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correspondence
B
D(A) ’ D(A, B) ’ ’ D(B)

allows one to lift any vector ¬eld on M up to a π-projectable ¬eld on E.
Conversely, if is such a correspondence, then we can construct a natural
transformation • of the functors D(A, •) and D(•) due to the fact that
for smooth ¬nite-dimensional manifolds one has D(A, P ) = P —A D(A) and
D(P ) = P —B D(P ) for an arbitrary B-module P . We use the notation
= B in the sequel.
Definition 4.12. Let • be a connection in (A, B, •) and consider two
derivations X, Y ∈ D(A, B). The curvature form of the connection • on
the pair X, Y is de¬ned by
R (X, Y ) = [ (X), (Y )] ’ ( (X) —¦ Y ’ (Y ) —¦ X). (4.46)
(X) —¦ Y ’ (Y ) —¦ X is a B-valued
Note that (4.46) makes sense, since
derivation of A.
Consider now the de Rham di¬erential d = dB : B ’ Λ1 (B). Then the
composition dB —¦ • : A ’ B is a derivation. Consequently, we may consider
the derivation (dB —¦ •) ∈ D(Λ1 (B)).
Definition 4.13. The element U ∈ D(Λ1 (B)) de¬ned by
(dB —¦ •) ’ dB
U= (4.47)
is called the connection form of .
Directly from the de¬nition we obtain the following
Lemma 4.23. The equality
iX (U ) = X ’ ( X|A ) (4.48)
holds for any X ∈ D(B).
Using this result, we now prove
Proposition 4.24. If B is a smooth algebra, then
iY iX [[U , U ]]fn = 2R ( X|A , Y |A ) (4.49)
for any X, Y ∈ D(B).
Proof. First note that deg U = 1. Then using (4.44) and (4.41) we
obtain

iX [[U , U ]]fn = [[iX U , U ]]fn + [[U , iX U ]]fn ’ i[[X,U ’ i[[X,U
]]fn U ]]fn U
fn
= 2 [[iX U , U ]] ’ i[[X,U ]]fn U .
178 4. BRACKETS

Applying iY to the last expression and using (4.42) and (4.44), we get now
iY iX [[U , U ]]fn = 2 [[iX U , iY U ]]fn ’ i[[X,Y ]]fn U .

But [[V, W ]]fn = [V, W ] for any V, W ∈ D(Λ0 (A)) = D(A). Hence, by (4.48),
we have
iY iX [[U , U ]]fn = 2 [X ’ ( X|A ), Y ’ ( Y |A )] ’ ([X, Y ] ’ ([X, Y ]|A )) .
( X|A )|A = X|A and [X, Y ]|A = X —¦
It only remains to note now that
Y |A ’ Y —¦ X|A .
Definition 4.14. A connection in (A, B, •) is called ¬‚at, if R = 0.


Fix an algebra A and let us introduce the category FC(A), whose objects
are triples (A, B, •) endowed with a connection • while morphisms are
de¬ned as follows. Let O = (A, B, •, • ) and O = (A, B, •, • ) be two
objects of FC(A). Then a morphism from O to O is a mapping f : B ’ B
such that:
(i) f is an A-algebra homomorphism, i.e., the diagram
f
’B
B
















A
is commutative, and
(ii) for any B-module P (which can be considered as a B-module as well
due to the homomorphism f the diagram
D(B, f )
’ D(B, P )
D(B, P )






P
P




D(A, P )
is commutative, where D(B, f )(X) = X —¦ f for any derivation
X : B ’ P.
Due to Proposition 4.24, for ¬‚at connections we have
[[U , U ]]fn = 0. (4.50)
Let U ∈ D(Λ1 (B)) be an element satisfying equation (4.50). Then from
the graded Jacobi identity (4.42) we obtain
2[[U, [[U, X]]fn ]]fn = [[[[U, U ]]fn , X]]fn = 0
2. NIJENHUIS BRACKET 179

for any X ∈ D(Λ— (A)). Consequently, the operator
‚U = [[U, ·]]fn : D(Λi (B)) ’ D(Λi+1 (B))
de¬ned by the equality ‚U (X) = [[U, X]]fn satis¬es the identity ‚U —¦ ‚U = 0.
Consider now the case U = U , where is a ¬‚at connection.
Definition 4.15. An element X ∈ D(Λ— (B)) is called vertical, if
X(a) = 0 for any a ∈ A. Denote the B-submodule of such elements by
Dv (Λ— (B)).
Lemma 4.25. Let be a connection in (A, B, •). Then
(1) an element X ∈ D(Λ— (B)) is vertical if and only if iX U = X;
(2) the connection form U is vertical, U ∈ Dv (Λ1 (B));
(3) the mapping ‚U preserves verticality, i.e., for all i one has the em-
beddings ‚U (Dv (Λi (B))) ‚ D v (Λi+1 (B)).
Proof. To prove (1), use Lemma 4.23: from (4.48) it follows that
iX U = X if and only if ( X|A ) = 0. But ( X|A )|A = X|A . The second
statements follows from the same lemma and from the ¬rst one:
iU U = U ’ ( U |A ) = U ’ (U ’ ( U |A ))|A = U .
Finally, (3) is a consequence of (4.44).
Definition 4.16. Denote the restriction ‚U |Dv (Λ— (A)) by ‚ and call
the complex
‚ ‚
0 ’ D v (B) ’’ D v (Λ1 (B)) ’ · · · ’ D v (Λi (B)) ’’ D v (Λi+1 (B)) ’ · · ·
(4.51)
the -complex of the triple (A, B, •). The corresponding cohomology is de-
noted by H — (B; A, •) = i≥0 H i (B; A, •) and is called the -cohomology
of the triple (A, B, •).
Introduce the notation
dv = LU : Λi (B) ’ Λi+1 (B). (4.52)
Proposition 4.26. Let be a ¬‚at connection in a triple (A, B, •) and
B be a smooth (or ¬nitely smooth) algebra. Then for any X, Y ∈ D v (Λ— (A))
and ω ∈ Λ— (A) one has
‚ [[X, Y ]]fn = [[‚ X, Y ]]fn + (’1)X [[X, ‚ Y ]]fn , (4.53)
[iX , ‚ ] = (’1)X i‚ X, (4.54)
‚ (ω § X) = (dv ’ d)(ω) § X + (’1)ω ω § ‚ X, (4.55)
[dv , iX ] = i‚ + (’1)X LX . (4.56)
X

Proof. Equality (4.53) is a direct consequence of (4.42). Equality
(4.54) follows from (4.44). Equality (4.55) follows from (4.45) and (4.48).
Finally, (4.56) is obtained from (4.43).
180 4. BRACKETS

Corollary 4.27. The module H — (B; A, •) inherits the graded Lie al-
gebra structure with respect to the Fr¨licher“Nijenhuis bracket [[·, ·]]fn , as well
o
as the contraction operation.
Proof. Note that D v (Λ— (A)) is closed with respect to the Fr¨licher“
o
Nijenhuis bracket: to prove this fact, it su¬ces to apply (4.44). Then the
¬rst statement follows from (4.53). The second one is a consequence of
(4.54).
Remark 4.11. We preserve the same notations for the inherited struc-
tures. Note, in particular, that H 0 (B; A, •) is a Lie algebra with respect to
the Fr¨licher“Nijenhuis bracket (which reduces to the ordinary Lie bracket
o
in this case). Moreover, H 1 (B; A, •) is an associative algebra with respect
to the inherited contraction, while the action
X ∈ H 0 (B; A, •), „¦ ∈ H 1 (B; A, •)
R„¦ : X ’ iX „¦,
is a representation of this algebra as endomorphisms of H 0 (B; A, •).
Consider now the mapping dv : Λ— (B) ’ Λ— (B) de¬ned by (4.52) and
de¬ne dh = dB ’ dv .
Proposition 4.28. Let B be a (¬nitely) smooth algebra and be a ¬‚at
connection in the triple (B; A, •). Then
(1) The pair (dh , dv ) forms a bicomplex, i.e.,
dv —¦ dv = 0, dh —¦ dh = 0, dh —¦ dv + dv —¦ dh = 0. (4.57)
(2) The di¬erential dh possesses the following properties
[dh , iX ] = ’i‚ X, (4.58)
‚ (ω § X) = ’dh (ω) § X + (’1)ω ω § ‚ X, (4.59)
where ω ∈ Λ— (B), X ∈ D v (Λ— (B)).
Proof. (1) Since deg dv = 1, we have
2dv —¦ dv = [dv , dv ] = [LU , LU ] = L[[U = 0.
,U ]]fn
Since dv = LU , the identity [dB , dv ] = 0 holds (see (4.38)), and it concludes
the proof of the ¬rst part.
(2) To prove (4.58), note that
[dh , iX ] = [dB ’ dh , iX ] = (’1)X LX ’ [dv , iX ],
and (4.58) holds due to (4.56). Finally, (4.59) is just the other form of
(4.55).
Definition 4.17. Let be a connection in (A, B, •).
(1) The bicomplex (B, dh , dv ) is called the variational bicomplex associ-
ated to the connection .
(2) The corresponding spectral sequence is called the -spectral sequence
of the triple (A, B, •).
3. STRUCTURE OF SYMMETRY ALGEBRAS 181

Obviously, the -spectral sequence converges to the de Rham cohomology
of B.
To ¬nish this section, note the following. Since the module Λ1 (B) is
generated by the image of the operator dB : B ’ Λ1 (B) while the graded
algebra Λ— (B) is generated by Λ1 (B), we have the direct sum decomposition

Λp (B) — Λq (B),
Λ— (B) = v h
i≥0 p+q=i

where

Λq (B) = Λ1 (B) § · · · § Λ1 (B),
Λp (B) = Λ1 (B) § · · · § Λ1 (B),
v v v h h
h
p times q times

while the submodules Λ1 (B) ‚ Λ1 (B), Λ1 (B) ‚ Λ1 (B) are spanned in
v h
1 (B) by the images of the di¬erentials dv and dh respectively. Obviously,
Λ
we have the following embeddings:

dh Λp (B) — Λq (B) ‚ Λp (B) — Λq+1 (B),
v v
h h
dv Λp (B) — Λq (B) ‚ Λp+1 (B) — Λq (B).
v v
h h

Denote by D p,q (B) the module D v (Λp (B) — Λq (B)). Then, obviously,
v h
v (B) = p,q (B), while from equalities (4.58) and (4.59) we
D D
i≥0 p+q=i
obtain

Dp,q (B) ‚ Dp,q+1 (B).


Consequently, the module H — (B; A, •) is split as

H p,q (B; A, •)
H — (B; A, •) = (4.60)
i≥0 p+q=i

with the obvious meaning of the notation H p,q (B; A, •).

Proposition 4.29. If O = (B, ) is an object of the category FC(A),
then

H p,0 (B) = ker ‚ .
v
D1 (C p Λ(B))



3. Structure of symmetry algebras
Here we expose the theory of symmetries and recursion operators in the
categories FC(A). Detailed motivations for the de¬nition can be found in
previous chapters as well as in Chapter 5. A brief discussion concerning rela-
tions of this algebraic scheme to further applications to di¬erential equations
the reader will ¬nd in concluding remarks below.
182 4. BRACKETS

3.1. Recursion operators and structure of symmetry algebras.
We start with the following
Definition 4.18. Let O = (B, ) be an object of the category FC(A).
(i) The elements of H 0,0 (B) = H 0 (B) are called symmetries of O.
(ii) The elements of H 1,0 (B) are called recursion operators of O.
We use the notations
def
Sym = H 0,0 (B)
and
def
Rec = H 1,0 (B).
From Corollary 4.27 and Proposition 4.29 one obtains
Theorem 4.30. For any object O = (B, ) of the category FC(A) the
following facts take place:
(i) Sym is a Lie algebra with respect to commutator of derivations.
(ii) Rec is an associative algebra with respect to contraction, U being the
unit of this algebra.
(iii) The mapping R : Rec ’ Endk (Sym), where
R„¦ (X) = iX („¦), „¦ ∈ Rec, X ∈ Sym,
is a representation of this algebra and hence
(iv) i(Sym) (Rec) ‚ Sym .
In what follows we shall need a simple consequence of basic de¬nitions:
Proposition 4.31. For any object O = (B, ) of FC(A)
[[ Sym, Rec]] ‚ Rec
and
[[Rec, Rec]] ‚ H 2,0 (B).
Corollary 4.32. If H 2,0 (B) = 0, then all recursion operators of the
object O = (B, ) commute with each other with respect to the Fr¨licher“
o
Nijenhuis bracket.
We call the objects satisfying the conditions of the previous corollary
2-trivial. To simplify notations we denote
R„¦ (X) = „¦(X), „¦ ∈ Rec, X ∈ Sym .
From Proposition 4.31 and equality (4.42) one gets
Proposition 4.33. Consider an object O = (B, ) of FC(A) and let
X, Y ∈ Sym, „¦, θ ∈ Rec. Then
[[„¦, θ]](X, Y ) = [„¦(X), θ(Y )] + [θ(X), „¦(Y )] ’ „¦([θ(X), Y ]
+ [X, θ(Y )]) ’ θ([„¦(X), Y ] + [X, „¦(Y )]) + („¦ —¦ θ + θ —¦ „¦) [X, Y ].
3. STRUCTURE OF SYMMETRY ALGEBRAS 183

In particular, for „¦ = θ one has
1
[[„¦, „¦]](X, Y ) = [„¦(X), „¦(Y )]
2
’ „¦([„¦(X), Y ]) ’ „¦([X, „¦(Y )]) + „¦(„¦([X, Y ])). (4.61)
The proof of this statement is similar to that of Proposition 4.24. The
right-hand side of (4.61) is called the Nijenhuis torsion of „¦ (cf. [49]).
Corollary 4.34. If O is a 2-trivial object, then
[„¦(X), „¦(Y )] = „¦ ([„¦(X), Y ] + [X, „¦(Y )] ’ „¦[X, Y ]) . (4.62)
Choose a recursion operator „¦ ∈ Rec and for any symmetry X ∈ Sym
denote „¦i (X) = Ri (X) by Xi . Then (4.62) can be rewritten as
„¦
[X1 , Y1 ] = [X1 , Y ]1 + [X, Y1 ]1 ’ [X, Y ]2 . (4.63)
Using (4.63) as the induction base, one can prove the following
Proposition 4.35. For any 2-trivial object O and m, n ≥ 1 one has
[Xm , Yn ] = [Xm , Y ]n + [X, Yn ]m ’ [X, Y ]m+n .
Let, as before, X be a symmetry and „¦ be a recursion operator. Then
def
„¦X = [[X, „¦]] is a recursion operator again (Proposition 4.31). Due to
(4.42), its action on Y ∈ Sym can be expressed as
„¦X (Y ) = [X, „¦(Y )] ’ „¦[X, Y ]. (4.64)
From (4.64) one has
Proposition 4.36. For any 2-trivial object O, symmetries X, Y ∈ Sym,
a recursion „¦ ∈ Rec, and integers m, n ≥ 1 one has
n’1
[X, Yn ] = [X, Y ]n + („¦X Yi )n’i’1
i=0
and
m’1
[Xm , Y ] = [X, Y ]m ’ („¦Y Xj )m’j’1 .
j=0

From the last two results one obtains
Theorem 4.37 (the structure of a Lie algebra for Sym). For any 2-
trivial object O, its symmetries X, Y ∈ Sym, a recursion operator „¦ ∈ Rec,
and integers m, n ≥ 1 one has
n’1 m’1
(„¦X Yi )m+n’i’1 ’
[Xm , Yn ] = [X, Y ]m+n + („¦Y Xj )m+n’j’1 .
i=0 j=0

Corollary 4.38. If X, Y ∈ Sym are such that „¦X and „¦Y commute
with „¦ ∈ Rec with respect to the Richardson“Nijenhuis bracket, then
[Xm , Yn ] = [X, Y ]m+n + n(„¦X Y )m+n’1 ’ m(„¦Y X)m+n’1 .
184 4. BRACKETS

We say that a recursion operator „¦ ∈ Rec is X-invariant, if „¦X = 0.
Corollary 4.39 (on in¬nite series of commuting symmetries). If O is
a 2-trivial object and if a recursion operator „¦ ∈ Rec is X-invariant,
X ∈ Sym, then a hierarchy {Xn }, n = 0, 1, . . . , generated by X and „¦
is commutative:
[Xm , Xn ] = 0
for all m, n.
3.2. Concluding remarks. Here we brie¬‚y discuss relations of the
above exposed algebraic scheme to geometry of partial di¬erential equations
exposed in the previous chapters and the theory of recursion operators dis-
cussed in Chapters 5“7.
First recall that correspondence between algebraic approach and geo-
metrical picture is established by identifying the category of vector bun-
dles over a smooth manifold M with the category of geometrical mod-
ules over A = C ∞ (M ), see [60]. In the case of di¬erential equations, M
plays the role of the manifold of independent variables while B = ± B±
is the function algebra on the in¬nite prolongation of the equation E and
B± = C ∞ (E ± ), where E ± , ± = 0, 1, . . . , ∞, is the ±-prolongation of E. The
mapping • : A ’ B is dual to the natural projection π∞ : E ∞ ’ M and
thus in applications to di¬erential equations it su¬ces to consider the case
A = ± B± .
If E is a formally integrable equation, the bundle π∞ : E ∞ ’ M pos-
sesses a natural connection (the Cartan connection C) which takes a vector
¬eld X on M to corresponding total derivative on E ∞ . Consequently, the
category of di¬erential equations [100] is embedded to the category of alge-
bras with ¬‚at connections FC(C ∞ (M )). Under this identi¬cation the spec-
tral sequence de¬ned in De¬nition 4.17 coincides with A. Vinogradov™s C-
spectral sequence [102] (or variational bicomplex), the module Sym, where
O = (C ∞ (M ), C ∞ (E ∞ ), C), is the Lie algebra of higher symmetries for the
equation E and, in principle, Rec consists of recursion operators for these
symmetries. This last statement should be clari¬ed.
In fact, as we shall see later, if one tries to compute the algebra Rec
straightforwardly, the results will be trivial usually ” even for equations
which really possess recursion operators. The reason lies in nonlocal char-
acter of recursion operators for majority of interesting equations [1, 31, 4].
Thus extension of the algebra C ∞ (E ∞ ) with nonlocal variables (see 3) is the
way to obtain nontrivial solutions ” and actual computation show that all
known (as well as new ones!) recursion operators can be obtained in such
a way (see examples below and in [58, 40]). In practice, it usually su¬ces
to extend C ∞ (E ∞ ) by integrals of conservation laws (of a su¬ciently high
order).
The algorithm of computations becomes rather simple due to the follow-
ing fact. It will shown that for non-overdetermined equations all cohomology
3. STRUCTURE OF SYMMETRY ALGEBRAS 185

p,q
groups HC (E) are trivial except for the cases q = 0, 1 while the di¬erential
‚C : D1 (C p (E)) ’ D1 (C p (E) § Λh (E)) coincides with the universal lineariza-
v v
1
tion operator E of the equation E extended to the module of Cartan forms.
p,0
Therefore, the modules HC (E) coincide with ker( E ) (see 4.29)
p,0
HC (E) = ker( E ) (4.65)
and thus can be computed e¬ciently.
In particular, it will shown that for scalar evolution equations all coho-
p,0
mologies HC (E), p ≥ 2, vanish and consequently equations of this type are
2-trivial and satisfy the conditions of Theorem 4.37 which explains commu-
tativity of some series of higher symmetries (e.g., for the KdV equation).
186 4. BRACKETS
CHAPTER 5


Deformations and recursion operators

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