where Di (θ) is the Lie derivative of the form θ ∈ Λ— (E) along the vector ¬eld

Di ∈ D(E).

—

As it follows from the above said, the cohomology module HC (E) inherits

from Λ— — D1 the structure of the graded Lie algebra with respect to the

Fr¨licher“Nijenhuis bracket. In the case when U = U is the connection

o

form of a connection : D(M ) ’ D(P ), additional algebraic structures

arise in the cohomology modules H — (P ) = i H i (P ) of the corresponding

vertical complex.

First of all note that for any element „¦ ∈ Λ— (P ) — D v (P ) the identity

„¦ U =„¦ (5.16)

holds. Hence, if ˜ ∈ Λ— (P ) — D v (P ) is a vertical element too, then equality

(4.31) on p. 173 acquires the form

‚ ˜ + (’1)„¦ ‚ („¦

„¦ ˜) = ‚ („¦) ˜. (5.17)

From (5.17) it follows that

ker(‚ ) ‚ ker(‚ ),

ker(‚ )

1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 195

im(‚ ) ‚ im(‚ ),

ker(‚ )

ker(‚ ) ‚ im(‚ ).

im(‚ )

Therefore, the contraction operation

: Λi (P ) — D v (P ) — Λj (P ) — D v (P ) ’ Λi+j’1 (P ) — D v (P )

induces an operation

: H i (P ) —R H j (P ) ’ H i+j’1 (P ),

which is de¬ned by posing

[„¦] [˜] = [„¦ ˜],

where [·] denotes the cohomological class of the corresponding element.

In particular, H 1 (P ) is closed with respect to the contraction operation,

and due to (4.31) this operation determines in H 1 (P ) an associative algebra

structure. Consider elements φ ∈ H 0 (P ) and ˜ ∈ H 1 (P ). Then one can

de¬ne an action of ˜ on φ by posing

˜ ∈ H 0 (P ).

R˜ (φ) = φ (5.18)

Thus we have a mapping

R : H 1 (P ) ’ EndR (H 0 (P ))

which is a homomorphism of associative algebras due to (4.31) on p. 173.

In particular, taking P = E ∞ and ξ = π∞ , we obtain the following

Proposition 5.5. For any formally integrable equation E ‚ J k (π) the

1

module HC (E) is an associative algebra with respect to the contraction opera-

0

tion . This algebra acts on HC (E) = sym(E) by means of the representation

R de¬ned by (5.18).

When (5.16) takes place, equality (4.55), see p. 179, acquires the form

‚ (ρ § „¦) = (LU ’ dρ) § „¦ + (’1)ρ ρ § ‚ („¦). (5.19)

Let us set

dh = d ’ L U (5.20)

and note that

(dh )2 = (LU )2 ’ LU —¦ d ’ d —¦ LU + d2 = ’LU —¦ d ’ d —¦ LU .

But

L„¦ —¦ d = (’1)„¦ d —¦ L„¦ (5.21)

and, therefore, (dh )2 = 0. Thus we have the di¬erential

dh : Λi (P ) ’ Λi+1 (P ), i = 0, 1, . . . ,

and the corresponding cohomologies

H h,i (P ) = ker(dh,i+1 )/im(dh,i ).

196 5. DEFORMATIONS AND RECURSION OPERATORS

From (5.19) it follows that

ker(dh ) § ker(‚ ) ‚ ker(‚ ),

im(dh ) § ker(‚ ) ‚ im(‚ ),

ker(dh ) § im(‚ ) ‚ im(‚ ),

and hence a well-de¬ned wedge product

§ : H h,i (P ) —R H j (P ) ’ H i+j (P ).

Moreover, from (5.20) and (5.21) it follows that

L„¦ —¦ dh = L[[„¦,U + (’1)„¦ dh —¦ L„¦

]]fn

for any „¦ ∈ Λ— (P ) — D v (P ). It means that by posing

ω ∈ Λ— (P )

L[„¦] [ω] = [L„¦ ω],

we get a well-de¬ned homomorphism of graded Lie algebras

L : H — (P ) ’ D gr (H h,— (P )),

where H h,— (P ) = i H h,i (P ).

If (x1 , . . . , xn , y 1 , . . . , y s ) are local coordinates in P , then an easy com-

putation shows that

dh (f ) = i (f )dxi ,

i

dh (dxi ) = 0,

j

dh (dy j ) = § dxi ,

d (5.22)

i

i

where f ∈ C ∞ (P ), i = 1, . . . , n, j = 1, . . . , s, while the coe¬cients j

i

and vector ¬elds i are given by (5.2). Obviously, the di¬erential dH is

completely de¬ned by (5.22).

2. Spectral sequences and graded evolutionary derivations

In this section, we construct three spectral sequences associated with

C-cohomologies of in¬nitely prolonged equations. One of them is used to

compute the algebra HC (π) = HC (J ∞ (π)) of the “empty” equation. The

— —

result obtained leads naturally to the notion of graded evolutionary deriva-

tions which seem to play an important role in the geometry of di¬erential

equations.

The ¬rst of spectral sequences to be de¬ned originates from a ¬ltration

in Λ— (E) — D v (E) associated with the notion of the degree of horizontality.

Namely, an element ˜ ∈ Λp (E) — D v (E) is said to be i-horizontal if

X1 (X2 . . . (Xp’i+1 ˜) . . . ) = 0

p

for any X1 , . . . , Xp’i+1 ∈ Dv (E). Denote by Hi (E) the set of all such ele-

p p

ments. Obviously, Hi (E) ⊃ Hi+1 (E).

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 197

Proposition 5.6. For any equation E, the embedding

p p+1

‚C (Hi (E)) ‚ Hi+1 (E)

takes place.

To prove this we need some auxiliary facts.

Lemma 5.7. For any vector ¬elds X1 , . . . , Xp ∈ Dv (E) and an element

˜ ∈ Λ— (E) — D v (E) the equality

‚C (˜) = (’1)p ‚C (X1

X1 ... Xp ... Xp ˜)

p

(’1)p+i X1

+ ... Xi’1 ‚C (Xi ) Xi+1 ... Xp ˜ (5.23)

i=1

holds.

Proof. Recall that for any „¦ ∈ Λ— (E) — D v (E) one has

„¦ UE = „¦ (5.24)

and, by (5.17)

˜ ’ (’1)„¦ ‚C („¦

„¦ ‚C (˜) = ‚C („¦) ˜). (5.25)

In particular, taking „¦ = X ∈ D v (E), we get

˜ ’ ‚C (X

X ‚C ˜ = ‚C (X) ˜). (5.26)

This proves (5.23) for p = 1. The proof is ¬nished by induction on p starting

with (5.26).

Lemma 5.8. Consider vertical vector ¬elds X1 , . . . , Xp+1 ∈ Dv (E) and

p

an element ˜ ∈ H0 (E) = Λp (E) — D v (E). Then

X1 ... Xp+1 (‚C ˜) = 0,

p p+1

i.e., ‚C (H0 (E)) ‚ H1 (E).

This result is a direct consequence of (5.23).

Recall that a form θ ∈ Λp (E) is said to be horizontal if the identity

X θ = 0 holds for any X ∈ D v (E); the set of such forms is denoted by

Λp (E). It is easy to see that Hi (E) = Λi (E) § H0 (E), i.e., any element

p p’i

h

h

p

˜ ∈ Hi (E) can be represented as

ρs § ˜ s ,

˜= (5.27)

s

where ρs ∈ Λp (E), ˜s ∈ Λp’i (E) — D v (E). Applying (5.19) and (5.20) to

h

is the Cartan connection C, we get

(5.27) in the case when

’dh (ρs ) § ˜s + (’1)i ρs § ‚C (˜s ) .

‚C (˜) = (5.28)

C

s

198 5. DEFORMATIONS AND RECURSION OPERATORS

Lemma 5.9. 1 Let ξ : P ’ M be a ¬ber bundle with a ¬‚at connection

: D(M ) ’ D(P ) and

Λ— (P ) = {ρ ∈ Λ— (P ) | Y ρ = 0, Y ∈ D v (P )}

h

be the module of horizontal forms on P . Then for any form ρ ∈ Λi (P ) one

0

has

dh (ρ) ∈ Λi+1 (P ).

h

Proof. Let „¦ ∈ Λ— (P ) — D(P ), ρ ∈ Λ— (P ), and Y ∈ D(P ). Then

standard computations show that

ρ) ’ (’1)„¦ [[„¦, Y ]]fn

+ (’1)„¦ L„¦ (Y

Y (L„¦ ρ) = L(Y „¦) ρ ρ. (5.29)

and Y ∈ D v (P ), using (5.16) one has

In particular, if „¦ = U

(LU ρ) = Y (ρ) ’ LU (Y

Y ρ) + ‚ (Y ) ρ,

from where it follows that

dh (ρ) = ’dh (Y ρ) ’ ‚ (Y )

Y ρ,

since, by de¬nition, dh = d ’ LU .

Hence, if Y ∈ D v (P ) and ρ ∈ Λ— (P ), then one has ‚ (Y ) ∈ Λ1 (P ) —

h

v (P ) and Y h (ρ) = 0.

D d

Proposition 5.6 now follows from Lemmas 5.8, 5.9 and identity (5.28).

Remark 5.4. From the de¬nition of the di¬erential dh it immediately

C

follows that its restriction on Λ— (E), denoted by dh , coincides with the hor-

h

izontal de Rham complex of the equation E (see Chapter 2). As it follows

from (5.22), in local coordinates this restriction is completely determined by

the equalities

dh (f ) = Di (f ) dxi , dh (dxi ) = 0, (5.30)

i

where i = 1, . . . , n, f ∈ F(E) and D1 , . . . , Dn are total derivatives. One can

— —

show that the action L of HC (E) can be restricted onto the module Hh (E)

of horizontal cohomologies. In fact, if ρ ∈ Λ— (E) and X, Y ∈ D v (E), then

0

X Y (ρ) = Y (X ρ) + [X, Y ] ρ = 0.

On the other hand, if „¦ ∈ Λ— (E) — D v (E), then from (5.29) it follows that

Y (L„¦ ρ) = L(Y „¦) ρ.

Hence, by induction,

(Λ— (E)) ‚ Λ— (E).

L h h

Λ— (E)—D v (E)

On the other hand, the operator LUE is exactly the Cartan di¬erential of

the equation E (see also Chapter 2).

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 199

Let us now de¬ne a ¬ltration in Λ— (E) — D v (E) by setting

p

F l (Λp (E) — D v (E)) = Hp+l (E). (5.31)

Obviously,

F l (Λp (E) — D v (E)) ⊃ F l+1 (Λp (E) — D v (E))

and Proposition 5.6 is equivalent to the fact that

‚C F l (Λp (E) — D v (E)) ‚ F l (Λp+1 (E) — D v (E)).

Thus (5.31) de¬nes a spectral sequence for the complex (5.9) which we call

H-spectral. Its term E0 is of the form

p,q p+q p+q

E0 = H2p+q (E)/H2p+q+1 (E), (5.32)

where p = 0, ’1, . . . , q = ’2p, . . . , ’2p + n.

p,q

To express E0 in more suitable terms, let us recall the splitting

Λ1 (E) = Λ1 (E) • CΛ1 (E),

h

where CΛ1 (E) is the set of all 1-forms vanishing on the Cartan distribution

on E. Let

C i Λ(E) = CΛ1 (E) § · · · § CΛ1 (E) .

i times

Then for any p the module Λp (E) can be represented as

p

p

C p’i Λ(E) § Λi (E).

Λ (E) = h

i=0

Thus

p

p

C p’i Λ(E) § Λi (E) — Dv (E)

Hi (E) = h

i=0

from where it follows that

E0 = C ’p Λ(E) § Λ2p+q (E) — D v (E).

p,q

h

The con¬guration of the term E0 for the H-spectral sequence is presented

on Fig. 5.1, where D v = Dv (E), Λi = Λi (E), etc.

h h

The second spectral sequence to be de¬ned is in a sense complementary

to the ¬rst one. Namely, we say that an element ˜ ∈ Λp (E) — D v (E) is

(p ’ i + 1)-Cartan, if X1 . . . Xi ˜ = 0 for any X1 , . . . , Xi ∈ CD(E), and

p

denote the set of all such elements by Ci (E) ‚ Λp (E) — D v (E). Obviously,

p p

Ci (E) ‚ Ci+1 (E).

Proposition 5.10. For any equation E ‚ J k (π) one has

p p+1

‚C (Ci (E)) ‚ Ci+1 (E).

To prove this proposition, we need some preliminary facts.

200 5. DEFORMATIONS AND RECURSION OPERATORS

C 2 Λ § Λn — D v

h

C 2 Λ § Λn’1 — Dv

h

C 2 Λ § Λn’2 — Dv C 1 Λ § Λn — D v

h

h

C 1 Λ § Λn’1 — Dv

... h

C 1 Λ § Λn’2 — Dv Λn — D v

... q=n

h

h

Λn’1 — Dv q = n’1

... ... h

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

C 2 Λ § Λ1 — D v ... ... ...

h

C 2 Λ — Dv C 1 Λ § Λ2 — D v ... ...

h

C 1 Λ § Λ1 — D v ... ...

h

C 1 Λ — Dv Λ2 — D v q=2

h

Λ1 — D v q=1

h

Dv q=0

p = ’2 p = ’1 p=0

Figure 5.1. The H-spectral sequence con¬guration (term E0 ).

Lemma 5.11. For any vector ¬elds X1 , . . . , Xp ∈ CD(E) and an element

˜ ∈ Λ— (E) — D v (E) the equality

‚C (˜) = (’1)p ‚C (X1

X1 ... Xp ... Xp ˜)

p

˜]]fn

(’1)p+i+1 X1

+ ... Xi’1 [[Xi , Xi+1 ... Xp UE . (5.33)

i=1

holds.

Proof. We proceed by induction on p. Let X ∈ CD(E). Then, since

X UE = 0 and [[X, UE ]]fn = 0, from equality (4.45) on p. 175 it follows that

˜) ’ [[X, ˜]]fn

‚C (˜) = ’‚C (X

X UE , (5.34)

which gives us the starting point of induction.

Suppose now that (5.33) is proved for all s ¤ r. Then by (5.34) we have

X1 X2 ... Xr+1 ‚C (˜) = X1 (X2 ... Xr+1 ‚C (˜))

= (’1)r X1 ‚C (X2 ... Xr+1 ˜)

r+1

˜]]fn

(’1)r+i X2

+ X1 ... Xi’1 [[Xi , Xi+1 ... Xr+1 UE

i=2

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 201

˜]]fn

= (’1)r ’‚C (X1 ˜) ’ [[X1 , X2

... Xr+1 ... Xr+1 UE

r+1

˜]]fn

(’1)r+i X1

+ X2 ... Xi’1 [[Xi , Xi+1 ... Xr+1 UE

i=2

= (’1)r+1 ‚C (X1 ... Xr+1 ˜)

r+1

˜]]fn

(’1)r+i+1 X1

+ ... Xi’1 [[Xi , Xi+1 ... Xr+1 UE ,

i=1

which ¬nishes the proof of lemma.

p

Lemma 5.12. For any X ∈ CD(E) and ˜ ∈ Ci (E) we have

(i)

p’1

˜ ∈ Ci’1 (E)

X

and

(ii)

p

[[X, ˜]]fn ∈ Ci (E).

Proof. The ¬rst statement is obvious. To prove the second one, note

that from equality (4.45) on p. 175 it follows that for any X, X1 ∈ D(E) and

˜ ∈ Λ— (E) — D v (E) one has

[[X, ˜]]fn = [[X, X1 ˜]]fn + [[X1 , X]]fn

X1 ˜.

Now, by an elementary induction one can conclude that

[[X, ˜]]fn = [[X, X1 ˜]]fn

X1 ... Xi ... Xi

i

[[Xs , X]]fn

+ X1 ... Xs’1 Xs ... Xi ˜ (5.35)

s=1

for any X1 , . . . , Xi ∈ D(E).

p

Consider vector ¬elds X, X1 , . . . , Xi ∈ CD(E) and an element ˜ ∈ Ci (E).

Then, since [[Xs , X]]fn = [Xs , X] ∈ CD(E), all the summands on the right-

hand side of (5.35) vanish.

p

Proof of Proposition 5.10. Consider an element ˜ ∈ Ci (E) and

¬elds X1 , . . . , Xi+1 ∈ CD(E). Then, by (5.33), one has

‚C (˜) = (’1)i+1 ‚C (X1

X1 ... Xi+1 ... Xi+1 ˜)

i+1

˜]]fn

(’1)i+s X1

+ ... Xs’1 [[Xs , Xs+1 ... Xi+1 UE . (5.36)

s=1

The ¬rst summand on the right-hand side vanishes by de¬nition while the

rest of them, due to equality (4.31) on p. 173 and since UE ∈ Λ1 (E) — D v (E),

can be represented in the form