(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 („¦).