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