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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

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