Taking into account the last result, one has the following decomposition

p,q

r

HI (E) = HI (E),

p+q=r

where

p,q p,q p,q’1

HI (E) = ker(‚I )/im(‚I ),

where ‚ i,j : C i Λ(FI ) § Λj (FI ) — Dv (FI ) ’ C i (FI ) § Λj+1 (FI ) — Dv (FI ).

k h

In particular,

0,1 1,0

1

HI (E) = HI (E) • HI (E). (6.24)

1 0

Note now that from the point of view of HI (E)-action on HI (E) =

symI E, the ¬rst summand in (6.24) is of no interest, since

Dv (FI ) Λ1 (FI ) = 0.

h

—,0 1,0

—

We call HI (E) the Cartan part of HI (E), while the elements of HI (E)

are called recursion operators for the extension (FI , CDI (E)). One has the

following

p,0 p,0

Proposition 6.20. HI (E) = ker ‚I .

Proof. In fact, from Proposition 6.17 one has

im(‚I ) © (C — Λ(FI ) — Dv (FI )) = 0,

which proves the result.

260 6. SUPER AND GRADED THEORIES

—,0

Note that HI (E) inherits an associative graded algebra structure with

1,0

respect to contraction, HI (E) being its subalgebra.

2.8. Commutativity theorem. In this subsection we prove the fol-

lowing

1,0 1,0 2,0

Theorem 6.21. [[HI (E), HI (E)]]fn ‚ HI (E).

The proof is based on the following

Lemma 6.22. For any ω ∈ C 1 Λ(FI ) one has

UI ω = ω. (6.25)

Proof of Lemma 6.22. It is su¬cient to prove (6.25) for the genera-

tors of the module C 1 Λ(FI ) which are of the form

g ∈ FI .

ω = LI (g),

From (6.10) one has

L I —¦ i UI ’ i UI —¦ L I + L UI = i[[UI ,UI ]]fn ,

UI

or

LI —¦ iUI ’ iUI —¦ LI + LI = 0. (6.26)

Applying (6.26) to some g ∈ FI , one sees that

UI LI (g) = LI (g).

1,0

Proof of Theorem 6.21. Let „¦, ˜ ∈ HI (E), i.e., „¦, ˜ ∈ C 1 Λ(FI )

and ‚I „¦ = ‚I ˜ = 0. Then from (6.11) it follows that

[[„¦, ˜]]fn = [[UI „¦, ˜]]fn + [[„¦, UI ˜]]fn ,

UI

or, due to Lemma 6.22,

[[„¦, ˜]]fn = 2[[„¦, ˜]]fn .

UI

Hence,

1 1

[[„¦, ˜]]fn = UI [[„¦, ˜]]fn = UI (UI [[„¦, ˜]]fn )

2 4

1

= ((UI UI ) [[„¦, ˜]]fn ’ (UI § UI ) [[„¦, ˜]]fn )

4

1

= (UI [[„¦, ˜]]fn ’ (UI § UI ) [[„¦, ˜]]fn )

4

1 1

= [[„¦, ˜]]fn ’ (UI § UI ) [[„¦, ˜]]fn ,

2 4

or

1

[[„¦, ˜]]fn = ’ (UI § UI ) [[„¦, ˜]]fn .

2

But UI ∈ C 1 Λ(FI ) — Dv (FI ) which ¬nishes the proof.

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 261

Corollary 6.23. The element UI is a unit of the associative algebra

1,0

HI (E).

Proof. The result follows from the de¬nition of the element UI and

from Lemma 6.22.

2,0

Corollary 6.24. Under the assumption HI (E) = 0, all recursion op-

erators for the graded extension (FI , CDI (E)) commute with respect to the

Fr¨licher“Nijenhuis bracket.

o

1,0 0

Let „¦ ∈ HI (E) be a recursion operator. Denote its action on HI (E) =

0

symI (E) by „¦(X) = X „¦, X ∈ HI (E). Then, from (6.11) it follows that

[[„¦, ˜]]fn = (’1)X·Y (’1)Y ·„¦ [„¦(X), ˜(Y )]

Y X

+ (’1)(Y +„¦)·˜ [˜(X), „¦(Y )]

’ (’1)„¦·˜ „¦((’1)Y ·˜ [˜(X), Y ] + [X, ˜(Y )])

’ ˜((’1)Y ·„¦ [„¦(X), Y ] + [X, „¦(Y )])

+ ((’1)„¦·˜ „¦ —¦ ˜ + ˜ —¦ „¦)[X, Y ] , (6.27)

1,0

for all X, Y ∈ symI (E), „¦, ˜ ∈ HI (E).

2,0

Corollary 6.25. If HI (E) = 0, then for any symmetries X, Y ∈

1,0

symI (E) and recursion operators „¦, ˜ ∈ HI (E) one has

(’1)Y ·„¦ [„¦(X), ˜(Y )] + (’1)(Y +„¦)·˜ [˜(X), „¦(Y )]

= (’1)„¦·˜ „¦((’1)Y ·„¦ [˜(X), Y ] + [X, ˜(Y )]) + ˜((’1)Y ·„¦ [„¦(X), Y ]

+ [X, „¦(Y )]) + ((’1)„¦—¦˜ „¦ —¦ ˜ + ˜ —¦ „¦)[X, Y ]. (6.28)

In particular,

(1 + (’1)„¦·„¦ ) (’1)Y ·„¦ [„¦(X), „¦(Y )]

’ (’1)Y ·„¦ „¦[„¦(X), Y ] ’ „¦[X, „¦(Y )] + „¦2 [X, Y ] = 0,

and if „¦ · „¦ is even, then

[„¦(X), „¦(Y )] = „¦([„¦(X), Y ] + (’1)Y „¦ [X, „¦(Y )] ’ (’1)Y „¦ „¦[X, Y ]).

(6.29)

Using Corollary 6.25, one can describe a Lie algebra structure of sym I E

in a way similar to Section 3 of Chapter 4.

3. Nonlocal theory and the case of evolution equations

Here we extend the theory of coverings and that of nonlocal symmetries

(see Chapter 3 to the case of graded equations (cf. [87]). We con¬ne our-

selves to evolution equations though the results obtained, at least partially,

are applicable to more general cases. For any graded equation the notion of

262 6. SUPER AND GRADED THEORIES

its tangent covering (an add analog of the Cartan covering, see Example 3.2

on p. 100) is introduced which reduces computation of recursion operators to

computations of special nonlocal symmetries. In this setting, we also solve

the problem of extending “shadows” of recursion operators up to real ones.

3.1. The GDE(M ) category. Let M be a smooth manifold and A =

C ∞ (M ). We de¬ne the GDE(M ) category of graded di¬erential equations

over M as follows. The objects of GDE(M ) are pairs (F, F ), where F is

a commutative n-graded A-algebra (the case n = ∞ is included) endowed

with a ¬ltration

A = F’∞ ‚ . . . ‚ Fi ‚ Fi+1 ‚ . . . , Fi = F, (6.30)

i

while F is a ¬‚at (A, F)-connection (see Subsection 2.2), i.e.,

(i) F ∈ homF (D(A, F), D(F)),

(ii) F (X)(a) = X(a), X ∈ D(A, F), a ∈ A,

(iii) [ F (X), F (Y )] = F ( F (X) —¦ Y ’ F (Y ) —¦ X), X, Y ∈ D(A, F).

From the de¬nition it follows that the grading of F is 0, and we also

suppose that for any X ∈ D(A, F) the derivation F (X) agrees with the

¬ltration (6.30), i.e.,

‚ Fi+s

F (X)(Fi )

for some s = s(X) and all i large enough.

Let (F, F ) and (G, G ) be two objects and • : F ’ G be a graded

¬ltered homomorphism. Then for any X ∈ D(A, F) the composition • —¦ X

lies in D(A, G). We say that it is a morphism of the object (F, F ) to

(G, G ) if the diagram

•

F ’G

—¦ X)

F (X) G (•

“ “

•

F ’G

is commutative for all X ∈ D(A, F). If • is a monomorphism, we say that

it represents a covering of (G, G ) over (F, F ).

Remark 6.7. Let E be an equation in some bundle over M . Then all

graded extensions of E are obviously objects of GDE(M ).

Remark 6.8. The theory of the previous section can be literally applied

to the objects of GDE(M ) as well.

3.2. Local representation. In what follows, we shall deal with the

following kinds of objects of the category GDE(M ):

(i) in¬nite prolongations of di¬erential equations;

(ii) their graded extensions;

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 263

(iii) coverings over (i) and (ii).

For particular applications local versions of these objects will be consid-

ered. It means the following:

(i) In a neighborhood O ‚ M local coordinates x = (x1 , . . . , xn ) are

chosen (independent variables);

(ii) the bundle π : E ’ M in which E is de¬ned is supposed to be a

vector bundle, and it trivializes over O. If (e1 , . . . , em ) is a basis

of local sections of π over O, then f = u1 e1 + · · · + um em for any

f ∈ “(π|O ), and u1 , . . . , um play the role of dependent variables for

the equation E;

(iii) the equation E is represented by a system of relations

±

F1 (x, . . . , uj , . . . ) = 0,

σ

.....................

F1 (x, . . . , uj , . . . ) = 0,

σ

where uj = ‚ |σ| uj /‚xσ , σ = (i1 , . . . , in ), |σ| = i1 + · · · + in ¤ k, are

σ

coordinates in the manifold of k-jets J k (π), k being the order of E;

(iv) a graded extension F of F(E) (see Subsection 2.3 is freely generated

over F(E) by homogeneous elements v 1 , v 2 , . . . . It means that F∞ is

j j

generated by v„ , where v0 = v j and

j j

v(i1 ,...,is +1,...,in ) = [Ds , v(i1 ,...,in ) ],

Ds being the total derivative on E ∞ corresponding to ‚/‚xs . In this

setting any graded extension of E can be represented as

± j j j

F1 (x, . . . , uσ , . . . ) + φ1 (x, . . . , uσ , . . . , v„ , . . . ) = 0,

................................................

F (x, . . . , uj , . . . ) + φ (x, . . . , uj , . . . , v„ , . . . ) = 0,

j

r

σ σ

r

................................................

φr+1 (x, . . . , uj , . . . v„ , . . . ) = 0,

j

σ

.............................

φr+l (x, . . . , uj , . . . v„ , . . . ) = 0,

j

σ

j

where φ1 , . . . , φr are functions such that φ1 = 0, . . . , φr = 0 for v„ = 0.

(v) for any covering • : F ’ G of the graded extension F by an object

(G, G ) we assume that G is freely generated over F by homogeneous

elements w 1 , w2 , . . . and

‚ ‚ def ˜

Xis

= Di + = Di

G

‚ws

‚xi s

with

‚ ‚

˜˜ s

Xis

[Di , Dj ] = [Di , Xj ]+[ , Dj ]

‚ws ‚ws

s s

264 6. SUPER AND GRADED THEORIES

‚ ‚

Xis s

+[ , Xj ] = 0,

‚ws ‚ws

s s

where i, j = 1, . . . , n, Xis ∈ G, and D1 , . . . , Dn are total derivatives

extended onto F: Di = F (‚/‚xi ). Elements w 1 , w2 , . . . are called

nonlocal variables related to the covering •, the number of nonlocal

variables being called the dimension of •.

3.3. Evolution equations. Below we deal with super (Z2 -graded) evo-

lution equations E in two independent variables x and t:

±

u1 = f 1 (x, t, u1 , . . . , um , . . . , u1 , . . . , um ),

t k k

(6.31)

.......................................

m

ut = f m (x, t, u1 , . . . , um , . . . , u1 , . . . , um ),

k k

where u1 , . . . , um are either of even or of odd grading, and uj denotes s

j /‚xs . We take x, t, u1 , . . . , um , . . . , u1 , . . . uj , . . . for the internal coor-

‚u 0 0 i i

dinates on E ∞ . The total derivatives Dx and Dt restricted onto the in¬nite

prolongation of (6.31) are of the form

∞ m

‚ ‚

uj

Dx = + ,

i+1

‚uj

‚x i

i=0 j=1

∞m

‚ ‚

Dx (f j )

i

Dt = + . (6.32)

‚uj

‚t i

i=0 j=1

In the chosen local coordinates, the structural element U = UE of the

equation E is represented as

∞ m

‚

(duj ’ uj dx ’ D i (f j ) dt) —

U= . (6.33)

i i+1

‚uj

i

i=0 j=1

Then for a basis of the module C 1 Λ(E) one can choose the forms

ωi = LU (uj ) = duj ’ uj dx ’ D i (f j ) dt,

j

i i i+1

while (6.33) is rewritten as

∞ m

‚

j

ωi —

U= . (6.34)

‚uj

i

i=0 j=1

j

— ‚/‚uj ∈ Λp (E) — D v (E). Then from (6.6) one

∞ m

Let ˜ = j=1 θi

i=0 i

has

∞ m

j j

fn

dx § (θi+1 ’ Dx (θi ))

‚E (˜) = [[U, ˜]] =

i=0 j=1

∞ m ij

± ‚Dx f ‚

j

+ dt § ’ Dt (θi ) —

θβ . (6.35)

‚u± ‚uj

β i

β=0 ±=1

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 265

From (6.35) one easily gets the following

p,0

Theorem 6.26. Let E be an equation of the form (6.31). Then HC (E)

consists of the elements

∞ m

‚

Dx (θj ) —

i

= ,

θ

‚uj

i

i=0 j=1

where θ = (θ 1 , . . . , θm ), θj ∈ C p Λ(E), is a vector-valued form satisfying the

equations

k m l

i j ‚f

Dx (θ ) j = 0, l = 1, . . . , m, (6.36)

‚ui

i=0 j=1

or in short,

(p)

p,0

HC (E) = ker E,

(p)

where E is the extension of the operator of universal linearization operator

onto the module C p Λ(E) —R Rm :

k m l

i j ‚f

(p) l

( E (θ)) = Dx (θ ) j , l = 1, . . . , m. (6.37)

‚ui

i=0 j=1

3.4. Nonlocal setting and shadows. Let now • be a covering of

equation (6.31) determined by nonlocal variables w 1 , w2 , . . . with the ex-

tended total derivatives of the form

‚

˜

Dx = D x + Xs ,

‚ws

s

‚

˜

Dt = D t + Ts , (6.38)

‚ws

s

satisfying the identity

˜˜

[Dx , Dt ] = 0. (6.39)

Denote by F(E• ) the corresponding algebra of functions and by Λ— (E• )

and D(E• ) the modules of di¬erential forms and vector ¬elds on F(E• )

respectively. Then the structural element of the covering object is

‚

(dωs ’ Xs dx ’ Ts dt) —

U• = U +

‚ws

s

and the identity