E∞

But a morphism of this type, as it can be easily checked, is exactly a shadow

of a nonlocal symmetry in the covering P (cf. Section 2). And as we know,

action of the Lenard operator on the scaling symmetry of the KdV equation

results in a shadow which can be reconstructed using the methods of Section

3.

We conclude this section with discussing the problem of inversion of re-

cursion operators. This nontrivial, from analytical point of view, procedure,

becomes quite trivial in the geometrical setting.

In fact, to invert a recursion operator (R, K, L) just amounts to changing

arrows in the corresponding diagram:

R

L

L

=

=

K

K

←

’

V E∞ V E∞

v

„

v

„

’

←

E∞

152 3. NONLOCAL THEORY

We shall illustrate the procedure using the example of the modi¬ed KdV

equation (mKdV).

Example 3.5 (see also [28, 27, 29]). Consider the mKdV eqiation

written in the form

ut = uxxx ’ u2 ux .

Then the corresponding Cartan covering is given by the pair of equations

ut = uxxx ’ u2 ux ,

vt = vxxx ’ u2 vx ’ 2uux v,

while the recursion operator for the mKdV equation comes out of the cov-

ering R of the form

wx = uv,

wt = uvx x ’ ux vx + uxx v ’ u3 v

and is of the form L : z = vxx ’ 2 u2 v ’ 2 ux w, where z stands for the nonlocal

3 3

coordinate in the second copy of V E ∞ .

To invert L, it needs to reconstruct the covering over the second copy

of V E ∞ using the above information. From the form of L we obtain vxx =

z + 2 u2 v + 2 ux w, from where it follows that the needed nonlocal variables

3 3

are v, w, and s satisfying the relations

wx = uv,

vx = s,

2 2

sx = ux w + u2 v + z

3 3

and

2 1

wt = uux w + uxx ’ u3 z ’ ux s + uz,

3 3

2 1

vt = uxx w ’ u2 s + zx ,

3 3

2 2 2 2 2 1

uxxx ’ u2 ux w + uuxx ’ u4 v ’ uux s + zx ’ u2 z.

st =

3 9 3 9 3 3

Consequently, we got the covering L : R ’ V E ∞ with (w, v, s) ’ v, and it

is natural to identify the triple (R = R, L = K, K = L) with the inverted

recursion operator.

It should be noted that the covering R can be simpli¬ed in the following

way: set

3 3 2

p’ = w ’

p+ = w + q = ’ uw + s.

, ,

2 2 3

¨

8. BACKLUND TRANSFORMATIONS AND RECURSION OPERATORS 153

Then we get

2± 3

p± = ± up ± q,

x

3 2

qx = z,

√

2 1 62 3

p± = ± uxx ’ u3 p± ’ ux ± q±

u zx + uz,

t

3 3 6 2

qt = zxx ’ u2 z,

while K acquires the form v = p+ ’ p’ .

154 3. NONLOCAL THEORY

CHAPTER 4

Brackets

This chapter is of a purely algebraic nature. Following [99] (see also

[60, Ch. 1]), we construct di¬erential calculus in the category of modules

over a unitary commutative K-algebra A, K being a commutative ring with

unit (in the corresponding geometrical setting K is usually the ¬eld R and

A = C ∞ (M ) for a smooth manifold M ). Properly understood, this calculus

is a system of special functors, together with their natural transformations

and representative objects.

In the framework of the calculus constructed, we study form-valued

derivations and deduce, in particular, two types of brackets: the Richardson“

Nijenhuis and Fr¨licher“Nijenhuis ones. If a derivation is integrable in the

o

sense of the second one, a cohomology theory can be related to it. A source

of integrable elements are algebras with ¬‚at connections.

These algebras serve as an adequate model for in¬nitely prolonged dif-

ferential equations, and we shall also show that all basic conceptual con-

structions introduced on E ∞ in previous chapters are also valid for algebras

with ¬‚at connections, becoming much more transparent. In particular, the

notions of a symmetry and a recursion operator for an algebra with ¬‚at

connection are introduced in cohomological terms and the structure of sym-

metry Lie algebras is analyzed. Later (in Chapter 5) we specify all these

results for the case of the bundle E ∞ ’ M .

1. Di¬erential calculus over commutative algebras

Throughout this section, K is a commutative ring with unit, A is a

commutative K-algebra, P, Q, . . . are modules over A. We introduce linear

di¬erential operators ∆ : P ’ Q, modules of jets J k (P ), derivations, and

di¬erential forms Λi (A).

1.1. Linear di¬erential operators. Consider two A-modules P and

Q and the K-module homK (P, Q). Then there exist two A-module struc-

tures in homK (P, Q): the left one

a ∈ A, f ∈ homK (P, Q), p ∈ P,

(la f )(p) = af (p),

and the right one

a ∈ A, f ∈ homK (P, Q), p ∈ P.

(ra f )(p) = f (ap),

Let us introduce the notation δa = la ’ ra .

155

156 4. BRACKETS

Definition 4.1. A linear di¬erential operator of order ¤ k acting from

an A-module P to an A-module Q is a mapping ∆ ∈ homK (P, Q) satisfying

the identity

(δa0 —¦ · · · —¦ δak )∆ = 0 (4.1)

for all a0 , . . . ak ∈ A.

For any a, b ∈ A, one has

la —¦ r b = r b —¦ l a

and consequently the set of all di¬erential operators of order ¤ k

(i) is stable under both left and right multiplication and

(ii) forms an A-bimodule.

(+)

This bimodule is denoted by Diff k (P, Q), while the left and the right

multiplications in it are denoted by a∆ and a+ ∆ respectively, a ∈ A, ∆ ∈

(+) (+)

Diff k (P, Q). When P = A, we use the notation Diff k (Q).

Obviously, one has embeddings of A-bimodules

(+) (+)

Diff k (P, Q) ’ Diff k (P, Q)

for any k ¤ k and we can de¬ne the module

def

(+) (+)

Diff — (P, Q) = Diff k (P, Q).

k≥0

(+)

Note also that for k = 0 we have Diff 0 (P, Q) = homA (P, Q).

Let P, Q, R be A-modules and ∆ : P ’ Q, ∆ : Q ’ R be di¬erential

operators of orders k and k respectively. Then the composition ∆ —¦∆ : P ’

R is de¬ned.

Proposition 4.1. The composition ∆ —¦ ∆ is a di¬erential operator of

order ¤ k + k .

Proof. In fact, by de¬nition we have

δa (∆ —¦ ∆) = δa (∆ ) —¦ ∆ + ∆ —¦ δa (∆). (4.2)

for any a ∈ A. Let a = {a0 , . . . , as } be a set of elements of the algebra

A. Say that two subsets ar = {ai1 , . . . , air } and as’r+1 = {aj1 , . . . , ajs’r+1 }

form an unshu¬„e of a, if i1 < · · · < ir , j1 < · · · < js’r+1 . Denote the set

def

of all unshu¬„es of a by unshu¬„e(a) and set δa = δa0 —¦ · · · —¦ δas . Then from

(4.2) it follows that

δa (∆ —¦ ∆ ) = δar (∆) —¦ δas’r+1 (∆ ) (4.3)

(ar ,as’r+1 )∈unshu¬„e(a)

for any ∆, ∆ . Hence, if s ≥ k + k + 1, both summands in (4.3) vanish

which ¬nishes the proof.

1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 157

Remark 4.1. Let M be a smooth manifold, π, ξ be vector bundles over

M and P = “(π), Q = “(ξ). Then ∆ is a di¬erential operator in the sense

of De¬nition 4.1 if and only if it is a linear di¬erential operator acting from

sections of π to those of ξ.

First note that it su¬ces to consider the case M = Rn , π and ξ being

trivial one-dimensional bundles over M . Obviously, any linear di¬erential

operator in a usual analytical sense satis¬es De¬nition 4.1. Conversely, let

∆ : C ∞ (M ) ’ C ∞ (M ) satisfy De¬nition 4.1 and be an operator of order

k. Consider a function f ∈ C ∞ (M ) and a point x0 ∈ M . Then in a

neighborhood of x0 the function f is represented in the form

(x ’ x0 )σ ‚ |σ| f

(x ’ x0 )σ gσ (x),

f (x) = +

‚x|σ|

σ!

x=x0

|σ|¤k |σ|=k+1

where (x ’ x0 )σ = (x1 ’ x0 )i1 . . . (xn ’ x0 )in , σ! = i1 ! . . . in !, and gσ are some

n

1

smooth functions. Introduce the notation

(x ’ x0 )σ

∆σ = ∆ ;

σ!

then

«

‚ |σ| f

(x ’ x0 )σ gσ (x) .

+ ∆

∆(f ) = ∆σ (4.4)

‚x|σ|

x=x0

|σ|¤k |σ|=k+1

Due to the fact that ∆ is a k-th order operator, from equality (4.3) it

follows that the last summand in (4.4) vanishes. Hence, ∆f is completely

determined by the values of partial derivatives of f up to order k and depends

on these derivatives linearly.

Consider a di¬erential operator ∆ : P ’ Q and A-module homomor-

phisms f : Q ’ R and f : R ’ P . Then from De¬nition 4.1 it follows that

both f —¦ ∆ : P ’ R and ∆ —¦ f : R ’ Q are di¬erential operators of order

(+)

ord ∆. Thus the correspondence (P, Q) ’ Diff k (P, Q), k = 0, 1, . . . , —, is

a bifunctor from the category of A-modules to the category of A-bimodules.

Proposition 4.2. Let us ¬x a module Q. Then the functor Diff + (•, Q)

k

is representable in the category of A-modules. Moreover, for any di¬eren-

tial operator ∆ : P ’ Q of order k there exists a unique homomorphism

f∆ : P ’ Diff + (Q) such that the diagram

k

∆

’Q

P

(4.5)

f∆

k

D

’

←

Diff + (Q)

k

158 4. BRACKETS

def

∈

is commutative, where the operator Dk is de¬ned by Dk ( ) = (1),

Diff + (Q).

k

def

Proof. Let p ∈ P, a ∈ A and set (f∆ (p))(a) = ∆(ap). It is easily seen

that it is the mapping we are looking for.

Definition 4.2. Let ∆ : P ’ Q be a k-th order di¬erential operator.

def

The composition ∆(l) = Dl —¦ ∆ : P ’ Diff + (Q) is called the l-th Diff-

l

prolongation of ∆.

Consider, in particular, the l-th prolongation of the operator Dk . By

de¬nition, we have the following commutative diagram

Dl

Diff + (P ) ’ Diff + (P )

l,k k

(D

)(l

k

cl,k Dk

)

“

’“

Dk+l

Diff + (P ) ’P

l+k

def def

where Diff + ,...,in = Diff + —¦ · · · —¦ Diff + and cl,k = fDk —¦Dl . The mapping

i1 i1 in

cl,k = cl,k (P ) : Diff l,k (P ) ’ Diff l+k (P ) is called the gluing homomorphism

while the correspondence P ’ cl,k (P ) is a natural transformation of functors

called the gluing transformation.

Let ∆ : P ’ Q, : Q ’ R be di¬erential operators of orders k and l

respectively. The A-module homomorphisms

f ∆ : P ’ Diff + (Q), P ’ Diff + (R), f : Q ’ Diff + (R)

f —¦∆ :

k k+l l

are de¬ned. On the other hand, since Diff + (•) is a functor, we have the

k

homomorphism Diff k (f ) : Diff k (Q) ’ Diff + (Diff + (R)).

+ +

k l

Proposition 4.3. The diagram

f —¦∆

’ Diff + (R)

P k+l

‘

f∆ ck,l (4.6)

“

Diff + (f )

k

Diff + (Q) ’ Diff + (R)

k k,l

is commutative.

By this reason, the transformation ck,l is also called the universal com-

position transformation.

1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 159

1.2. Jets. Let us now study representability of the functors Diff k (P, •).

Consider an A-module P and the tensor product A —K P endowed with

two A-module structures

la (b — p) = (ab) — p, ra (b — p) = b — (ap), a, b ∈ A, p ∈ P.

We also set δ a = la ’ra and denote by µk the submodule1 in A—K P spanned

by all elements of the form

(δ a0 —¦ · · · —¦ δ as )(a — p), a0 , . . . , as ∈ A, s ≥ k.

def

Definition 4.3. The module J k (P ) = (A—K P )/µk is called the mod-

ule of k-jets for the module P . The correspondence

jk : P ’ J k (P ), p ’ (1 — p) mod µk ,

is called the k-jet operator.

Proposition 4.4. The mapping jk is a linear di¬erential operator of

order ¤ k. Moreover, for any linear di¬erential operator ∆ : P ’ Q there

exists a uniquely de¬ned homomorphism f ∆ : J k (P ) ’ Q such that the

diagram

jk

’ J k (P )

P

∆

∆

f

’

←

Q

is commutative.

Hence, Diff k (P, •) is a representable functor. Note also that J k (P )

carries two structures of an A-module (with respect to la and ra ) and the

correspondence P ’ J k (P ) is a functor from the category of A-modules to

the category of A-bimodules.

Note that by de¬nition we have short exact sequences of A-modules

νk+1,k

0 ’ µk+1 /µk ’ J k+1 (P ) ’ ’ ’ J k (P ) ’ 0

’’

and thus we are able to de¬ne the A-module

def

J ∞ (P ) = proj lim J k (P )

{νk+1,k }