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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 }

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