1

J (A) ’P

’

’

←

∆

ν

•

’

Λ1 (A)

∆

j1

‘

d

A

Since ∆ is a ¬rst order di¬erential operator, there exists a homomorphism

f ∆ : J 1 (A) ’ P satisfying the equality ∆ = f ∆ —¦ j1 . But ∆ is a derivation,

i.e., ∆(1) = 0, which means that ker(f ∆ ) contains A · j1 (1). Hence, there

exists a unique mapping •∆ such that the above diagram is commutative.

Remark 4.3. From the de¬nition it follows that Λ1 (A), as an A-module,

is generated by the elements da, a ∈ A, with the relations

d(±a + βb) = ±da + βdb, d(ab) = adb + bda,

1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 165

±, β ∈ K, a, b ∈ A, while the de Rham di¬erential takes a to the coset

a mod (A · j1 (1)).

Let us set now

Λi (A) = Λ1 (A) § · · · § Λ1 (A) . (4.16)

i times

The elements of Λi (A) are called di¬erential i-forms of the algebra A. We

def

also formally set Λ0 (A) = A.

Proposition 4.10. The modules Λi (A), i ≥ 0, are representative ob-

jects for the functors Di (•).

Proof. The case i = 0 is trivial while the case i = 1 was proved already

(see Proposition 4.9). Let now i > 1 and a ∈ A. De¬ne the mappings

»a : homA (Λi (A), P ) ’ homA (Λi’1 (A), P ), ia : Di (P ) ’ Di’1 (P )

by setting

def def

(»a •)(ω) = •(da § ω), ia ∆ = ∆(a),

where ω ∈ Λi’1 (A), • ∈ homA (Λi (A), P ), and ∆ ∈ Di (P ).

Using induction on i, let us construct isomorphisms

ψi : homA (Λi (A), P ) ’ Di (P )

in such a way that the diagrams

ψi

homA (Λi (A), P ) ’ Di (P )

»a ia (4.17)

“ “

ψi’1

i’1

’ Di’1 (P )

homA (Λ (A), P )

are commutative for all a ∈ A.

The case i = 1 reduces to Proposition 4.9. Let now i > 1 and assume

that for i ’ 1 the statement is valid. Then from (4.17) we should have

• ∈ homA (Λi (A), P ),

(ψi (•))(a) = ψi’1 (»a (•)),

which completely determines ψi . From the de¬nition of the mapping »a it

follows that

»a —¦ »b = ’»b —¦ »a , a, b ∈ A,

»ab = a»b + b»a ,

i.e., im ψi ∈ Di (P ) (see Proposition 4.5).

Let us now show that ψi constructed in such a way is an isomorphism.

Take ∆ ∈ Di (P ), a1 , . . . , ai and set

def

¯ ’1

ψi (da1 § . . . dai ) = ψi’1 (X(a1 )) (da2 § · · · § dai ).

166 4. BRACKETS

’1

It may be done since ψi’1 exists by the induction assumption. Directly from

¯ ¯

de¬nitions one obtains that ψi —¦ ψi = id, ψi —¦ ψi = id. It is also obvious that

the isomorphisms ψi are natural, i.e., the diagrams

ψi

homA (Λi (A), P ) ’ Di (P )

homA (Λi (A), f ) Di (f )

“ “

ψi

homA (Λi (A), Q) ’ Di (Q)

are commutative for all homomorphisms f ∈ homA (P, Q).

From the result proved we obtain the pairing

·, · : Di (P ) —A Λi (A) ’ P (4.18)

de¬ned by

def

’1

ω ∈ Λi (A), ∆ ∈ Di (P ).

∆, ω = ψi (∆) (ω),

A direct consequence of the proof of Proposition 4.10 is the following

Corollary 4.11. The identity

∆, da § ω = ∆(a), ω (4.19)

holds for any ω ∈ Λi (A), ∆ ∈ Di+1 (A), a ∈ A.

Let us de¬ne the mappings d = di : Λi’1 (A) ’ Λi (A) by taking the ¬rst

de Rham di¬erential for d1 and setting

def

di (a0 da1 § · · · § dai ) = da0 § da1 § · · · § dai

for i > 1. From (4.16) and Remark 4.3 it follows that the mappings d i are

well de¬ned.

Proposition 4.12. The mappings di possess the following properties:

(i) di is a ¬rst order di¬erential operator acting from Λi’1 (A) to Λi (A);

(ii) d(ω § θ) = d(ω) § θ + (’1)i ω § d(θ) for any ω ∈ Λi (A), θ ∈ Λj (A);

(iii) di —¦ di’1 = 0.

The proof is trivial.

In particular, (iii) means that the sequence of mappings

d

d

0 ’ A ’1 Λ1 (A) ’ · · · ’ Λi’1 (A) ’i Λi (A) ’ · · ·

’ ’ (4.20)

is a complex.

Definition 4.6. The mapping di is called the (i-th) de Rham di¬er-

ential. The sequence (4.20) is called the de Rham complex of the algebra

A.

1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 167

Remark 4.4. Before proceeding with further exposition, let us make

some important comments on the relation between algebraic and geometrical

settings. As we saw above, the algebraic de¬nition of a linear di¬erential

operator is in full accordance with the analytical one. The same is true if we

compare algebraic “vector ¬elds” (i.e., elements of the module D(A)) with

vector ¬elds on a smooth manifold M : derivations of the algebra C ∞ (M )

are identical to vector ¬elds on M .

This situation changes, when we pass to representative objects. A simple

example illustrates this e¬ect. Let M = R and A = C ∞ (M ). Consider the

di¬erential one-form ω = dex ’ ex dx ∈ Λ1 (A). This form is nontrivial as an

element of the module Λ1 (A). On the other hand, for any A-module P let

us de¬ne the value of an element p ∈ P at point x ∈ M as follows. Denote

by µx the ideal

def

µx = {f ∈ C ∞ (M ) | f (x) = 0} ‚ C ∞ (M )

def

and set px = p mod µx . In particular, if P = A, thus de¬ned value coincides

with the value of a function f at a point. One can easily see that ωx = 0

for any x ∈ M . Thus, ω is a kind of a “ghost”, not observable at any point

of the manifold. The reader will easily construct similar examples for the

modules J k (A). In other words, we can state that

Λi (M ) = Λi (C ∞ (M )), “(πk ) = J k (“(π))

for an arbitrary smooth manifold M and a vector bundle π : E ’ M .

Let us say that C ∞ (M )-module P is geometrical, if

µx · P = 0.

x∈M

Obviously, all modules of the form “(π) are geometrical. We can introduce

the geometrization functor by setting

def

µx · P.

G(P ) = P/

x∈M

Then the following result is valid:

Proposition 4.13. Let M be a smooth manifold and π : E ’ M be a

smooth vector bundle. Denote by A the algebra C ∞ (M ) and by P the module

“(π). then:

(ii) The functor Di (•) is representable in the category of geometrical A-

modules and one has

Di (Q) = homA (G(Λi (A)), Q)

for any geometrical module Q.

(i) The functor Diff(P, •) is representable in the category of geometrical

A-modules and one has

Diff k (P, Q) = homA (G(J k (P )), Q)

for any geometrical module Q.

168 4. BRACKETS

In particular,

Λi (M ) = G(Λi (C ∞ (M ))), “(πk ) = G(J k (“(π))).

1.5. Smooth algebras. Let us introduce a class of algebras which

plays an important role in geometrical theory.

Definition 4.7. A commutative algebra A is called smooth, if Λ1 (A)

is a projective A-module of ¬nite type while A itself is an algebra over the

¬eld of rational numbers Q.

Denote by S i (P ) the i-th symmetric power of an A-module P .

Lemma 4.14. Let A be a smooth algebra. Then both S i (Λ1 (A)) and

Λi (A) are projective modules of ¬nite type.

def

Proof. Denote by T i = T i (Λ1 (A)) the i-th tensor power of Λ1 (A).

Since the module Λ1 (A) is projective, then it can be represented as a direct

summand in a free module, say P . Consequently, T i is a direct summand

in the free module T i (P ) and thus is projective with ¬nite number of gen-

erators.

On the other hand, since A is a Q-algebra, both S i (Λi (A)) and Λi (A)

are direct summands in T i which ¬nishes the proof.

Proposition 4.15. If A is a smooth algebra, then the following isomor-

phisms are valid :

(i) Di (A) D1 (A) § · · · § D1 (A),

i times

(ii) Di (P ) Di (A) —A P ,

where P is an arbitrary A-module.

Proof. The result follows from Lemma 4.14 combined with Proposition

4.10

For smooth algebras, one can also e¬ciently describe the modules

J k (A). Namely, the following statement is valid:

Proposition 4.16. If A is a smooth algebra, then all the modules J k (A)

are projective of ¬nite type and the isomorphisms

J k (A) S i (Λ1 (A))

i¤k

take place.

Proof. We shall use induction on k. First note that the mapping a ’

aj1 (1) splits the exact sequence

ν1,0

0 ’ ker(ν1,0 ) ’ J 1 (A) ’ ’ J 0 (A) = A ’ 0.

’

But by de¬nition, ker(ν1,0 ) = Λ1 (A) and thus J 1 (A) = A • Λ1 (A).

Let now k > 1 and assume that for k ’ 1 the statement is true. By

de¬nition, ker(νk,k’1 ) = µk’1 /µk , where µi ‚ A —K A are the submodules

1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 169

introduced in Subsection 1.2. Note that the identity a—b = a(b—1)’aδ b (1—

1) implies the direct sum decomposition µk’1 = µk • (µk’1 /µk ) and thus

the quotient module µk’1 /µk is identi¬ed with the submodule in A —K A

spanned by

(δ a1 —¦ · · · —¦ δ ak ) (1 — 1), a0 , . . . , ak ∈ A.

Consequently, any a ∈ A determines the homomorphism

δ a : µk’2 /µk’1 ’ µk’1 /µk

by

δ a : a — a ’ aa — a ’ a — aa .

But one has δ ab = aδ b + bδ a and hence δ : a ’ δ a is an element of the

module D1 (homA (µk’2 /µk’1 , µk’1 /µk )). Consider the corresponding ho-

momorphism

• = •δ ∈ homA (Λ1 (A), homA (µk’2 /µk’1 , µk’1 /µk )).

Due to the canonical isomorphism

homA (Λ1 (A), homA (µk’2 /µk’1 , µk’1 /µk ))

homA (Λ1 (A) —A µk’2 /µk’1 , µk’1 /µk ),

we obtain the mapping

• : Λ1 (A) —A µk’2 /µk’1 ’ µk’1 /µk ,

and repeating the procedure, get eventually the mapping • : T k ’ µk’1 /µk .

Due to the identity δa —¦δb = δb —¦δa , this mapping induces the homomorphism

•S : S k (Λ1 (A)) ’ µk’1 /µk which, in terms of generators, acts as

•S (da1 · · · · · dak ) = (δ a1 —¦ . . . —¦ δ ak ) (1 — 1)

and thus is epimorphic.

Consider the dual monomorphism

•— : µk’1 /µk = Diff k (A)/ Diff k’1 (A) ’ (S k (Λ1 (A)))— = S k (D1 (A)).

S

Let σ ∈ Diff k (A)/ Diff k’1 (A) and ∆ ∈ Diff k (A) be a representative of the

class σ. Then

(•— (σ))(da1 · · · · · dak ) = (δa1 —¦ · · · —¦ δak ) (∆).

S

But, on the other hand, it is not di¬cult to see that the mapping

1

•— : X1 · . . . Xk ’ [X1 —¦ . . . Xk ],

¯S

k!

•— : S k (D1 (A)) ’ Diff k (A)/ Diff k’1 (A), where [∆] denotes the coset of

¯S

the operator ∆ ∈ Diff k (A) in the quotient module Diff k (A)/ Diff k’1 (A), is

inverse to •— . Thus, •— is an isomorphism. Then the mapping

S S

µk’1 /µk ’ (µk’1 /µk )—— S k (Λ1 (A)),

where the ¬rst arrow is the natural homomorphism, is the inverse to •S .

170 4. BRACKETS

S k (Λ1 (A)) and we have

From the above said it follows that µk’1 /µk

the exact sequence

0 ’ S k (Λ1 (A)) ’ J k (A) ’ J k’1 (A) ’ 0.

But, by the induction assumption, J k’1 (A) is a projective module isomor-

phic to i¤k’1 S i (Λ1 (A)). Hence,

J k (A) S k (Λ1 (A)) • J k’1 (A) S i (Λ1 (A))

i¤k

which ¬nishes the proof.

def

Definition 4.8. Let P be an A-module. The module Smbl— (P ) =

k≥0 Smblk (P ), where

def

Smblk (P ) = Diff k (P )/ Diff k’1 (P ),

is called the module of symbols for P . The coset of ∆ ∈ Diff k (P ) in Smblk (P )

is called the symbol of the operator ∆.

Let σ ∈ Smbli (A) and σ ∈ Smblj (A) and assume that ∆ ∈ Diff i (A) and

∆ ∈ Diff j (A) are representatives of σ, σ respectively. De¬ne the product

σσ as the coset of ∆ —¦ ∆ in Diff i+j (A). It is easily checked that Smbl— (A)

forms a commutative A-algebra with respect to thus de¬ned multiplication.

As a direct consequence of the last proposition and of Proposition 4.4,

we obtain

Corollary 4.17. If A is a smooth algebra, then the following state-

ments are valid :

(i) Diff k (P ) Diff k (A) —A P ,

(ii) Diff — (A), as an associative algebra, is generated by A = Diff 0 (A) and

D1 (A) ‚ Diff 1 (A),

(iii) Smblk (P ) Smblk (A) —A P ,

(iv) Smbl— (A), as a commutative algebra, is isomorphic to the symmetric

tensor algebra of D1 (A).

Remark 4.5. It should be noted that Smbl— A is more than just a com-

mutative algebra. In fact, in the case A = C ∞ (M ), as it can be easily seen,

elements of Smbl— A can be naturally identi¬ed with smooth functions on

T — M polynomial along the ¬bers of the natural projection T — M ’ M . The

manifold T — M is symplectic and, in particular, the algebra C ∞ (T — M ) pos-

sesses a Poisson bracket which induces a bracket in Smbl— A ‚ C ∞ (T — M ).

This bracket, as it happens, is of a purely algebraic nature.

Let us consider two symbols σ1 ∈ Smbli1 A, σ2 ∈ Smbli2 A such that

σr = ∆r mod Diff ir ’1 A, r = 1, 2, and set

def

{σ1 , σ2 } = [∆1 , ∆2 ] mod Diff i1 +i2 ’2 . (4.21)

The operation {·, ·} de¬ned by (4.21) is called the Poisson bracket in the

algebra of symbols and in the case A = C ∞ (M ) coincides with the classical

2. NIJENHUIS BRACKET 171

Poisson bracket on the cotangent space. It possesses the usual properties,

i.e.,

{σ1 , σ2 } + {σ2 , σ1 } = 0,

{σ1 , {σ2 , σ3 }} + {σ2 , {σ3 , σ1 }} + {σ3 , {σ1 , σ2 }} = 0,

{σ1 , σ2 σ3 } = {σ1 , σ2 }σ3 + σ2 {σ1 , σ3 }

and, in particular, Smbl— A becomes a Lie K-algebra with respect to this

bracket. This is a starting point to construct Hamiltonian formalism in a

general algebraic setting. For details and generalizations see [104, 53, 54].

2. Fr¨licher“Nijenhuis bracket

o

We still consider the general algebraic setting of the previous section

and extend standard constructions of calculus to form-valued derivations.

It allows us to de¬ne Fr¨licher“Nijenhuis brackets and introduce a coho-

o

mology theory ( -cohomologies) associated to commutative algebras with

¬‚at connections. In the next chapter, applying this theory to in¬nitely

prolonged partial di¬erential equations, we obtain an algebraic and analyt-

ical description of recursion operators for symmetries and describe e¬cient

tools to compute these operators. These and related results, together with

their generalizations, were ¬rst published in the papers [55, 56, 57] and

[59, 58, 40].

2.1. Calculus in form-valued derivations. Let k be a ¬eld of char-

acteristic zero and A be a commutative unitary k-algebra. Let us recall the

basic notations:

• D(P ) is the module of P -valued derivations A ’ P , where P is an

A-module;

• Di (P ) is the module of P -valued skew-symmetric i-derivations. In

particular, D1 (P ) = D(P );

• Λi (A) is the module of di¬erential i-forms of the algebra A;

• d : Λi (A) ’ Λi+1 (A) is the de Rham di¬erential.

Recall also that the modules Λi (A) are representative objects for the

functors Di : P ’ Di (P ), i.e., Di (P ) = HomA (Λi (A), P ). The isomorphism

D(P ) = HomA (Λ1 (A), P ) can be expressed in more exact terms: for any

derivation X : A ’ P , there exists a uniquely de¬ned A-module homomor-

phism •X : Λ1 (A) ’ P satisfying the equality X = •X —¦ d. Denote by

Z, ω ∈ P the value of the derivation Z ∈ Di (P ) at ω ∈ Λi (A).

Both Λ— (A) = i≥0 Λi (A) and D— (A) = i≥0 Di (A) are endowed with

the structures of superalgebras with respect to the wedge product operations

§ : Λi (A) — Λj (A) ’ Λi+j (A),

§ : Di (A) — Dj (A) ’ Di+j (A),

the de Rham di¬erential d : Λ— (A) ’ Λ— (A) becoming a derivation of Λ— (A).

Note also that D— (P ) = i≥0 Di (P ) is a D— (A)-module.

172 4. BRACKETS

Using the paring ·, · and the wedge product, we de¬ne the inner product

(or contraction) iX ω ∈ Λj’i (A) of X ∈ Di (A) and ω ∈ Λj (A), i ¤ j, by

setting

Y, iX ω = (’1)i(j’i) X § Y, ω , (4.22)

where Y is an arbitrary element of Dj’i (P ), P being an A-module. We

formally set iX ω = 0 for i > j. When i = 1, this de¬nition coincides with

the one given in Section 1. Recall that the following duality is valid:

X, da § ω = X(a), ω , (4.23)

where ω ∈ Λi (A), X ∈ Di+1 (P ), and a ∈ A (see Corollary 4.11). Using the

property (4.23), one can show that

iX (ω § θ) = iX (ω) § θ + (’1)Xω ω § iX (ω)

for any ω, θ ∈ Λ— (A), where (as everywhere below) the symbol of a graded

object used as the exponent of (’1) denotes the degree of that object.

We now de¬ne the Lie derivative of ω ∈ Λ— (A) along X ∈ D— (A) as

LX ω = iX —¦ d ’ (’1)X d —¦ iX ω = [iX , d]ω, (4.24)

where [·, ·] denotes the graded (or super) commutator: if ∆, ∆ : Λ— (A) ’

Λ— (A) are graded derivations, then

[∆, ∆ ] = ∆ —¦ ∆ ’ (’1)∆∆ ∆ —¦ ∆.

For X ∈ D(A) this de¬nition coincides with the ordinary commutator of

derivations.

Consider now the graded module D(Λ— (A)) of Λ— (A)-valued deriva-

tions A ’ Λ— (A) (corresponding to form-valued vector ¬elds ” or, which

is the same ” vector-valued di¬erential forms on a smooth manifold).

Note that the graded structure in D(Λ— (A)) is determined by the splitting

D(Λ— (A)) = i≥0 D(Λi (A)) and thus elements of grading i are derivations

X such that im X ‚ Λi (A). We shall need three algebraic structures asso-

ciated to D(Λ— (A)).

First note that D(Λ— (A)) is a graded Λ— (A)-module: for any X ∈

D(Λ— (A)), ω ∈ Λ— (A) and a ∈ A we set (ω § X)a = ω § X(a). Second,

we can de¬ne the inner product iX ω ∈ Λi+j’1 (A) of X ∈ D(Λi (A)) and

ω ∈ Λj (A) in the following way. If j = 0, we set iX ω = 0. Then, by induc-

tion on j and using the fact that Λ— (A) as a graded A-algebra is generated

by the elements of the form da, a ∈ A, we set

iX (da § ω) = X(a) § ω ’ (’1)X da § iX (ω), a ∈ A. (4.25)

Finally, we can contract elements of D(Λ— (A)) with each other in the fol-

lowing way:

X, Y ∈ D(Λ— (A)), a ∈ A.

(iX Y )a = iX (Y a), (4.26)

Three properties of contractions are essential in the sequel.

2. NIJENHUIS BRACKET 173

Proposition 4.18. Let X, Y ∈ D(Λ— (A)) and ω, θ ∈ Λ— (A). Then

iX (ω § θ) = iX (ω) § θ + (’1)ω(X’1) ω § iX (θ), (4.27)

iX (ω § Y ) = iX (ω) § Y + (’1)ω(X’1) ω § iX (Y ), (4.28)

[iX , iY ] = i[[X,Y ]]rn , (4.29)

where

[[X, Y ]]rn = iX (Y ) ’ (’1)(X’1)(Y ’1) iY (X). (4.30)

Proof. Equality (4.27) is a direct consequence of (4.25). To prove

(4.28), it su¬ces to use the de¬nition and expressions (4.26) and (4.27).

Let us prove (4.29) now. To do this, note ¬rst that due to (4.26), the

equality is su¬cient to be checked for elements ω ∈ Λj (A). Let us use

induction on j. For j = 0 it holds in a trivial way. Let a ∈ A; then one has

[iX , iY ](da § ω) = iX —¦ iY ’ (’1)(X’1)(Y ’1) iY —¦ iX (da § ω)

= iX (iY (da § ω)) ’ (’1)(X’1)(Y ’1) iY (iX (da § ω)).

But

iX (iY (da § ω)) = iX (Y (a) § ω ’ (’1)Y da § iY ω)

= iX (Y (a)) § ω + (’1)(X’1)Y Y (a) § iX ω ’ (’1)Y (X(a) § iY ω

’ (’1)X da § iX (iY ω)),

while

iY (iX (da § ω) = iY (X(a) § ω ’ (’1)X da § iX ω)

= iY (X(a)) § ω + (’1)X(Y ’1) X(a) § iY ω ’ (’1)X (Y (a) § iX ω

’ (’1)Y da § iY (iX ω)).

Hence,

[iX , iY ](da § ω) = iX (Y (a)) ’ (’1)(X’1)(Y ’1) iY (X(a)) § ω

+ (’1)X+Y da § iX (iY ω) ’ (’1)(X’1)(Y ’1) iY (iX ω) .

But, by de¬nition,

iX (Y (a)) ’ (’1)(X’1)(Y ’1) iY (X(a))

= (iX Y ’ (’1)(X’1)(Y ’1) iY X)(a) = [[X, Y ]]rn (a),

whereas

iX (iY ω) ’ (’1)(X’1)(Y ’1) iY (iX ω) = i[[X,Y ]]rn (ω)

by induction hypothesis.

Note also that the following identity is valid for any X, Y, Z ∈ D(Λ— (A)):

Z + (’1)X (X § Y )

X (Y Z) = (X Y) Z. (4.31)

174 4. BRACKETS

Definition 4.9. The element [[X, Y ]]rn de¬ned by (4.30) is called the

Richardson“Nijenhuis bracket of elements X and Y .

Directly from Proposition 4.18 we obtain the following

Proposition 4.19. For any derivations X, Y, Z ∈ D(Λ— (A)) and a form

ω ∈ Λ— (A) one has

[[X, Y ]]rn + (’1)(X+1)(Y +1) [[Y, X]]rn = 0, (4.32)

(’1)(Y +1)(X+Z) [[[[X, Y ]]rn , Z]]rn = 0, (4.33)

[[X, ω § Y ]]rn = iX (ω) § Y + (’1)(X+1)ω ω § [[X, Y ]]rn . (4.34)

Here and below the symbol denotes the sum of cyclic permutations.

Remark 4.6. Note that Proposition 4.19 means that D(Λ— (A))“ is a

Gerstenhaber algebra with respect to the Richardson“Nijenhuis bracket [48].

Here the superscript “ denotes the shift of grading by 1.

Similarly to (4.24), let us de¬ne the Lie derivative of ω ∈ Λ— (A) along

X ∈ D(Λ— (A)) by

LX ω = (iX —¦ d ’ (’1)X’1 d —¦ iX )ω = [iX , d]ω (4.35)

Remark 4.7. Let us clarify the change of sign in (4.35) with respect to

formula (4.24). If A is a commutative algebra, then the module D— (Λ— (A))

is a bigraded module: if ∆ ∈ Di (Λj (A)), then bigrading of this element is

def

(i, j). We can also consider the total grading by setting deg ∆ = i + j. In

this sense, if X ∈ Di (A), then deg X = i, and for X ∈ D1 (Λj (A)), then

deg X = j + 1. This also explains shift of grading in Remark 4.6.

From the properties of iX and d we obtain

Proposition 4.20. For any X ∈ D(Λ— (A)) and ω, θ ∈ Λ— (A), one has

the following identities:

LX (ω § θ) = LX (ω) § θ + (’1)Xω ω § LX (θ), (4.36)

Lω§X = ω § LX + (’1)ω+X d(ω) § iX , (4.37)

[LX , d] = 0. (4.38)

Our main concern now is to analyze the commutator [LX , LY ] of two Lie

derivatives. It may be done e¬ciently for smooth algebras (see De¬nition

4.7).

Proposition 4.21. Let A be a smooth algebra. Then for any derivations

X, Y ∈ D(Λ— (A)) there exists a uniquely determined element [[X, Y ]]fn ∈

D(Λ— (A)) such that

[LX , LY ] = L[[X,Y ]]fn . (4.39)

2. NIJENHUIS BRACKET 175

Proof. To prove existence, recall that for smooth algebras one has

Di (P ) = HomA (Λi (A), P ) = P —A HomA (Λi (A), A) = P —A Di (A)

for any A-module P and integer i ≥ 0. Using this identi¬cation, let us

represent elements X, Y ∈ D(Λ— (A)) in the form

X = ω — X and Y = θ — Y for ω, θ ∈ Λ— (A), X , Y ∈ D(A).

Then it is easily checked that the element

Z = ω § θ — [X , Y ] + ω § LX θ — Y + (’1)ω dω § iX θ — Y

’ (’1)ωθ θ § LY ω — X ’ (’1)(ω+1)θ dθ § iY ω — X

= ω § θ — [X , Y ] + LX θ — Y ’ (’1)ωθ LY ω — X (4.40)

satis¬es (4.39).

Uniqueness follows from the fact that LX (a) = X(a) for any a ∈ A.

Definition 4.10. The element [[X, Y ]]fn ∈ Di+j (Λ— (A)) de¬ned by for-

mula (4.39) (or by (4.40)) is called the Fr¨licher“Nijenhuis bracket of form-

o

i (Λ— (A)) and Y ∈ D j (Λ— (A)).

valued derivations X ∈ D

The basic properties of this bracket are summarized in the following

Proposition 4.22. Let A be a smooth algebra, X, Y, Z ∈ D(Λ— (A)) be

derivations and ω ∈ Λ— (A) be a di¬erential form. Then the following iden-

tities are valid :

[[X, Y ]]fn + (’1)XY [[Y, X]]fn = 0, (4.41)

(’1)Y (X+Z) [[X, [[Y, Z]]fn ]]fn = 0, (4.42)

i[[X,Y ]]fn = [LX , iY ] + (’1)X(Y +1) LiY X , (4.43)

iZ [[X, Y ]]fn = [[iZ X, Y ]]fn + (’1)X(Z+1) [[X, iZ Y ]]fn

(4.44)

+ (’1)X i[[Z,X]]fn Y ’ (’1)(X+1)Y i[[Z,Y ]]fn X,

[[X, ω § Y ]]fn = LX ω § Y ’ (’1)(X+1)(Y +ω) dω § iY X

(4.45)

+ (’1)Xω ω § [[X, Y ]]fn .

Note that the ¬rst two equalities in the previous proposition mean that

the module D(Λ— (A)) is a Lie superalgebra with respect to the Fr¨licher“

o

Nijenhuis bracket.

Remark 4.8. The above exposed algebraic scheme has a geometrical

realization, if one takes A = C ∞ (M ), M being a smooth ¬nite-dimensional

manifold. The algebra A = C ∞ (M ) is smooth in this case. However,

in the geometrical theory of di¬erential equations we have to work with

in¬nite-dimensional manifolds3 of the form N = proj lim{πk+1,k } Nk , where

3

In¬nite jets, in¬nite prolongations of di¬erential equations, total spaces of coverings,

etc.

176 4. BRACKETS

all the mappings πk+1,k : Nk+1 ’ Nk are surjections of ¬nite-dimensional

smooth manifolds. The corresponding algebraic object is a ¬ltered algebra

A = k∈Z Ak , Ak ‚ Ak+1 , where all Ak are subalgebras in A. As it was al-

ready noted, self-contained di¬erential calculus over A is constructed, if one

considers the category of all ¬ltered A-modules with ¬ltered homomorphisms

for morphisms between them. Then all functors of di¬erential calculus in

this category become ¬ltered, as well as their representative objects.

In particular, the A-modules Λi (A) are ¬ltered by Ak -modules Λi (Ak ).

We say that the algebra A is ¬nitely smooth, if Λ1 (Ak ) is a projective Ak -

module of ¬nite type for any k ∈ Z. For ¬nitely smooth algebras, elements

of D(P ) may be represented as formal in¬nite sums k pk — Xk , such that

any ¬nite sum Sn = k¤n pk — Xk is a derivation An ’ Pn+s for some ¬xed

s ∈ Z. Any derivation X is completely determined by the system {Sn } and

Proposition 4.22 obviously remains valid.

Remark 4.9. In fact, the Fr¨licher“Nijenhuis bracket can be de¬ned in

o

a completely general situation, with no additional assumption on the algebra

A. To do this, it su¬ces to de¬ne [[X, Y ]]fn = [X, Y ], when X, Y ∈ D1 (A)

and then use equality (4.44) as inductive de¬nition. Gaining in generality,

we then loose of course in simplicity of proofs.

2.2. Algebras with ¬‚at connections and cohomology. We now

introduce the second object of our interest. Let A be an k-algebra, k being

a ¬eld of zero characteristic, and B be an algebra over A. We shall assume

that the corresponding homomorphism • : A ’ B is an embedding. Let P

be a B-module; then it is an A-module as well and we can consider the B-

module D(A, P ) of P -valued derivations A ’ P .

Definition 4.11. Let • : D(A, •) ’ D(•) be a natural transforma-

tions of the functors D(A, •) : A ’ D(A, P ) and D(•) : P ’ D(P ) in the

category of B-modules, i.e., a system of homomorphisms P : D(A, P ) ’

D(P ) such that the diagram

P

’ D(P )

D(A, P )

D(A, f ) D(f )

“ “

Q

’ D(Q)

D(A, Q)

is commutative for any B-homomorphism f : P ’ Q. We say that • is a

connection in the triad (A, B, •), if P (X) A = X for any X ∈ D(A, P ).

Here and below we use the notation Y |A = Y —¦ • for any derivation

Y ∈ D(P ).

Remark 4.10. When A = C ∞ (M ), B = C ∞ (E), • = π — , where M and

E are smooth manifolds and π : E ’ M is a smooth ¬ber bundle, De¬nition

2. NIJENHUIS BRACKET 177

4.11 reduces to the ordinary de¬nition of a connection in the bundle π. In

fact, if we have a connection • in the sense of De¬nition 4.11, then the

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

In this chapter, we apply the algebraic formalism of Chapter 4 to the

speci¬c case of partial di¬erential equations. Namely, we consider a formally

integrable equation E ‚ J k (π), π : E ’ M , taking the associated triple

(C ∞ (M ), F(E), π∞ ) for the algebra with ¬‚at connection, where F(E) =

—

∞ ∞ ’ M is the

k Fk (E) is the algebra of smooth functions on E , π∞ : E

natural projection and the Cartan connection C plays the role of .

We compute the corresponding cohomology groups for the case E ∞ =

J ∞ (π) and deduce de¬ning equations for a general E. We also establish

relations between in¬nitesimal deformations of the equation structure and

recursion operators for symmetries and consider several illustrative exam-

ples.

We start with repeating some de¬nitions and proofs of the previous

chapter in the geometrical situation.

1. C-cohomologies of partial di¬erential equations

Here we introduce cohomological invariants of partial di¬erential equa-

tions based on the results of Sections 1, 2 of Chapter 4. We call these in-

variants C-cohomologies since they are determined by the Cartan connection

C on E ∞ . We follow the scheme from the classical paper by Nijenhuis and

Richardson [78], especially in interpretation of the cohomology in question.

Let ξ : P ’ M be a ¬ber bundle with a connection , which is considered

as a C ∞ (M )-homomorphism : D(M ) ’ D(P ) taking a ¬eld X ∈ D(M ) to

the ¬eld (X) = X ∈ D(P ) and satisfying the condition X (ξ — f ) = X(f )

for any f ∈ C ∞ (M ).

Let y ∈ P , ξ(y) = x ∈ M , and denote by Py = ξ ’1 (x) the ¬ber of

the projection ξ passing through y. Then determines a linear mapping

y : Tx (M ) ’ Ty (P ) such that ξ—,y ( y (v)) = v for any v ∈ Tx (M ). Thus

with any point y ∈ P a linear subspace y (Tx (M )) ‚ Ty (P ) is associated. It

determines a distribution D on P which is called the horizontal distribution

of the connection . If is ¬‚at, then D is integrable.

As it is well known (see, for example, [46, 47]), the connection form

U = U ∈ Λ1 (P ) — D(P ) can be de¬ned as follows. Let y ∈ P , Y ∈ D(P ),

Yy ∈ Ty (P ) and v = ξ—,y (Yy ). Then we set

U )y = Y y ’

(Y y (v). (5.1)

187

188 5. DEFORMATIONS AND RECURSION OPERATORS

In other words, the value of U at the vector Yy ∈ Ty (P ) is the projection of

Yy onto the tangent plane Ty (P ) along the horizontal plane1 passing through

y ∈ P.

If (x1 , . . . , xn ) are local coordinates in M and (y 1 , . . . , y s ) are coordinates

along the ¬ber of ξ (the case s = ∞ is included), we can de¬ne by the

following equalities

s

‚ ‚ ‚

j

= + = i. (5.2)

i

‚y j

‚xi ‚xi

j=1

Then U is of the form

s n

‚

j

j

dy ’ dxi —

U= , (5.3)

i

‚y j

j=1 i=1

From equality (4.40) on p. 175 it follows that

‚k ‚k ‚

j j

fn ±

dxi § dxj — k .

[[U , U ]] = 2 + (5.4)

i

‚y ±

‚xi ‚y

±

i,j,k

Recall that the curvature form R of the connection is de¬ned by the

equality

’ X, Y ∈ D(M ).

R (X, Y ) = [ X, Y] [X,Y ] ,

We shall express the element [[U , U ]]fn in terms of the form R now

(cf. Proposition 4.24). Let us consider a ¬eld X ∈ D(P ) and represent it in

the form

X = X v + X h, (5.5)

where, by de¬nition,

Xv = X Xh = X ’ Xv

U,

are vertical and horizontal components of X respectively. In the same

manner one can de¬ne vertical and horizontal components of any element

„¦ ∈ Λ— (P ) — D(P ).

Obviously, X v ∈ Dv (P ), where

Dv (P ) = {X ∈ D(P ) | Xξ — (f ) = 0, f ∈ C ∞ (M )},

while the component X h is of the form

fi ∈ C ∞ (P ), Xi ∈ D(M ),

Xh = fi Xi ,

i

and lies in the distribution D .

1

With respect to .

1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 189

Proposition 5.1. Let : D(M ) ’ D(P ) be a connection in the ¬ber

bundle ξ : P ’ M . Then for any ξ-vertical vector ¬eld X v one has

[[U , U ]]fn = 0.

Xv

fi , gj ∈ C ∞ (P ), Xi , Yj ∈ D(M ),

If X h = i fi Xi , Y h = gj Yj ,

j

are horizontal vector ¬elds, then

[[U , U ]]fn = 2

Yh Xh fi gj R (Xi , Yj ), (5.6)

i,j

or, to be short,

[[U , U ]]fn = 2R .

Proof. Let X ∈ D(P ). Then from equality (4.45) on p. 175 it follows

that

[[U, U ]]fn = 2([[U, X]]fn U ]]fn ),

U ’ [[U, X

X

where U = U . Hence, if X = X v is a vertical ¬eld, then

[[U, U ]]fn = 2([[U, X v ]]fn U ]]fn ) = ’2([[U, X v ]]fn )h .

Xv U ’ [[U, X v

But the left-hand side of this equality is vertical (see (5.4)) and thus vanishes.

This proves the ¬rst part of the proposition.

Let now X = X h be a horizontal vector ¬eld. Then

[[U, U ]]fn = 2[[U, X h ]]fn U = 2([[U, X h ]]fn )v .

Xh

Hence, if Y h is another horizontal ¬eld, then, by (4.31) on p. 173, one has

[[U, U ]]fn ) = 2Y h ([[U, X h ]]fn [[U, X h ]]fn )

Yh (X h U ) = 2(Y h U.

But from (4.45) (see p. 175) it follows that

[[U, X h ]]fn = [[X h , Y h ]]fn

Yh U = [X h , Y h ] U.

Therefore,

[[U, U ]]fn = 2[X h , Y h ]

Yh Xh U = 2([X h , Y h ] ’ [X h , Y h ]h )

h

’[

=2 fi gi ([ Xi , Yj ] Xi , Yj ] )

i,j

But obviously, for any f ∈ C ∞ (M )

for X = i fi and Y = gj Yj .

Xi j

one has

[ Xi , Yj ](f ) = [Xi , Yj ](f )

and, consequently,

h

[ Xi , Yj ] = [Xi ,Yj ] ,

which ¬nishes the proof.

190 5. DEFORMATIONS AND RECURSION OPERATORS

From equality (5.6) and from the considerations in the end of Section 2

of Chapter 4 it follows that if the connection in question is ¬‚at, i.e. R = 0,

then the element U determines a complex

‚0

0 ’ D(P ) ’’ Λ1 (P ) — D(P ) ’ · · ·

‚i

’ Λ (P ) — D(P ) ’’ Λi+1 (P ) — D(P ) ’ · · · ,

i

(5.7)

where ‚ = ‚ i = [[U , · ]]fn .

Remark 5.1. Horizontal vector ¬elds X h are de¬ned by the condition

X h U = 0. Denote the module of such ¬elds by D h (P ):

Dh (P ) = {X ∈ D(P ) | X U = 0}.

Then, by setting ˜ = U = U in (4.31) on p. 173, one can see that

‚ („¦ U ) = ‚ („¦) U

for any „¦ ∈ Λ— (P ) — D(P ). Hence,

‚ (Λ— (P ) — D v (P )) ‚ Λ— (P ) — D v (P )

and

‚ (Λ— (P ) — D h (P )) ‚ Λ— (P ) — D h (P ).

Considering a direct sum decomposition

Λ— (P ) — D(P ) = Λ— (P ) — D v (P ) • Λ— (P ) — D h (P )

one can see that

‚ = ‚v • ‚h ,

where

‚ v = ‚ |Λ— (P )—Dv (P ) , ‚ h = ‚ |Λ— (P )—Dh (P ) .

To proceed further let us compute 0-cohomology of the complex (5.7).

From equality (4.31) on p. 173 it follows that for any two vector ¬elds

Y, Z ∈ D(P ) the equality

‚ 0 Y + [Z, Y ]

Z U = [Z U ,Y ]

holds. Thus Y ∈ ker(‚ 0 ) if and only if

[Z, Y ] U = [Z U ,Y ]

for any Z ∈ D(P ). Using decomposition (5.5) for the ¬elds Y and Z and

substituting it into the last equation, we get that the condition Y ∈ ker(‚ 0 )

is equivalent to the system of equations

[Z v , Y h ] U = [Z v , Y h ], [Z h , Y v ] U = 0. (5.8)

1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 191

Let Y h = i fi (see above). Then from the ¬rst equality of (5.8) it

Xi

follows that

Z v (fi ) v

= fi [ Xi , Z ].

Xi

i i

But the left-hand side of this equation is a horizontal vector ¬eld while the

right-hand side is always vertical. Hence,

Z v (fi ) =0

Xi

i

for any vertical ¬eld Z v . Choosing locally independent vector ¬elds Xi , we

see that the functions fi actually lie in C ∞ (M ) (or, strictly speaking, in

ξ — (C ∞ (M )) ‚ C ∞ (P )). It means that, at least locally, Y h is of the form

Yh = X ∈ D(M ).

X,

But since X = X if and only if X = X , the ¬eld X is well de¬ned on

the whole manifold M .

On the other hand, from the second equality of (5.8) we see that Y v ∈

ker(‚ 0 ) if and only if the commutator [Z h , Y v ] is a horizontal ¬eld for any

horizontal Z h . Thus we get the following result:

Proposition 5.2. A direct sum decomposition

ker(‚ 0 ) = Dv (P ) • (D(M ))

: D(M ) ’ D(P )

(D(M )) is the image of the mapping

takes place, where

and

Dv (P ) = {Y ∈ D v (P ) | [Y, D h (P )] ‚ D h (P )}.

One can see now that D v (P ) consists of nontrivial in¬nitesimal sym-

metries of the distribution D while the elements of (D(M )) are trivial

symmetries (in the sense that the corresponding transformations slide inte-

gral manifolds of D along themselves). To skip this trivial part of ker(‚ 0 ),

note that

(i) U ∈ Λ1 (P ) — D v (P ),

and (see Remark 5.1)

(ii) ‚ i Λi (P ) — D v (P ) ‚ Λi+1 (P ) — D v (P ).

Thus we have a vertical complex

‚0

0 ’ D (P ) ’’ Λ1 (P ) — D v (P ) ’ · · ·

v

‚i

’ Λi (P ) — D v (P ) ’’ Λi+1 (P ) — D v (P ) ’ · · · ,

the i-th cohomology of which is denoted by H i (P ). From the above said it

follows that H 0 (P ) coincides with the Lie algebra of nontrivial in¬nitesimal

symmetries for the distribution D .

192 5. DEFORMATIONS AND RECURSION OPERATORS

Consider now an in¬nitely prolonged equation E ∞ ‚ J ∞ (π) and the

Cartan connection C = CE in the ¬ber bundle π∞ : E ∞ ’ M . The corre-

sponding connection form U , where = C, will be denoted by UE in this

case. Knowing the form UE , one can reconstruct the Cartan distribution on

E ∞ . Since this distribution contains all essential information about solutions

of E, one can state that UE determines the equation structure on E ∞ (see

De¬nition 2.4 in Chapter 2).

By rewriting the vertical complex de¬ned above in the case ξ = π∞ , we

get a complex

0

‚C

0 ’ D (E) ’ Λ1 (E) — D v (E) ’ · · ·

v

’

i

‚C

’ Λ (E) — D (E) ’ Λi+1 (E) — D v (E) ’ · · · ,

i v

’ (5.9)

where, for the sake of simplicity, Λi (E) stands for Λi (E ∞ ). The cohomologies

i

of (5.9) are denoted by HC (E) and are called C-cohomologies of the equation

E.

From the de¬nition of the Lie algebra sym(E) and from the previous

considerations we get the following

Theorem 5.3. For any formally integrable equation E one has the iso-

morphism

0

HC (E) = sym(E).

1

To obtain an interpretation of the group HC (E), consider the element

U = UE ∈ Λ1 (E) — D v (E) and its deformation U (µ), U (0) = U , where µ ∈ R

is a small parameter. It is natural to expect this deformation to satisfy the

following conditions:

(i)

U (µ) ∈ Λ1 (E) — D v (E) (verticality)

and

(ii)

[[U (µ), U (µ)]]fn = 0 (integrability). (5.10)

Let us expand U (µ) into a formal series in µ,

U (µ) = U0 + U1 µ + · · · + Ui µi + · · · , (5.11)

and substitute (5.11) into (i) and (ii). Then one can see that U1 ∈ Λ1 (E) —

Dv (E) and

[[U0 , U1 ]]fn = 0.

1 1

Since U0 = U (0) = U , it follows that U1 ∈ ker(‚E ). Thus ker(‚E ) con-

sists of all (vertical) in¬nitesimal deformations of U preserving the natural

conditions (i) and (ii).

1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 193

0 0

On the other hand, im(‚E ) consists of elements of the form ‚E (X) =

[[U, X]]fn , X ∈ Dv (E). Such elements can be viewed as in¬nitesimal defor-

mations of U originating from transformations of E ∞ which are trivial on

M (i.e., ¬ber-wise transformations of the bundle π∞ : E ∞ ’ M ). In fact,

let P be a manifold and At : P ’ P , t ∈ R, A0 = id, be a one-parameter

group of di¬eomorphisms with

d

(At ) = X ∈ D(P ).

dt t=0

Then for any ˜ ∈ Λ— (P ) — D(P ) one can consider the element At,— (L˜ )

de¬ned by means of the commutative diagram

L˜

Λ— (P ) ’ Λ— (P )

A— A— (5.12)

t t

“ “

At,— (L˜ ) —

—

’ Λ (P )

Λ (P )

Then, obviously, for any homogeneous element ˜ = θ — Y ∈ Λ— (P ) — D(P )

and a form ω ∈ Λ— (P ) we have

d

At,— (L˜ )(ω)

dt t=0

d

A— (θ) § A— Y A— ω + (’1)θ dA— θ § A— (Y A— ω)

= ’t ’t

t t t t

dt t=0

= X(θ) § Y (ω) + θ § [X, Y ](ω) + (’1)θ dX(θ) § (Y ω)

+ (’1)θ dθ § [X, Y ] ω = L([[X,θ—Y ]]fn ) (ω).

— Yi ∈ Λ— (P ) — D(P ) and sets

Thus, if one takes ˜ = i θi

A— (˜) = ˜(t) = A— (θi ) — A— Yi A— , (5.13)

’t

t t t

i

then

˜(t) = ˜ + [[X, ˜]]fn t + o(t).

In other words, [[X, ˜]]fn is the velocity of the transformation of ˜ with re-

spect to At . Taking P = E ∞ and ˜ = U , one can see that the elements

V = [[U, X]]fn are in¬nitesimal transformations of U arising from transfor-

mations At : E ∞ ’ E ∞ . If π∞ —¦ At = π∞ , then X ∈ D v (E) and V ∈ im(‚E ).

0

It is natural to call such deformations of U trivial.

Since, as it was pointed out above, the element U determines the struc-

ture of di¬erential equation on the manifold E ∞ we obtain the following

result.

194 5. DEFORMATIONS AND RECURSION OPERATORS

1

Theorem 5.4. The elements of HC (E) are in one-to-one correspondence

with the classes of nontrivial in¬nitesimal vertical deformations of the equa-

tion E.

Remark 5.2. One can consider deformations of UE not preserving the

verticality condition. Then classes of the corresponding in¬nitesimal defor-

mations are identi¬ed with the elements of the ¬rst cohomology module of

the complex (5.7) (for P = E ∞ and = C). The theory of such deformations

is quite interesting but lies beyond the scope of the present book.

Remark 5.3. Since the operation [[·, ·]]fn de¬ned on HC (E ∞ ) takes its

1

2 2

values in HC (E) the elements of the module HC (E) (or a part of them at least)

can be interpreted as the obstructions for the deformations of E (cf. [78]).

Local coordinate expressions for the element UE and for the di¬erentials

‚C = ‚E in the case E ∞ = J ∞ (π) look as follows.

i

Let (x1 , . . . , xn , u1 , . . . , um ) be local coordinates in J 0 (π) and pj , j =

σ

1, . . . , m, |σ| ≥ 0, be the corresponding canonical coordinates in J ∞ (π).

Then from equality (1.35) on p. 26 and (5.3) it follows that

‚

j

ωσ —

U= , (5.14)

‚pj

σ

j,σ

j

where ωσ are the Cartan forms on J ∞ (π) given by (1.27) (see p. 18).

Consider an element ˜ = j,σ θσ — ‚/‚pj ∈ Λ— (π) — D v (π). Then, due

j

σ

to (5.14) and (4.40), p. 175, we have

n m

‚

j j

dxi § θσ+1i ’ Di (θσ ) —

‚π (˜) = , (5.15)

‚pj

σ

i=1 j=1 |σ|≥0

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-