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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

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