; I)-module.

—

(ii) HI (E) is a graded Lie algebra with respect to the Fr¨licher“Nijenhuis

o

— (F ) — D v (F ).

bracket inherited from Λ I I

— (E) inherits from Λ— (F ) — D v (F ) the contraction operation

(iii) HI I I

r+s’1

r s

HI (E) ‚ HI

HI (E) (E),

—

and HI (E), with the shifted grading, is a graded Lie algebra with re-

spect to the inherited Richardson“Nijenhuis bracket.

2.6. Symmetries and deformations. Skipping standard reasoning,

we de¬ne in¬nitesimal symmetries of a graded extension (FI (E), CDI (E)) as

DCI (E) = {X ∈ DI (E) | [X, CDI (E)] ‚ CDI (E)};

DCI (E) forms an n-graded Lie algebra while CDI (E) is its graded ideal con-

sisting of trivial symmetries. Thus, a Lie algebra of nontrivial symmetries

is

symI E = DCI (E)/CDI (E).

If the extension at hand is C-natural, then, due to the connection CI , one

has the direct sum decompositions

D(FI ) = Dv (FI ) • CDI (E), v

DCI (E) = DCI (E) • CDI (E), (6.19)

where

DCI (E) = {X ∈ DI (E) | [X, CDI (E)] = 0} = D v (FI ) © DCI (E),

v v

and symI E is identi¬ed with the ¬rst summand in (6.19).

Let µ ∈ R be a small parameter and UI (µ) ∈ Λ1 (FI ) — Dv (FI ) be a

smooth family such that

(i) UI (0) = UI ,

(ii) [[UI (µ), UI (µ)]]fn = 0 for all µ.

Then UI ( ) is a (vertical) deformation of a graded extension structure,

and if

1

UI (µ) = UI + UI · µ + o(µ),

1

then UI is called (vertical) in¬nitesimal deformation of UI . Again, skipping

motivations and literally repeating corresponding proof from Chapter 5, we

have the following

Theorem 6.15. For any C-natural graded extension (FI , CDI (E)) of the

equation E one has

0

(i) HI (E) = symI (E);

2. GRADED EXTENSIONS 257

1

(ii) The module HI (E) consists of the classes of nontrivial in¬nitesimal

vertical deformations of the graded extension structure UI .

The following result is an immediate consequence of the results of pre-

vious subsection:

Theorem 6.16. Let (FI , CDI (E)) be a graded extension. Then

1

(i) The module HI (E) is an associative algebra with respect to contrac-

tion.

(ii) The mapping

1 0

R : HI (E) ’ EndR (HI (E)),

where

0 1

R„¦ (X) = X X ∈ HI (E), „¦ ∈ HI (E),

„¦,

is a representation of this algebra. And consequently,

(iii)

1

(symI E) HI (E) ‚ symI E.

2.7. Recursion operators. The ¬rst equality in (6.19) gives us the

dual decomposition

Λ1 (FI ) = CΛ1 (FI ) • Λ1 (FI ), (6.20)

h

where

CΛ1 (FI ) = {ω ∈ Λ1 (FI ) | CDI (E) ω = 0},

Λ1 (FI ) = {ω ∈ Λ1 (FI ) | Dv (FI ) ω = 0}.

h

± f± dg± , f± , g± ∈ FI , be a one-form. Then, since by

In fact, let ω =

de¬nition d = dh + LI , one has

ω= f± (dh g± + LI (g± )).

±

D v (F

Let X ∈ I ). Then from Proposition 6.13 (ii) it follows that

LI (g) = ’LI (X g ∈ FI .

X g) + ‚I (X) g + LX (g) = X(g),

Hence,

(d ’ LI )g = X(g) ’ X(g) = 0.

X dh g = X

On the other hand,

LI (g) = UI dg,

and if Y ∈ CDI (E), then

Y LI (g) = Y (UI dg) = (Y UI ) dg

due to Proposition 6.5 (v); but Y UI = 0 for any Y ∈ CDI (E).

Thus, similar to the nongraded case, one has the decomposition

C p Λ(FI ) § Λq (FI ),

Λr (FI ) = (6.21)

h

p+q=r

258 6. SUPER AND GRADED THEORIES

where

C p Λ(FI ) = CΛ1 (FI ) § · · · § CΛ1 (FI ),

p times

and

Λq (FI ) = Λ1 (FI ) § · · · § Λ1 ,

h h

h

q times

and the wedge product § is taken in the graded sense (see Subsection 1.2).

Remark 6.6. The summands in (6.21) can also be described in the fol-

lowing way

C p Λ(FI ) § Λq (FI ) = {ω ∈ Λp+q (FI ) | X1 ... Xp+1 ω = 0,

h

ω = 0 for all X± ∈ Dv (FI ), Yβ ∈ CDI (E)}.

Y1 ... Yq+1

Proposition 6.17. Let (FI , CDI (E)) be a C-natural extension. Then

one has

‚I (C p Λ(FI ) § Λq (FI ) — Dv (FI )) ‚ C p Λ(FI ) § Λq+1 (FI ) — Dv (FI )

h h

for all p, q ≥ 0.

The proof is based on two lemmas.

Lemma 6.18. dh C 1 Λ(FI ) ‚ C 1 Λ(FI ) § Λ1 (FI ).

h

Proof of Lemma 6.18. Due to Remark 6.6, it is su¬cient to show that

Xv Yv X v , Y v ∈ Dv (FI ),

dh ω = 0, (6.22)

and

Xh Yh X h , Y h ∈ CDI (E),

dh ω = 0, (6.23)

where ω ∈ C 1 Λ(FI ). Obviously, we can restrict ourselves to the case ω =

LI (g), g ∈ FI :

Yv dh ω = Y v dh LI (g) = ’Y v LI d h g

= LI (Y v dh g) + LY v (dh g) = dh Y v (g).

Hence,

Xv Yv dh ω = X v dh Y v (g) = 0,

which proves (6.22). Now,

Yh dh ω = ’Y h LI d h g = Y h dh g) ’ UI

(d(UI d(dh g)).

But UI is a vertical element, i.e., UI ∈ Λ1 (FI ) — Dv (FI ). Therefore,

UI dh g = 0

and

Yh dh ω = ’Y h UI d(dh g)

2. GRADED EXTENSIONS 259

= ’Y h d(dh g) ’ (Y h § UI )

UI ) d(dh g).

The ¬rst summand in the right-hand side of the last equality vanishes, since,

by de¬nition, Y h UI = 0 for any Y h ∈ CDI (E). Hence,

Xh Yh dh ω = ’X h (Y h § UI ) d(dh g)

= ’(X h (Y h § UI )) d(dh g) ’ (X h § Y h § UI ) d(dh g)

= ’(X h § Y h § UI ) d(dh g).

But X h § Y h § UI is a (form valued) 3-vector while d(dh g) is a 2-form;

hence

Xh Yh dh ω = 0,

which ¬nishes the proof of Lemma 6.18.

Lemma 6.19. ‚I Dv (FI ) ‚ Λ1 — Dv (FI ).

h

Proof of Lemma 6.19. One can easily see that it immediately follows

from Proposition 6.13 (iii).

Proof of Proposition 6.17. The result follows from previous lem-

mas and Proposition 6.13 (i) which can be rewritten as

‚I (ρ § „¦) = ’dh (ρ) § „¦ + (’1)ρ1 ρ § ‚I („¦).

Taking into account the last result, one has the following decomposition

p,q

r

HI (E) = HI (E),

p+q=r

where

p,q p,q p,q’1

HI (E) = ker(‚I )/im(‚I ),

where ‚ i,j : C i Λ(FI ) § Λj (FI ) — Dv (FI ) ’ C i (FI ) § Λj+1 (FI ) — Dv (FI ).

k h

In particular,

0,1 1,0

1

HI (E) = HI (E) • HI (E). (6.24)

1 0

Note now that from the point of view of HI (E)-action on HI (E) =

symI E, the ¬rst summand in (6.24) is of no interest, since

Dv (FI ) Λ1 (FI ) = 0.

h

—,0 1,0

—

We call HI (E) the Cartan part of HI (E), while the elements of HI (E)

are called recursion operators for the extension (FI , CDI (E)). One has the

following

p,0 p,0

Proposition 6.20. HI (E) = ker ‚I .

Proof. In fact, from Proposition 6.17 one has

im(‚I ) © (C — Λ(FI ) — Dv (FI )) = 0,

which proves the result.

260 6. SUPER AND GRADED THEORIES

—,0

Note that HI (E) inherits an associative graded algebra structure with

1,0

respect to contraction, HI (E) being its subalgebra.

2.8. Commutativity theorem. In this subsection we prove the fol-

lowing

1,0 1,0 2,0

Theorem 6.21. [[HI (E), HI (E)]]fn ‚ HI (E).

The proof is based on the following

Lemma 6.22. For any ω ∈ C 1 Λ(FI ) one has

UI ω = ω. (6.25)

Proof of Lemma 6.22. It is su¬cient to prove (6.25) for the genera-

tors of the module C 1 Λ(FI ) which are of the form

g ∈ FI .

ω = LI (g),

From (6.10) one has

L I —¦ i UI ’ i UI —¦ L I + L UI = i[[UI ,UI ]]fn ,

UI

or

LI —¦ iUI ’ iUI —¦ LI + LI = 0. (6.26)

Applying (6.26) to some g ∈ FI , one sees that

UI LI (g) = LI (g).

1,0

Proof of Theorem 6.21. Let „¦, ˜ ∈ HI (E), i.e., „¦, ˜ ∈ C 1 Λ(FI )

and ‚I „¦ = ‚I ˜ = 0. Then from (6.11) it follows that

[[„¦, ˜]]fn = [[UI „¦, ˜]]fn + [[„¦, UI ˜]]fn ,

UI

or, due to Lemma 6.22,

[[„¦, ˜]]fn = 2[[„¦, ˜]]fn .

UI

Hence,

1 1

[[„¦, ˜]]fn = UI [[„¦, ˜]]fn = UI (UI [[„¦, ˜]]fn )

2 4

1

= ((UI UI ) [[„¦, ˜]]fn ’ (UI § UI ) [[„¦, ˜]]fn )

4

1

= (UI [[„¦, ˜]]fn ’ (UI § UI ) [[„¦, ˜]]fn )

4

1 1

= [[„¦, ˜]]fn ’ (UI § UI ) [[„¦, ˜]]fn ,

2 4

or

1

[[„¦, ˜]]fn = ’ (UI § UI ) [[„¦, ˜]]fn .

2

But UI ∈ C 1 Λ(FI ) — Dv (FI ) which ¬nishes the proof.

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 261

Corollary 6.23. The element UI is a unit of the associative algebra

1,0

HI (E).

Proof. The result follows from the de¬nition of the element UI and

from Lemma 6.22.

2,0

Corollary 6.24. Under the assumption HI (E) = 0, all recursion op-

erators for the graded extension (FI , CDI (E)) commute with respect to the

Fr¨licher“Nijenhuis bracket.

o

1,0 0

Let „¦ ∈ HI (E) be a recursion operator. Denote its action on HI (E) =

0

symI (E) by „¦(X) = X „¦, X ∈ HI (E). Then, from (6.11) it follows that

[[„¦, ˜]]fn = (’1)X·Y (’1)Y ·„¦ [„¦(X), ˜(Y )]

Y X

+ (’1)(Y +„¦)·˜ [˜(X), „¦(Y )]

’ (’1)„¦·˜ „¦((’1)Y ·˜ [˜(X), Y ] + [X, ˜(Y )])

’ ˜((’1)Y ·„¦ [„¦(X), Y ] + [X, „¦(Y )])

+ ((’1)„¦·˜ „¦ —¦ ˜ + ˜ —¦ „¦)[X, Y ] , (6.27)

1,0

for all X, Y ∈ symI (E), „¦, ˜ ∈ HI (E).

2,0

Corollary 6.25. If HI (E) = 0, then for any symmetries X, Y ∈

1,0

symI (E) and recursion operators „¦, ˜ ∈ HI (E) one has

(’1)Y ·„¦ [„¦(X), ˜(Y )] + (’1)(Y +„¦)·˜ [˜(X), „¦(Y )]

= (’1)„¦·˜ „¦((’1)Y ·„¦ [˜(X), Y ] + [X, ˜(Y )]) + ˜((’1)Y ·„¦ [„¦(X), Y ]

+ [X, „¦(Y )]) + ((’1)„¦—¦˜ „¦ —¦ ˜ + ˜ —¦ „¦)[X, Y ]. (6.28)

In particular,

(1 + (’1)„¦·„¦ ) (’1)Y ·„¦ [„¦(X), „¦(Y )]

’ (’1)Y ·„¦ „¦[„¦(X), Y ] ’ „¦[X, „¦(Y )] + „¦2 [X, Y ] = 0,

and if „¦ · „¦ is even, then

[„¦(X), „¦(Y )] = „¦([„¦(X), Y ] + (’1)Y „¦ [X, „¦(Y )] ’ (’1)Y „¦ „¦[X, Y ]).

(6.29)

Using Corollary 6.25, one can describe a Lie algebra structure of sym I E

in a way similar to Section 3 of Chapter 4.

3. Nonlocal theory and the case of evolution equations

Here we extend the theory of coverings and that of nonlocal symmetries

(see Chapter 3 to the case of graded equations (cf. [87]). We con¬ne our-

selves to evolution equations though the results obtained, at least partially,

are applicable to more general cases. For any graded equation the notion of

262 6. SUPER AND GRADED THEORIES

its tangent covering (an add analog of the Cartan covering, see Example 3.2

on p. 100) is introduced which reduces computation of recursion operators to

computations of special nonlocal symmetries. In this setting, we also solve

the problem of extending “shadows” of recursion operators up to real ones.

3.1. The GDE(M ) category. Let M be a smooth manifold and A =

C ∞ (M ). We de¬ne the GDE(M ) category of graded di¬erential equations

over M as follows. The objects of GDE(M ) are pairs (F, F ), where F is

a commutative n-graded A-algebra (the case n = ∞ is included) endowed

with a ¬ltration

A = F’∞ ‚ . . . ‚ Fi ‚ Fi+1 ‚ . . . , Fi = F, (6.30)

i

while F is a ¬‚at (A, F)-connection (see Subsection 2.2), i.e.,

(i) F ∈ homF (D(A, F), D(F)),

(ii) F (X)(a) = X(a), X ∈ D(A, F), a ∈ A,

(iii) [ F (X), F (Y )] = F ( F (X) —¦ Y ’ F (Y ) —¦ X), X, Y ∈ D(A, F).

From the de¬nition it follows that the grading of F is 0, and we also

suppose that for any X ∈ D(A, F) the derivation F (X) agrees with the

¬ltration (6.30), i.e.,

‚ Fi+s

F (X)(Fi )

for some s = s(X) and all i large enough.

Let (F, F ) and (G, G ) be two objects and • : F ’ G be a graded

¬ltered homomorphism. Then for any X ∈ D(A, F) the composition • —¦ X

lies in D(A, G). We say that it is a morphism of the object (F, F ) to

(G, G ) if the diagram

•

F ’G

—¦ X)

F (X) G (•

“ “

•

F ’G

is commutative for all X ∈ D(A, F). If • is a monomorphism, we say that

it represents a covering of (G, G ) over (F, F ).

Remark 6.7. Let E be an equation in some bundle over M . Then all

graded extensions of E are obviously objects of GDE(M ).

Remark 6.8. The theory of the previous section can be literally applied

to the objects of GDE(M ) as well.

3.2. Local representation. In what follows, we shall deal with the

following kinds of objects of the category GDE(M ):

(i) in¬nite prolongations of di¬erential equations;

(ii) their graded extensions;

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 263

(iii) coverings over (i) and (ii).

For particular applications local versions of these objects will be consid-

ered. It means the following:

(i) In a neighborhood O ‚ M local coordinates x = (x1 , . . . , xn ) are

chosen (independent variables);

(ii) the bundle π : E ’ M in which E is de¬ned is supposed to be a

vector bundle, and it trivializes over O. If (e1 , . . . , em ) is a basis

of local sections of π over O, then f = u1 e1 + · · · + um em for any

f ∈ “(π|O ), and u1 , . . . , um play the role of dependent variables for

the equation E;

(iii) the equation E is represented by a system of relations

±

F1 (x, . . . , uj , . . . ) = 0,

σ

.....................

F1 (x, . . . , uj , . . . ) = 0,

σ

where uj = ‚ |σ| uj /‚xσ , σ = (i1 , . . . , in ), |σ| = i1 + · · · + in ¤ k, are

σ

coordinates in the manifold of k-jets J k (π), k being the order of E;

(iv) a graded extension F of F(E) (see Subsection 2.3 is freely generated

over F(E) by homogeneous elements v 1 , v 2 , . . . . It means that F∞ is

j j

generated by v„ , where v0 = v j and

j j

v(i1 ,...,is +1,...,in ) = [Ds , v(i1 ,...,in ) ],

Ds being the total derivative on E ∞ corresponding to ‚/‚xs . In this

setting any graded extension of E can be represented as

± j j j

F1 (x, . . . , uσ , . . . ) + φ1 (x, . . . , uσ , . . . , v„ , . . . ) = 0,

................................................

F (x, . . . , uj , . . . ) + φ (x, . . . , uj , . . . , v„ , . . . ) = 0,

j

r

σ σ

r

................................................

φr+1 (x, . . . , uj , . . . v„ , . . . ) = 0,

j

σ

.............................

φr+l (x, . . . , uj , . . . v„ , . . . ) = 0,

j

σ

j

where φ1 , . . . , φr are functions such that φ1 = 0, . . . , φr = 0 for v„ = 0.

(v) for any covering • : F ’ G of the graded extension F by an object

(G, G ) we assume that G is freely generated over F by homogeneous

elements w 1 , w2 , . . . and

‚ ‚ def ˜

Xis

= Di + = Di

G

‚ws

‚xi s

with

‚ ‚

˜˜ s

Xis

[Di , Dj ] = [Di , Xj ]+[ , Dj ]

‚ws ‚ws

s s

264 6. SUPER AND GRADED THEORIES

‚ ‚

Xis s

+[ , Xj ] = 0,

‚ws ‚ws

s s

where i, j = 1, . . . , n, Xis ∈ G, and D1 , . . . , Dn are total derivatives

extended onto F: Di = F (‚/‚xi ). Elements w 1 , w2 , . . . are called

nonlocal variables related to the covering •, the number of nonlocal

variables being called the dimension of •.

3.3. Evolution equations. Below we deal with super (Z2 -graded) evo-

lution equations E in two independent variables x and t:

±

u1 = f 1 (x, t, u1 , . . . , um , . . . , u1 , . . . , um ),

t k k

(6.31)

.......................................

m

ut = f m (x, t, u1 , . . . , um , . . . , u1 , . . . , um ),

k k

where u1 , . . . , um are either of even or of odd grading, and uj denotes s

j /‚xs . We take x, t, u1 , . . . , um , . . . , u1 , . . . uj , . . . for the internal coor-

‚u 0 0 i i

dinates on E ∞ . The total derivatives Dx and Dt restricted onto the in¬nite

prolongation of (6.31) are of the form

∞ m

‚ ‚

uj

Dx = + ,

i+1

‚uj

‚x i

i=0 j=1

∞m

‚ ‚

Dx (f j )

i

Dt = + . (6.32)

‚uj

‚t i

i=0 j=1

In the chosen local coordinates, the structural element U = UE of the

equation E is represented as

∞ m

‚

(duj ’ uj dx ’ D i (f j ) dt) —

U= . (6.33)

i i+1

‚uj

i

i=0 j=1

Then for a basis of the module C 1 Λ(E) one can choose the forms

ωi = LU (uj ) = duj ’ uj dx ’ D i (f j ) dt,

j

i i i+1

while (6.33) is rewritten as

∞ m

‚

j

ωi —

U= . (6.34)

‚uj

i

i=0 j=1

j

— ‚/‚uj ∈ Λp (E) — D v (E). Then from (6.6) one

∞ m

Let ˜ = j=1 θi

i=0 i

has

∞ m

j j

fn

dx § (θi+1 ’ Dx (θi ))

‚E (˜) = [[U, ˜]] =

i=0 j=1

∞ m ij

± ‚Dx f ‚

j

+ dt § ’ Dt (θi ) —

θβ . (6.35)

‚u± ‚uj

β i

β=0 ±=1

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 265

From (6.35) one easily gets the following

p,0

Theorem 6.26. Let E be an equation of the form (6.31). Then HC (E)

consists of the elements

∞ m

‚

Dx (θj ) —

i

= ,

θ

‚uj

i

i=0 j=1

where θ = (θ 1 , . . . , θm ), θj ∈ C p Λ(E), is a vector-valued form satisfying the

equations

k m l

i j ‚f

Dx (θ ) j = 0, l = 1, . . . , m, (6.36)

‚ui

i=0 j=1

or in short,

(p)

p,0

HC (E) = ker E,

(p)

where E is the extension of the operator of universal linearization operator

onto the module C p Λ(E) —R Rm :

k m l

i j ‚f

(p) l

( E (θ)) = Dx (θ ) j , l = 1, . . . , m. (6.37)

‚ui

i=0 j=1

3.4. Nonlocal setting and shadows. Let now • be a covering of

equation (6.31) determined by nonlocal variables w 1 , w2 , . . . with the ex-

tended total derivatives of the form

‚

˜

Dx = D x + Xs ,

‚ws

s

‚

˜

Dt = D t + Ts , (6.38)

‚ws

s

satisfying the identity

˜˜

[Dx , Dt ] = 0. (6.39)

Denote by F(E• ) the corresponding algebra of functions and by Λ— (E• )

and D(E• ) the modules of di¬erential forms and vector ¬elds on F(E• )

respectively. Then the structural element of the covering object is

‚

(dωs ’ Xs dx ’ Ts dt) —

U• = U +

‚ws

s

and the identity

[[U• , U• ]]fn = 0

is ful¬lled due to (6.39).

If now

∞ m

‚ ‚

j

θi — ρs —

˜= +

‚uj ‚ws

s

i

i=0 j=1

266 6. SUPER AND GRADED THEORIES

is an element of the module Λp (E• ) — Dv (E• ), then one can easily see that

∞ m

j ˜j

fn

dx § θi+1 ’ Dx (θi )

‚• (˜) = [[U• , ˜]] =

i=0 j=1

∞ m ij

± ‚Dx f ‚

˜j

+ dt § ’ Dt (θi ) —

θβ

‚u± ‚uj

β i

β=0 ±=1

∞ m

‚Xs ‚Xs ˜

±

dx § ’ Dx (ρs )

+ θβ + ργ

‚u± ‚wγ

β

s γ

β=0 ±=1

∞m

± ‚Xs ‚Ts ‚

˜

+ dt § ’ Dt (ρs ) —

θβ ± + ργ . (6.40)

‚uβ ‚wγ ‚ws

γ

β=0 ±=1

Again, con¬ning oneself to the case ˜ ∈ C p Λ(E• ) — Dv (E• ), one gets the

following

Theorem 6.27. Let E be an equation of the form (6.31) and • be its

covering with nonlocal variables w1 , w2 , . . . and extended total derivatives

p,0

given by (6.38). Then the module HC (E• ) consists of the elements

∞ m

‚ ‚

˜i

Dx (θj ) — j + ρs —

= , (6.41)

θ,ρ

‚ws

‚ui s

i=0 j=1

where θ = (θ 1 , . . . , θm ) and ρ = (ρ1 , . . . , ρs , . . . ), θj , ρs ∈ C p Λ(E• ), are

vector-valued forms satisfying the equations

˜(p) (θ) = 0, (6.42)

E

and

∞ m

‚Xs ‚Xs

˜β ˜

Dx (θ± ) ± + ρj = Dx (ρs ),

‚uβ ‚wj

β=0 ±=1 j

∞m

‚Ts ‚Ts

˜β ˜

Dx (θ± ) ± + ρj = Dt (ρs ), (6.43)

‚uβ ‚wj

β=0 ±=1 j

(p) (p)

s = 1, 2, . . . , where ˜E is the natural extension of with Dx and Dt

E

˜ ˜

replaced by Dx and Dt in (6.37).

Similar to Chapter 5, we call (6.42) shadow equations and (6.42) relation

equations for the element (θ, ρ); solutions of (6.42) are called shadow solu-

tions, or simply shadows. Our main concern lies in reconstruction elements

p,0

of the module HC (E• ) from their shadows. Denote the set of such shadows

p,0

by SHC (E• ).

Remark 6.9. Let • be a covering. Consider horizontal one-forms

s

ω• = dh ws = Xs dx + Ts dt, s = 1, 2, . . . ,

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 267

where dh is the horizontal de Rham di¬erential associated to •. Then (6.42)

can be rewritten as

s

θ,ρ (ω• ) = d h ρs , s = 1, 2, . . . (6.44)

Remark 6.10. When Xs and Ts do not depend on nonlocal variables,

the conditions of • being a covering is equivalent to

s

dh ω• = 0, s = 1, 2, . . . ,

dh being the horizontal di¬erential on E. In particular, one-dimensional

coverings are identi¬ed with elements of ker(dh ). We say a one-dimensional

covering • to be trivial if corresponding form ω• is exact (for motivations see

Chapter 3). Thus, the set of classes of nontrivial one-dimensional coverings •

with ω• independent of nonlocal variables is identi¬ed with the cohomology

1

group Hh (E), or with the group of nontrivial conservation laws for E.

3.5. The functors K and T . Keeping in mind the problem of recon-

structing recursion operators from their shadows, we introduce two functors

in the category GDE(M ). One of them is known from the classical (non-

graded) theory (cf Chapter 3), the other is speci¬c to graded equations and

is a super counterpart of the Cartan even covering constructed in Chapter

3 (see also [97]).

1

Let (F, F ) be an object of the category GDE(M ) and Hh (F) be the

R-module of its ¬rst horizontal cohomology. Let {w± } be a set of generators

for Hh (F), each w± being the cohomology class of a form ω± ∈ Λ1 (F),

1

h

m i dx . We de¬ne the functor K : GDE(M ) ’ GDE(M ) of

ω± = i=1 X± i

1 (F) as follows.

killing Hh

The algebra KF is a graded commutative algebra freely generated by

i

{w± } over F with gr(w± ) = gr(X± ). The connection KF looks as

‚ ‚ ‚

i

= + X± .

F

KF

‚xi ‚xi ‚w±

±

1

From the fact that Hh is a covariant functor from GDE(M ) into the category

of R-modules it easily follows that K is a functor as well.

To de¬ne the functor T : GDE(M ) ’ GDE(M ), let us set T F = C — Λ(F),

where C — Λ— (F) = p≥0 C p Λ(F) is the module of all Cartan forms on F (see

Subsection 2.7). If F is n-graded, then T F carries an obvious structure of

(n + 1)-graded algebra. The action of vector ¬elds F (X), X ∈ D(M ), on

Λ— (F) by Lie derivatives preserves the submodule C — Λ(F). Since C — Λ(F),

as a graded algebra, is generated by the elements χ and dC ψ, χ, ψ ∈ F, this

action can be written down as

L χ= F (X)χ,

F (X)

L dC ψ = d C F (X)(ψ),

F (X)

(χ) · dC ψ + χL

L (χdC ψ) = L dC ψ.

F (X) F (X)

F

268 6. SUPER AND GRADED THEORIES

Moreover, for any X ∈ D(M ) and ω ∈ C — Λ(F) one has

F (X) ω = 0;

hence, for any θ ∈ C — Λ(F)

(θ § L )(ω)

F (X)

(ω) + (’1)θ1 dθ § (

=θ§L ω) = θ § L

F (X) , (ω),

F (X) F (X)

which means that we have a natural extension of the connection F in F

up to a connection T F in T F. It is easy to see that the correspondence

T : (F, F ) ’ (T F, T F ) is functorial. We call (T F, ∆T F ) the (odd ) Car-

tan covering of (F, F ).

In the case when (F, F ) is an evolution equation E of the form (6.31),

T (F, F ) is again an evolution equation T E with additional dependent vari-

ables v 1 , . . . , v m and additional relations

±

‚f 1 j

1

v = vi ,

t

‚uj

i,j i

(6.45)

...............

m

‚f j

m

v =

t vi .

‚uj

i,j i

Note that if a variable uj is of grading (i1 , . . . , in ), then the grading of v j is

(i1 , . . . , in , 1).

3.6. Reconstructing shadows. Computerized computations on non-

local objects, such as symmetries and recursion operators, can be e¬ectively

realized for shadows of these objects (see examples below). Here we describe

a setting which guarantees the existence of symmetries and, in general, el-

p,0

ements of HC (E) corresponding to the shadows computed. Below we still

consider evolution equations only.

Proposition 6.28. Let E be an evolution equation and • be its covering.

p,0

Let θ ∈ SHC (E• ). Then, if the coe¬cients Xs and Ts for the extensions of

total derivatives do not depend on nonlocal variables for all s, then

(i) for any extension θ,ρ of θ up to a vector ¬eld on E• the forms

s def

„¦s

θ,ρ (ω• ) =

(see Remark 6.9 in Subsection 3.4) are dh -closed on E• ;

p,0

(ii) the element θ is extendable up to an element of HC (E• ) if and only

if all „¦s are dh -exact forms.

Proof. To prove the ¬rst statement, note that using Proposition

6.13 (i) one has

‚

2 s

+ d h ρs ) —

0 = ‚• ( θ,ρ ) = ‚• ( θ,ρ (ω• )

‚ωs

s

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 269

‚

dh „¦ s —

=’ . (6.46)

‚ωs

s

The second statement immediately follows from (6.43).

Remark 6.11. If Xs , Ts depend on w1 , w2 , . . . , then (6.46) transforms

into

‚Xs ‚Ts ‚

dh „¦s ’ („¦s + dh ρs ) § —

dx + dt = 0. (6.47)

‚ws ‚ws ‚ws

s

p,0 q,0

Let now θ ∈ SHC (E• ) and ¦ ∈ HC (E• ). Then from Proposition

6.13 (iii) it follows that

[i¦ , ‚• ]θ = (’1)q (‚• ¦) θ = 0.

Hence, since by the de¬nition of shadows ‚• θ is a •-vertical element,

i¦ ‚• θ is vertical too. It means that ‚• i¦ θ is a •-vertical element, i.e.,

p+q’1

i• θ ∈ SHC (E• ). It proves the following result (cf. similar results of

Chapter 5):

p,0 q,0

Proposition 6.29. For any θ ∈ SHC (E• ) and ¦ ∈ HC (E• ) the ele-

p+q’1

ment ¦ θ lies in SHC (E• ). In particular, when applying a shadow of

a recursion operator to a symmetry, one gets a shadow of a symmetry.

The next result follows directly from the previous ones.

Theorem 6.30. Let E be an evolution equation of the form (6.31) and

E• be its covering constructed by in¬nite application of the functor K : E • =

K (∞) E, where

K (∞) E = inj lim(K n E), K n E = (K —¦ · · · —¦ K) E.

n’∞

n times

Then for any shadow R of a recursion operator in E• and a symmetry ¦ ∈

sym E• the shadow R(¦) can be extended up to a symmetry of E• . Thus, an

1,0

action of SHC (E• ) on sym(E• ) is de¬ned modulo “shadowless” symmetries.

1,0

To be sure that elements of SHC (E• ) can be extended up to recursion

operators in an appropriate setting, we prove the following two results.

Proposition 6.31. Let E be an equation and E• be its covering by means

of T E. Then there exists a natural embedding

—,0

Tsym : HC (E) ’ sym(T E)

of graded Lie algebras.

—,0

Proof. Let ¦ ∈ HC (E). Then L¦ acts on Λ— (E) and this action pre-

serves the submodule C — Λ(E) ‚ Λ— (E), since

[L¦ , dC ] = L[[¦,UE ]]fn = 0.

270 6. SUPER AND GRADED THEORIES

Let X ∈ CD(E). Then, due to (6.11), [[X, ¦]]fn UE = 0. But, using

(6.11) again, one can see that [[X, ¦]]fn is a vertical element. Hence,

[[X, ¦]]fn UE = [[X, ¦]]fn = 0.

—,0

Proposition 6.31 allows one to compute elements of HC (E) as nonlocal

symmetries in E• = T E. This is the base of computational technology used

in applications below.

The last result of this subsection follows from the previous ones.

Theorem 6.32. Let E be an evolution equation and E• be its covering

constructed by in¬nite application of the functor K —¦ T . Then any shadow

—,0 —,0

¦ ∈ SHC (E• ) can be extended up to an element of HC (E• ). In particular,

1,0

to any shadow SHC (E• ) a recursion operator corresponds in E• .

Remark 6.12. For “¬ne obstructions” to shadows reconstruction one

should use corresponding term of A.M. Vinogradov™s C-spectral sequence

([102], cf. [58]).

4. The Kupershmidt super KdV equation

As a ¬rst application of the graded calculus for symmetries of graded par-

tial di¬erential equations we discuss the symmetry structure of the so-called

Kupershmidt super KdV equation, which is an extension of the classical

KdV equation to the graded setting [24].

At this point we have already to make a remark. The equation under

consideration will be a super equation but not a supersymmetric equation

in the sense of Mathieu, Manin“Radul, where a supersymmetric equation

is an equation admitting and odd, or supersymmetry [74], [72]. The super

KdV equation is given as the following system of graded partial di¬erential

equations E for an even function u and an odd function • in J 3 (π; •), where

J 3 (π; •) is the space J 3 (π) for the bundle π : R — R2 ’ R2 , (u, x, t) ’ (x, t),

extended by the odd variable •:

ut = 6uux ’ uxxx + 3••xx ,

•t = 3ux • + 6u•x ’ 4•xxx , (6.48)

where subscripts denote partial derivatives with respect to x and t. As

usual, t is the time variable and x is the space variable. Here u, x, t, u, ux ,

ut , uxx , uxxx are even (commuting) variables, while •, •x , •xx , •xxx are

odd (anticommuting) variables. In the sequel we shall often use the term

“graded” instead of “super”.

We introduce the total derivative operators Dx and Dt on the space

∞ (π; •), by

J

‚ ‚ ‚ ‚ ‚

+ ··· ,

Dx = + ux + •x + uxx + •xx

‚x ‚u ‚• ‚ux ‚•x

4. THE KUPERSHMIDT SUPER KDV EQUATION 271

‚ ‚ ‚ ‚ ‚

+ ···

Dt = + ut + •t + utx + •tx (6.49)

‚t ‚u ‚• ‚ux ‚•x

The in¬nite prolongation E ∞ is the submanifold of J ∞ (π; •) de¬ned by

the graded system of partial di¬erential equations

nm

Dx Dt (ut ’ 6uux + uxxx ’ 3••xx ) = 0,

nm

Dx Dt (•t ’ 3ux • ’ 6u•x + 4•xxx ) = 0, (6.50)

where n, m ∈ N.

We choose internal coordinates on E ∞ as x, t, u, •, u1 , •1 , . . . , where we

introduced a further notation

ux = u 1 , •x = • 1 , uxx = u2 , •xx = •2 , . . . (6.51)

The restriction of the total derivative operators Dx and Dt to E ∞ , again

denoted by the same symbols, are then given by

‚ ‚ ‚

Dx = + (un+1 + •n+1 ),

‚x ‚un ‚•n

n≥0

‚ ‚ ‚

Dt = + ((un )t + (•n )t ). (6.52)

‚t ‚un ‚•n

n≥0

We note that (6.48) admits a scaling symmetry, which leads to the in-

troduction of a degree to each variable,

deg(x) = ’1, deg(t) = ’3,

deg(u) = 2, deg(u1 ) = 3, . . . ,

3 5

deg(•) = , deg(•1 ) = , . . . (6.53)

2 2

From this we see that each term in (6.48) is of degree 5 and 4 1 respectively.

2

4.1. Higher symmetries. We start the discussion of searching for

(higher) symmetries at the representation of vertical vector ¬elds,

‚ ‚ ‚ ‚

= ¦u + ¦• Dx (¦u )

n

+ Dx (¦• )

n

+ , (6.54)

¦

‚u ‚• ‚un ‚•n

n>0

where ¦ = (¦u , ¦• ) is the generating function of the vertical vector ¬eld ¦ .

We restrict our search for higher symmetries to even vector ¬elds, meaning

that ¦u is even, while ¦• is odd.

Moreover we restrict our search for higher symmetries to vector ¬elds

u •

¦ whose generating function ¦ = (¦ , ¦ ) depends on the variables x, t,

u, •, . . . , u5 , •5 . These requirements lead to a representation of the func-

tion ¦ = (¦u , ¦• ), ¦u , ¦• ∈ C ∞ (x, t, u, u1 , . . . , u5 ) — Λ(•, . . . , •5 ) in the

following form

¦u = f0 + f1 ••1 + f2 ••2 + f3 ••3 + f4 ••4 + f5 ••5 + f6 •1 •2

+ f7 •1 •3 + f8 •1 •4 + f9 •1 •5 + f10 •2 •3 + f11 •2 •4 + f12 •2 •5

+ f13 •3 •4 + f14 •3 •5 + f15 •4 •5 + f16 ••1 •2 •3 + f17 ••1 •2 •4

272 6. SUPER AND GRADED THEORIES

+ f18 ••1 •2 •5 + f19 ••1 •3 •4 + f20 ••1 •3 •5 + f21 ••1 •4 •5

+ f22 ••2 •3 •4 + f23 ••2 •3 •5 + f24 ••2 •4 •5 + f25 ••3 •4 •5

+ f26 •1 •2 •3 •4 + f27 •1 •2 •3 •5 + f28 •2 •3 •4 •5

+ f29 ••1 •2 •3 •4 •5 ,

¦ • = g 1 • + g 2 •1 + g 3 •2 + g 4 •3 + g 5 •4 + g 6 •5

+ g7 ••1 •2 + g8 ••1 •3 + g9 ••1 •4 + g10 ••1 •5 + g11 ••2 •3

+ g12 ••2 •4 + g13 ••2 •5 + g14 ••3 •4 + g15 ••3 •5 + g16 ••4 •5

+ g17 •1 •2 •3 + g18 •1 •2 •4 + g19 •1 •2 •5 + g20 •1 •3 •4 + g21 •1 •3 •5

+ g22 •1 •4 •5 + g23 •2 •3 •4 + g24 •2 •3 •5 + g25 •2 •4 •5 + g26 •3 •4 •5

+ g27 ••1 •2 •3 •4 + g28 ••1 •2 •3 •5 + g29 ••1 •2 •4 •5 + g30 ••1 •3 •4 •5

+ g31 ••2 •3 •4 •5 + g32 •1 •2 •3 •4 •5 , (6.55)

where f0 , . . . , f29 , g1 , . . . , g32 are functions depending on the even variables

x, t, u, u1 , . . . , u5 . We have to mention here that we are constructing

generic elements, even and odd explicitly, of the following exterior alge-

bra C ∞ (x, t, u, . . . , u5 ) — Λ(•, . . . , •5 ), where Λ(•, . . . , •5 ) is the (exterior)

algebra generated by •, . . . , •5 The symmetry condition (6.37) for p = 0

reads in this case to the system

Dt (¦u ) = ’ u3 + 3••2 ),

¦ (6uu1

Dt (¦• ) = + 6u•1 ’ 4•3 ),

¦ (3u1 • (6.56)

which results in equations

Dt (¦u ) ’ 6¦u u1 ’ 6uDx (¦u ) + Dx (¦u ) ’ 3¦• •2 ’ 3•Dx (¦• ) = 0,

3 2

Dt (¦• ) ’ 3Dx (¦u )• ’ 3u1 ¦• ’ 6¦u •1 ’ 6uDx (¦• ) + 4Dx (¦• ) = 0.

3

(6.57)

Substitution of the representation(6.55) of ¦ = (¦u , ¦• )), leads to an

overdetermined system of classical partial di¬erential equations for the co-

e¬cients f0 , . . . , f26 , g1 , . . . , g32 , which are, as mentioned above, functions

depending on the variables x, t, u, u1 , . . . , u5 .

The general solution of equations (6.57) and (6.55) is generated by the

functions

¦1 = (u1 , •1 );

¦2 = (6uu1 ’ u3 + 3••2 , 3u1 • + 6u•1 ’ 4•3 );

¦3 = (6tu1 + 1, 6t•1 );

¦4 = (3t(6uu1 ’ u3 + 3••2 ) + x(u1 ) + 2u,

3

3t(3u1 • + 6u•1 ’ 4•3 ) + x•1 + •);

2

¦5 = (u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 ’ 15••4 ’ 10•1 •3

+ 30u1 ••1 + 30u••2 ,

4. THE KUPERSHMIDT SUPER KDV EQUATION 273

16•5 ’ 40u•3 ’ 60u1 •2 ’ 50u2 •1 + 30u2 •1 + 30u1 u• ’ 15u3 •).

(6.58)

We note that the vector ¬elds ¦1 , ¦2 , ¦3 , ¦4 are equivalent to the

classical symmetries

‚

S1 = ,

‚x

‚

S2 = ,

‚t

‚ 1‚

’

S3 = t ,

‚x 6 ‚u

‚ ‚ ‚ 3‚

S4 = ’x ’ 3t + 2u +• . (6.59)

‚x ‚t ‚u 2 ‚•

In (6.59) S1 , S2 re¬‚ect space and time translation, S3 re¬‚ects Galilean

invariance, while S4 re¬‚ects the scaling as mentioned already. In (6.50), the

evolutionary vector ¬eld ¦5 is the ¬rst higher symmetry of the super KdV

equation and reduces to

‚

(u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 ) + ..., (6.60)

‚u

in the absence of odd variables •, •1 , . . . , being then just the classical ¬rst

higher symmetry of the KdV equation

ut = 6uu1 ’ u3 . (6.61)

4.2. A nonlocal symmetry. In this subsection we demonstrate the

existence and construction of nonlocal higher symmetries for the super KdV

equation (6.48). The construction runs exactly along the same lines as it is

for the classical equations.

So we start at the construction of conservation laws, conserved densities

and conserved quantities as discussed in Section 2. According to this con-

struction we arrive, amongst others, at the following two conservation laws,

i.e.,

Dt (u) = Dx (3u2 ’ u2 + 3••1 ),

Dt (u2 + 3••1 ) = Dx (4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 ), (6.62)

1

from which we obtain the nonlocal variables

x

p1 = u dx,

’∞

x

(u2 + 3••1 ) dx.

p3 = (6.63)

’∞

Now using these new nonlocal variables p1 , p3 , we de¬ne the augmented

system E of partial di¬erential equations for the variables u, p1 , p3 , •,

where u, p1 , p3 are even and • is odd,

ut = 6uux ’ uxxx + 3••xx ,

274 6. SUPER AND GRADED THEORIES

•t = 3ux • + 6u•x ’ 4•xxx ,

(p1 )x = u,

(p1 )t = 3u2 ’ u2 + 3••1 ,

(p3 )x = u2 + 3••1 ,

(p3 )t = 4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 . (6.64)

1

Internal coordinates for the in¬nite prolongation E ∞ of this augmented

system (6.64) are given as x, t, u, p1 , p3 , •, u1 , •1 , . . . . The total derivative

operators Dx and Dt on E ∞ are given by

‚ ‚

+ (u2 + 3••1 )

Dx = D x + u ,

‚p1 ‚p3

‚

Dt = Dt + (3u2 ’ u2 + 3••1 )

‚p1