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We de¬ne a contraction of a polyderivation ∆ ∈ Ds (A) into a form ω ∈ Λr
in the following way
i∆ ω ≡ ∆ ω = 0, if s > r,
i∆ ω = ∆(ω), if s = r, due to the de¬nition of Λr ,
ia ω = aω, if a ∈ A = D0 (A),
1
This distinction between ¬rst n gradings and additional (n + 1)-st one will be pre-
served both for graded forms and graded polyderivations throughout the whole chapter.
1. GRADED CALCULUS 247

and for r > s set by induction
i∆ (da § ω) = i∆(a) (ω) + (’1)∆·a+s da § i∆ (ω). (6.2)
Proposition 6.4. Let A be an n-graded commutative algebra.
(i) For any ∆ ∈ Ds (A) de¬nition (6.2) determines an (n + 1)-graded
di¬erential operator
i∆ : Λ — ’ Λ —
of the order s.
(ii) In particular, if ∆ ∈ D1 (A), then i∆ is a graded derivation of Λ— :
ω ∈ Λr , θ ∈ Λ— .
i∆ (ω § θ) = i∆ (ω) § θ + (’1)∆·ω+r ω § i∆ θ,
Now we consider tensor products of the form Λr —A Ds (A) and generalize
contraction and wedge product operations as follows
(ω — ∆) § (θ — ∆) = (’1)∆·θ (ω § θ) — (∆ § ),
iω—∆ (θ — ) = ω § i∆ (θ) — ,
where ω, θ ∈ Λ— , ∆, ∈ D— (A). Let us de¬ne the Richardson“Nijenhuis
bracket in Λ— — Ds (A) by setting
[[„¦, ˜]]rn = i„¦ (˜) ’ (’1)(ω+∆)·(˜+ )+(q’s)(r’s)
i˜ („¦), (6.3)
s
where „¦ = ω — ∆ ∈ Λr — Ds (A), ˜ = θ — ∈ Λq — Ds (A). In what follows,
we con¬ne ourselves with the case s = 1 and introduce an (n + 1)-graded
structure into Λ— — D1 (A) by setting
gr(ω — X) = (gr(ω) + gr(X), r), (6.4)
where gr(ω) and gr(X) are initial n-gradings of the elements ω ∈ Λr , X ∈
D1 (A). We also denote by „¦ and „¦1 the ¬rst n and (n + 1)-st gradings of
„¦ respectively in the powers of (’1).
Proposition 6.5. Let A be an n-graded commutative algebra. Then:
(i) For any two elements „¦, ˜ ∈ Λ— — D1 (A) one has
[i„¦ , i˜ ] = i[[„¦,˜]]rn .
1

Hence, the Richardson“Nijenhuis bracket [[·, ·]]rn = [[·, ·]]rn determines
1
— — D (A) the structure of (n + 1)-graded Lie algebra with respect
in Λ 1
to the grading in which (n+1)-st component is shifted by 1 with respect
to (6.4), i.e.,
(ii) [[„¦, ˜]]rn + (’1)„¦·˜+(„¦1 +1)(˜1 +1) [[˜, „¦]]rn = 0,
(iii) (’1)˜·(„¦+Ξ)+(˜1 +1)(„¦1 +Ξ1 ) [[[[„¦, ˜]]rn , Ξ]]rn = 0, where, as before,
denotes the sum of cyclic permutations.
(iv) Moreover, if ρ ∈ Λ— , then
[[„¦, ρ § θ]]rn = („¦ ρ) § ˜ + (’1)„¦·ρ § [[„¦, ˜]]rn .
(v) In conclusion, the composition of two contractions is expressed by
+ (’1)„¦1 i„¦§˜ .
i„¦ —¦ i ˜ = i „¦ ˜
248 6. SUPER AND GRADED THEORIES

1.4. De Rham complex and Lie derivatives. The de Rham di¬er-
ential d : Λr ’ Λr+1 is de¬ned as follows. For r = 0 it coincides with the
derivation d : A ’ Λ1 introduced in Proposition 6.2. For any adb ∈ Λ1 ,
a, b ∈ A, we set
d(adb) = da § db
and for a decomposable form ω = θ §ρ ∈ Λr , θ ∈ Λr , ρ ∈ Λr , r > 1, r , r <
r, set
dω = dθ § ρ + (’1)θ1 θ § dρ.
By de¬nition, d : Λ— ’ Λ— is a derivation of grading (0, 1) and, obviously,
d —¦ d = 0.
Thus, one gets a complex
d
0 ’ A ’ Λ1 ’ · · · ’ Λr ’ dΛr+1 ’ · · · ,

which is called the de Rham complex of A.
Let X ∈ D1 (A) be a derivation. A Lie derivative LX : Λ— ’ Λ— is
de¬ned as
LX = [iX , d] = iX —¦ d + d —¦ iX . (6.5)
Thus for any ω ∈ Λ— one has
LX ω = X dω + d(X ω).
The basic properties of LX are described by
Proposition 6.6. For any commutative n-graded algebra A one has
(i) If ω, θ ∈ Λ— , then
LX (ω § θ) = LX ω § θ + (’1)X·ω ω § LX θ,
i.e., LX really is a derivation of grading (gr(X), 0).
(ii) [LX , d] = LX —¦ d ’ d —¦ LX = 0.
(iii) For any a ∈ A and ω ∈ Λ— one has
LaX (ω) = aLX ω + da § iX (ω).
(iv) [LX , iY ] = [iX , LY ] = i[X,Y ] .
(v) [LX , LY ] = L[X,Y ] .
Now we extend the classical de¬nition of Lie derivative onto the elements
of Λ— — D1 (A) and for any „¦ ∈ Λ— — D1 (A) de¬ne
L„¦ = [i„¦ , d] = i„¦ —¦ d + (’1)„¦1 d —¦ i„¦ .
If „¦ = ω — X, then one has
Lω—X = ω § LX + (’1)ω1 dω § iX .
Proposition 6.7. For any n-graded commutative algebra A the follow-
ing statements are valid :
1. GRADED CALCULUS 249

(i) For any „¦ ∈ Λ— — D1 (A) one has
L„¦ (ρ § θ) = L„¦ (ρ) § θ + (’1)„¦·ρ+„¦1 ·ρ1 ρ § L„¦ θ, ρ, θ ∈ Λ— ,
i.e., L„¦ is a derivation of Λ— whose grading coincides with that of „¦.
(ii) [L„¦ , d] = L„¦ —¦ d ’ (’1)„¦1 d —¦ L„¦ = 0.
(iii) Lρ§„¦ = ρ § L„¦ + (’1)ρ1 +„¦1 dρ § i„¦ , ρ ∈ Λ— .
To formulate properties of L„¦ similar to (iv) and (v) of Proposition 6.6,
one needs a new notion.
1.5. Graded Fr¨licher“Nijenhuis bracket. We shall now study the
o
commutator of two Lie derivatives.
Proposition 6.8. Let, as before, A be an n-graded commutative alge-
bra.
(i) For any two elements „¦, ˜ ∈ Λ— — D1 (A), the commutator of
corresponding Lie derivatives [L„¦ , L˜ ] is of the form LΞ for some
Ξ ∈ Λ— — D1 (A).
(ii) The correspondence L : Λ— — D1 (A) ’ D1 (Λ— ), „¦ ’ L„¦ , is injec-
tive and hence Ξ in (i) is de¬ned uniquely. It is called the (graded)
Fr¨licher“Nijenhuis bracket of the elements „¦, ˜ and is denoted by
o
Ξ = [[„¦, ˜]]fn . Thus, by de¬nition, one has
[L„¦ , L˜ ] = L[[„¦,˜]]fn .
(iii) If „¦ and ˜ are of the form
ω, θ ∈ Λ— , X, Y ∈ D1 (A),
„¦ = ω — X, ˜ = θ — Y,
then
[[„¦, ˜]]fn = (’1)X·θ ω § θ — [X, Y ] + ω § LX θ — Y
+ (’1)„¦1 dω § (X θ) — Y
’ (’1)„¦·˜+„¦1 ·˜1 θ § LY ω — X
’ (’1)„¦·˜+(„¦1 +1)·˜1 dθ § (Y ω) — X
= (’1)X·θ ω § θ — [X, Y ] + L„¦ (θ) — Y
’ (’1)„¦·˜+„¦1 ·˜1 L˜ (ω) — X. (6.6)
(iv) If „¦ = X, ˜ = Y ∈ D1 (A) = Λ0 — D1 (A), then the graded Fr¨licher“
o
Nijenhuis bracket of „¦ and ˜ coincides with the graded commutator
of vector ¬elds:
[[X, Y ]]fn = [X, Y ].
The main properties of the Fr¨licher“Nijenhuis bracket are described by
o
Proposition 6.9. For any „¦, ˜, Ξ ∈ Λ— — D1 (A) and ρ ∈ Λ— one has
(i)
[[„¦, ˜]]fn + (’1)„¦·˜+„¦1 ·˜1 [[˜, „¦]]fn = 0. (6.7)
250 6. SUPER AND GRADED THEORIES

(ii)

(’1)(„¦+Ξ)·˜+(„¦1 +Ξ1 )·˜1 [[„¦, [[˜, Ξ]]fn ]]fn = 0, (6.8)

i.e., [[·, ·]]fn de¬nes a graded Lie algebra structure in Λ— — D1 (A).
(iii)

[[„¦, ρ § ˜]]fn = L„¦ (ρ) § ˜ ’ (’1)„¦·(˜+ρ)+(„¦1 +1)·(˜1 +ρ1 ) dρ § i˜ „¦
+ (’1)„¦·ρ+„¦1 ·ρ1 · ρ § [[„¦, ˜]]fn . (6.9)
(iv)
[L„¦ , i˜ ] + (’1)„¦·˜+„¦1 ·(˜1 +1) L˜ = i[[„¦,˜]]fn . (6.10)
„¦

(v)

iΞ [[„¦, ˜]]fn = [[iΞ „¦, ˜]]fn + (’1)„¦·Ξ+„¦1 ·(Ξ1 +1) [[„¦, iΞ ˜]]fn
+ (’1)„¦1 i[[Ξ,„¦]]fn ˜ ’ (’1)„¦·˜+(„¦1 +1)·˜1 i[[Ξ,˜]]fn „¦. (6.11)

Remark 6.5. Similar to the commutative case, identity (6.11) can be
taken for the inductive de¬nition of the graded Fr¨licher“Nijenhuis bracket.
o
Let now U be an element of Λ1 — D1 (A) and let us de¬ne the operator
‚U = [[U, ·]]fn : Λr — D1 (A) ’ Λr+1 — D1 (A). (6.12)
Then from the de¬nitions it follows that
‚U (U ) = [[U, U ]]fn = (1 + (’1)U ·U )LU —¦ LU (6.13)
and from (6.7) and (6.8) one has
(1 + (’1)U ·U )‚U (‚U „¦) + (’1)U ·U [[„¦, [[U, U ]]fn ]]fn = 0
for any „¦ ∈ Λ— — D1 (A).
We are interested in the case when (6.12) is a complex, i.e., ‚U —¦ ‚U = 0,
and give the following
Definition 6.1. An element U ∈ Λ1 — D1 (A) is said to be integrable, if
(i) [[U, U ]]fn = 0 and
(ii) (’1)U ·U equals 1.
From the above said it follows that for an integrable element U one has
‚U —¦ ‚U = 0, and we can introduce the corresponding cohomologies by
ker(‚U : Λr — D1 (A) ’ Λr+1 — D1 (A))
r
HU (A)
= .
im(‚U : Λr’1 — D1 (A) ’ Λr — D1 (A))
The main properties of ‚U are described by
Proposition 6.10. Let U ∈ Λ1 — D1 (A) be an integrable element and
„¦, ˜ ∈ Λ— — D1 (A), ρ ∈ Λ— . Then
(i) ‚U (ρ § „¦) = LU (ρ) § „¦ ’ (’1)U ·(„¦+ρ) dρ § i„¦ U + (’1)U ·ρ+ρ1 ρ § ‚U „¦.
2. GRADED EXTENSIONS 251

(ii) [LU , i„¦ ] = i‚U „¦ + (’1)U ·„¦+„¦1 L„¦ U .
(iii) [i„¦ , ‚U ]˜ + (’1)U ·˜ i[[„¦,˜]]fn U = [[i„¦ U , ˜]]fn + (’1)U ·„¦+„¦1 i‚U „¦ ˜.
(iv) ‚U [[„¦, ˜]]fn = [[‚U „¦, ˜]]fn + (’1)U ·„¦+„¦1 [[„¦, ‚U ˜]]fn .
From the last equality it follows that the Fr¨licher“Nijenhuis bracket is
o
— r
inherited by the module HU (A) = r≥0 HU (A) and thus the latter forms
an (n + 1)-graded Lie algebra with respect to this bracket.

2. Graded extensions
In this section, we adapt the cohomological theory of recursion opera-
tors constructed in Chapter 5 (see also [55, 58]) to the case of graded (in
particular, super) di¬erential equations. Our ¬rst step is an appropriate
de¬nition of graded equations (cf. [87] and the literature cited there). In
what follows, we still assume all the modules to be projective and of ¬nite
type over the main algebra A0 = C ∞ (M ) or to be ¬ltered by such modules
in a natural way.

2.1. General construction. Let R be a commutative ring with a unit
and A’1 ‚ A0 be two unitary associative commutative Zn -graded R-alge-
bras. Let D = D0 ‚ D(A’1 , A0 ) be an A0 -submodule in the module

D(A’1 , A0 ) = {‚ ∈ homR (A’1 , A0 ) | ‚(aa )
= ‚a · a + (’1)a·‚ a · ‚a , a, a ∈ A’1 }.
Let us de¬ne a Zn -graded A0 -algebra A1 by the generators
a ∈ A0 , ‚ ∈ D0 , gr[‚, a] = gr(‚) + gr(a),
[‚, a],
with the relations
[‚, a0 ] = ‚a0 ,
[‚, a + a ] = [‚, a] + [‚, a ],
[a ‚ + a ‚ , a] = a [‚ , a] + [‚ , a],
[‚, aa ] = [‚, a] · a + (’1)‚·a a · [‚, a ],
where a0 ∈ A’1 , a, a , a ∈ A, ‚, ‚ , ‚ ∈ D0 .
For any ‚ ∈ D0 we can de¬ne a derivation ‚ (1) ∈ D(A0 , A1 ) by setting
‚ (1) (a) = [‚, a], a ∈ A1 .
Obviously, ‚ (1) a = ‚a for a ∈ A0 . Denoting by D1 the A1 -submodule in
D(A0 , A1 ) generated by the elements of the form ‚ (1) , one gets the triple
{A0 , A1 , D1 }, A0 ‚ A1 , D1 ‚ D(A0 , A1 ),
which allows one to construct {A1 , A2 , D2 }, etc. and to get two in¬nite
sequences of embeddings
A’1 ’ A0 ’ · · · ’ Ai ’ Ai+1 ’ · · ·
252 6. SUPER AND GRADED THEORIES

and
D0 ’ D1 ’ · · · ’ Di ’ Di+1 ’ · · · ,
where Ai+1 = (Ai )1 , Di+1 = (Di )1 ‚ D(Ai’1 , Ai ), and Di ’ Di+1 is a
morphism of Ai+1 -modules.
Let us set
D∞ = inj lim Di .
A∞ = inj lim Ai ,
i’∞ i’∞

Then D∞ ‚ D(A∞ ) and any element ‚ ∈ D0 determines a derivation
D(‚) = ‚ (∞) ∈ D(A∞ ). The correspondence D : D0 ’ D(A∞ ) possesses
the following properties
D(X)(a) = X(a) for a ∈ A’1 ,
D(aX) = aD(X) for a ∈ A0 .
Moreover, by de¬nition one has
[D(X), D(Y )](a) = D(X)(Y (a)) ’ (’1)X·Y D(Y )(X(a)),
a ∈ A’1 , X, Y ∈ D0 .
2.2. Connections. Similar to Chapter 5, we introduce the notion of a
connection in the graded setting.
Let A and B be two n-graded algebras, A ‚ B. Consider modules the
of derivations D(A, B) and D(B) and a B-linear mapping
: D(A, B) ’ D(B).
The mapping is called a connection for the pair (A, B), or an (A, B)-
connection, if
(X)|A = X.
From the de¬nition it follows that is of degree 0 and that for any
derivations X, Y ∈ D(A, B) the element
(X) —¦ Y ’ (’1)X·Y (Y ) —¦ X
again lies in D(A, B). Thus one can de¬ne the element
( (X) —¦ Y ’ (’1)X·Y
R (X, Y ) = [ (X), (Y )] ’ (Y ) —¦ X)
which is called the curvature of the connection and possesses the following
properties
R (X, Y ) + (’1)X·Y R (Y, X) = 0, X, Y ∈ D(A, B),
a ∈ B,
R (aX, Y ) = aR (X, Y ),
R (X, bY ) = (’1)X·b bR (X, Y ), b ∈ B.
is called ¬‚at, if R (X, Y ) = 0 for all X, Y ∈ D(A, B).
A connection
Evidently, when the grading is trivial, the above introduced notions
coincide with the ones from Chapter 5.
2. GRADED EXTENSIONS 253

2.3. Graded extensions of di¬erential equations. Let now M be
a smooth manifold and π : E ’ M be a smooth locally trivial ¬bre bundle
over M . Let E ‚ J k (π) be a k-th order di¬erential equation represented as
a submanifold in the manifold of k-jets for the bundle π. We assume E to
be formally integrable and consider its in¬nite prolongation E i ‚ J ∞ (π).
Let F(E) be the algebra of smooth functions on E ∞ and CD(E) ‚
D(E) = D(F(E)) be the Lie algebra generated by total derivatives CX,
X ∈ D(M ), C : D(M ) ’ D(E) being the Cartan connection on E ∞ (see
Chapter 2).
Let F be an n-graded commutative algebra such that F0 = F(E). De-
note by CD0 (E) the F-submodule in D(F(E), F) generated by CD(E) and
consider the triple (F(E), F, CD0 (E)) as a starting point for the construc-
tion from Subsection 2.1. Then we shall get a pair (F∞ , CD∞ (E)), where
def
CD∞ (E) = (CD0 (E))∞ . We call the pair (F∞ , CD∞ (E)) a free di¬erential
F-extension of the equation E.
The algebra F∞ is ¬ltered by its graded subalgebras Fi , i = ’1, 0, 1, . . . ,
and we consider its ¬ltered graded CD∞ (E)-stable ideal I. Any vector ¬eld
(derivation) X ∈ CD∞ (E) determines a derivation XI ∈ D(FI ), where
FI = F/I. Let CDI (E) be an FI -submodule generated by such deriva-
tions. Obviously, it is closed with respect to the Lie bracket. We call the
pair (FI , CDI (E)) a graded extension of the equation E, if I © F(E) = 0,
where F(E) is considered as a subalgebra in F∞ .
Let F’∞ = C ∞ (M ). In an appropriate algebraic setting, the Cartan
connection C : D(F’∞ ) ’ D(F(E)) can be uniquely extended up to a con-
nection
CI : D(F’∞ , FI ) ’ CDI (E) ‚ D(FI ).
In what follows we call graded extensions which admit such a connection
C-natural. From the ¬‚atness of the Cartan connection and from the de¬ni-
tion of the algebra CD∞ (E) (see Subsection 2.1) it follows that CI is a ¬‚at
connection as well, i.e.,
RCI (X, Y ) = 0,
where X, Y ∈ D(F’∞ , FI ), for any C-natural graded extension
(FI , CDI (E)).
2.4. The structural element and C-cohomologies. Let us consider
a C-natural graded extension (FI , CDI (E)) and de¬ne a homomorphism UI ∈
homFI (D(FI ), D(FI )) by
UI (X) = X ’ CI (X’∞ ), X ∈ D(FI ), X’∞ = X|F’∞ . (6.14)
The element UI is called the structural element of the graded extension
(FI , CDI (E)).
Due to the assumptions formulated above, UI is an element of the module
D1 (Λ— (FI )), where FI is ¬nitely smooth (see Chapter 4) graded algebra, and
consequently can be treated in the same way as in the nongraded situation.
254 6. SUPER AND GRADED THEORIES

Theorem 6.11. For any C-natural graded extension (FI (E), CDI (E)),
the equation E being formally integrable, its structural element is integrable:
[[UI , UI ]]fn = 0.
Proof. Let X, Y ∈ D(FI ) and consider the bracket [[UI , UI ]]fn as an
element of the module homFI (DI (E) § DI (E), DI (E)). Then applying (6.11)
twice, one can see that

[[UI , UI ]]fn (X, Y ) = µ (’1)U ·Y [UI (X), UI (Y )] ’ (’1)U ·Y UI ([UI (X), Y ])
2
’ UI ([X, UI (Y )]) + UI ([X, Y ]) , (6.15)

where µ = (’1)X·Y (1 + (’1)U ·U ). Expression (6.15) can be called the graded
Nijenhuis torsion (cf. [49]).
From (6.14) if follows that the grading of UI is 0, and thus (6.15) trans-
forms to
[[UI , UI ]]fn (X, Y ) = (’1)X·Y · 2 [UI (X), UI (Y )] ’ UI [UI (X), Y ]
2
’ UI [X, UI (Y )] + UI [X, Y ] . (6.16)
Now, using de¬nition (6.14) of UI , one gets from (6.16):

[[UI , UI ]]fn (X, Y )
= (’1)X·Y · 2 [CI (X’∞ ), CI (Y’∞ )] ’ CI ([CI (X’∞ ), Y ]’∞ )

’ CI ([X, CI (Y’∞ ]’∞ ) + CI ((CI ([X, Y ]’∞ ))’∞ .
But for any vector ¬elds X, Y ∈ D(FI ) one has
(CI (X’∞ ))’∞ = X’∞ .
and
[X, Y ]’∞ = X —¦ Y’∞ ’ (’1)X·Y Y —¦ X∞ .
Hence,

[[UI , UI ]]fn (X, Y ) = (’1)X·Y · 2 [CI (X’∞ ), CI (Y’∞ )]

’ CI (CI (X’∞ ) —¦ Y’∞ ’ (’1)X·Y CI (Y’∞ ) —¦ X’∞ )
= (’1)X·Y 2RCI (X, Y ) = 0.


Hence, with any C-natural graded E-equation, in an appropriate alge-
braic setting, one can associate a complex
0 ’ D(FI ) ’ Λ1 (FI ) — D(FI ) ’ · · ·

· · · ’ Λr (FI ) — D(FI ) ’I Λr+1 (FI ) — D(FI ) ’ · · · ,
’ (6.17)
2. GRADED EXTENSIONS 255


where ‚I („¦) = [[UI , „¦]]fn , „¦ ∈ Λr (FI ) — D(FI ), with corresponding cohomol-
ogy modules.
Like in Chapters 4 and 5, we con¬ne ourselves with a subtheory of this
cohomological theory.
2.5. Vertical subtheory.
Definition 6.2. An element „¦ ∈ Λ— (FI ) — D(FI ) is called vertical, if
L„¦ (•) = 0 for any • ∈ F’∞ ‚ FI = Λ0 (FI ).
Denote by D v (FI ) the set of all vertical vector ¬elds from D(FI ) =
Λ0 (FI ) — D(FI ).
Proposition 6.12. Let (FI , CDI (E)) be a C-natural graded extension of
an equation E. Then
(i) The set of vertical elements in Λr (FI ) — D(FI ) coincides with the
module Λr (FI ) — Dv (FI ).
(ii) The module Λ— (FI ) — Dv (FI ) is closed with respect to the Fr¨licher“
o
Nijenhuis bracket as well as with respect to the contraction operation:
[[Λr (FI ) — Dv (FI ), Λs (FI ) — Dv (FI )]]fn ‚Λr+s (FI ) — Dv (FI ),
Λr (FI ) — Dv (FI ) Λs (FI ) — Dv (FI ) ‚Λr+s’1 (FI ) — Dv (FI ).
(iii) An element „¦ ∈ Λ— (FI ) — D(FI ) lies in Λ— (F) — D v (FI ) if and only
if
i„¦ (UI ) = „¦.
(iv) The structural element is vertical : UI ∈ Λ1 (FI ) — Dv (FI ).
From the last proposition it follows that complex (6.17) can be restricted
up to
0 ’ D v (FI ) ’ Λ1 (FI ) — Dv (FI ) ’ · · ·

· · · ’ Λr (FI ) — Dv (FI ) ’I Λr+1 (FI ) — Dv (FI ) ’ · · ·
’ (6.18)
Cohomologies
ker(‚I : Λr (FI ) — Dv (FI ) ’ Λr+1 (FI ) — Dv (FI ))
r
HI (E) =
im(‚I : Λr’1 (FI ) — Dv (FI ) ’ Λr (FI ) — Dv (FI ))
are called C-cohomologies of a graded extension. The basic properties of the
di¬erential ‚I in (6.18) are corollaries of Propositions 6.9 and 6.12:
Proposition 6.13. Let (FI (E), CDI (E)) be a C-natural graded extension
of the equation E and denote by LI the operator LUI . Then for any „¦, ˜ ∈
Λ— (FI ) — Dv (FI ) and ρ ∈ Λ— (FI ) one has
(i) ‚I (ρ § „¦) = (LI (ρ) ’ dρ) § „¦ + (’1)ρ1 · ρ § ‚I „¦,
(ii) [LI , i„¦ ] = i‚I „¦ + (’1)„¦1 L„¦ ,
(iii) [i„¦ , ‚I ]˜ = (’1)„¦1 (‚I „¦) ˜,
(iv) ‚I [[„¦, ˜]]fn = [[‚I „¦, ˜]]fn + (’1)„¦1 [[„¦, ‚I ˜]]fn .
256 6. SUPER AND GRADED THEORIES

Let dh = d ’ LI : Λ— (FI ) ’ Λ— (FI ). From (6.13) and Proposition 6.6 (ii)
it follows that dh —¦dh = 0. Similar to the nongraded case, we call dh the hori-
zontal di¬erential of the extension (FI , CDI (E)) and denote its cohomologies

by Hh (E; I).
Corollary 6.14. For any C-natural graded extension one has
— —
r
(i) The module HI (E) = r≥0 HI (E) is a graded Hh (E; I)-module.

(ii) HI (E) is a graded Lie algebra with respect to the Fr¨licher“Nijenhuis
o
— (F ) — D v (F ).
bracket inherited from Λ I I
— (E) inherits from Λ— (F ) — D v (F ) the contraction operation
(iii) HI I I
r+s’1
r s
HI (E) ‚ HI
HI (E) (E),

and HI (E), with the shifted grading, is a graded Lie algebra with re-
spect to the inherited Richardson“Nijenhuis bracket.
2.6. Symmetries and deformations. Skipping standard reasoning,
we de¬ne in¬nitesimal symmetries of a graded extension (FI (E), CDI (E)) as
DCI (E) = {X ∈ DI (E) | [X, CDI (E)] ‚ CDI (E)};
DCI (E) forms an n-graded Lie algebra while CDI (E) is its graded ideal con-
sisting of trivial symmetries. Thus, a Lie algebra of nontrivial symmetries
is
symI E = DCI (E)/CDI (E).
If the extension at hand is C-natural, then, due to the connection CI , one
has the direct sum decompositions
D(FI ) = Dv (FI ) • CDI (E), v
DCI (E) = DCI (E) • CDI (E), (6.19)
where
DCI (E) = {X ∈ DI (E) | [X, CDI (E)] = 0} = D v (FI ) © DCI (E),
v v

and symI E is identi¬ed with the ¬rst summand in (6.19).
Let µ ∈ R be a small parameter and UI (µ) ∈ Λ1 (FI ) — Dv (FI ) be a
smooth family such that
(i) UI (0) = UI ,
(ii) [[UI (µ), UI (µ)]]fn = 0 for all µ.
Then UI ( ) is a (vertical) deformation of a graded extension structure,
and if
1
UI (µ) = UI + UI · µ + o(µ),
1
then UI is called (vertical) in¬nitesimal deformation of UI . Again, skipping
motivations and literally repeating corresponding proof from Chapter 5, we
have the following
Theorem 6.15. For any C-natural graded extension (FI , CDI (E)) of the
equation E one has
0
(i) HI (E) = symI (E);
2. GRADED EXTENSIONS 257

1
(ii) The module HI (E) consists of the classes of nontrivial in¬nitesimal
vertical deformations of the graded extension structure UI .
The following result is an immediate consequence of the results of pre-
vious subsection:
Theorem 6.16. Let (FI , CDI (E)) be a graded extension. Then
1
(i) The module HI (E) is an associative algebra with respect to contrac-
tion.
(ii) The mapping
1 0
R : HI (E) ’ EndR (HI (E)),
where
0 1
R„¦ (X) = X X ∈ HI (E), „¦ ∈ HI (E),
„¦,
is a representation of this algebra. And consequently,
(iii)
1
(symI E) HI (E) ‚ symI E.
2.7. Recursion operators. The ¬rst equality in (6.19) gives us the
dual decomposition
Λ1 (FI ) = CΛ1 (FI ) • Λ1 (FI ), (6.20)
h
where
CΛ1 (FI ) = {ω ∈ Λ1 (FI ) | CDI (E) ω = 0},
Λ1 (FI ) = {ω ∈ Λ1 (FI ) | Dv (FI ) ω = 0}.
h

± f± dg± , f± , g± ∈ FI , be a one-form. Then, since by
In fact, let ω =
de¬nition d = dh + LI , one has
ω= f± (dh g± + LI (g± )).
±
D v (F
Let X ∈ I ). Then from Proposition 6.13 (ii) it follows that
LI (g) = ’LI (X g ∈ FI .
X g) + ‚I (X) g + LX (g) = X(g),
Hence,
(d ’ LI )g = X(g) ’ X(g) = 0.
X dh g = X
On the other hand,
LI (g) = UI dg,
and if Y ∈ CDI (E), then
Y LI (g) = Y (UI dg) = (Y UI ) dg
due to Proposition 6.5 (v); but Y UI = 0 for any Y ∈ CDI (E).
Thus, similar to the nongraded case, one has the decomposition
C p Λ(FI ) § Λq (FI ),
Λr (FI ) = (6.21)
h
p+q=r
258 6. SUPER AND GRADED THEORIES

where
C p Λ(FI ) = CΛ1 (FI ) § · · · § CΛ1 (FI ),
p times

and
Λq (FI ) = Λ1 (FI ) § · · · § Λ1 ,
h h
h
q times

and the wedge product § is taken in the graded sense (see Subsection 1.2).
Remark 6.6. The summands in (6.21) can also be described in the fol-
lowing way
C p Λ(FI ) § Λq (FI ) = {ω ∈ Λp+q (FI ) | X1 ... Xp+1 ω = 0,
h
ω = 0 for all X± ∈ Dv (FI ), Yβ ∈ CDI (E)}.
Y1 ... Yq+1
Proposition 6.17. Let (FI , CDI (E)) be a C-natural extension. Then
one has
‚I (C p Λ(FI ) § Λq (FI ) — Dv (FI )) ‚ C p Λ(FI ) § Λq+1 (FI ) — Dv (FI )
h h
for all p, q ≥ 0.
The proof is based on two lemmas.
Lemma 6.18. dh C 1 Λ(FI ) ‚ C 1 Λ(FI ) § Λ1 (FI ).
h

Proof of Lemma 6.18. Due to Remark 6.6, it is su¬cient to show that
Xv Yv X v , Y v ∈ Dv (FI ),
dh ω = 0, (6.22)
and
Xh Yh X h , Y h ∈ CDI (E),
dh ω = 0, (6.23)
where ω ∈ C 1 Λ(FI ). Obviously, we can restrict ourselves to the case ω =
LI (g), g ∈ FI :
Yv dh ω = Y v dh LI (g) = ’Y v LI d h g
= LI (Y v dh g) + LY v (dh g) = dh Y v (g).
Hence,
Xv Yv dh ω = X v dh Y v (g) = 0,
which proves (6.22). Now,
Yh dh ω = ’Y h LI d h g = Y h dh g) ’ UI
(d(UI d(dh g)).
But UI is a vertical element, i.e., UI ∈ Λ1 (FI ) — Dv (FI ). Therefore,
UI dh g = 0
and
Yh dh ω = ’Y h UI d(dh g)
2. GRADED EXTENSIONS 259

= ’Y h d(dh g) ’ (Y h § UI )
UI ) d(dh g).
The ¬rst summand in the right-hand side of the last equality vanishes, since,
by de¬nition, Y h UI = 0 for any Y h ∈ CDI (E). Hence,
Xh Yh dh ω = ’X h (Y h § UI ) d(dh g)
= ’(X h (Y h § UI )) d(dh g) ’ (X h § Y h § UI ) d(dh g)
= ’(X h § Y h § UI ) d(dh g).
But X h § Y h § UI is a (form valued) 3-vector while d(dh g) is a 2-form;
hence
Xh Yh dh ω = 0,
which ¬nishes the proof of Lemma 6.18.
Lemma 6.19. ‚I Dv (FI ) ‚ Λ1 — Dv (FI ).
h
Proof of Lemma 6.19. One can easily see that it immediately follows
from Proposition 6.13 (iii).
Proof of Proposition 6.17. The result follows from previous lem-
mas and Proposition 6.13 (i) which can be rewritten as
‚I (ρ § „¦) = ’dh (ρ) § „¦ + (’1)ρ1 ρ § ‚I („¦).


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
[[U• , U• ]]fn = 0
is ful¬lled due to (6.39).
If now
∞ m
‚ ‚
j
θi — ρs —
˜= +
‚uj ‚ws
s
i
i=0 j=1
266 6. SUPER AND GRADED THEORIES

is an element of the module Λp (E• ) — Dv (E• ), then one can easily see that
∞ m
j ˜j
fn
dx § θi+1 ’ Dx (θi )
‚• (˜) = [[U• , ˜]] =
i=0 j=1
∞ m ij
± ‚Dx f ‚
˜j
+ dt § ’ Dt (θi ) —
θβ
‚u± ‚uj
β i
β=0 ±=1
∞ m
‚Xs ‚Xs ˜
±
dx § ’ Dx (ρs )
+ θβ + ργ
‚u± ‚wγ
β
s γ
β=0 ±=1
∞m
± ‚Xs ‚Ts ‚
˜
+ dt § ’ Dt (ρs ) —
θβ ± + ργ . (6.40)
‚uβ ‚wγ ‚ws
γ
β=0 ±=1

Again, con¬ning oneself to the case ˜ ∈ C p Λ(E• ) — Dv (E• ), one gets the
following
Theorem 6.27. Let E be an equation of the form (6.31) and • be its
covering with nonlocal variables w1 , w2 , . . . and extended total derivatives
p,0
given by (6.38). Then the module HC (E• ) consists of the elements
∞ m
‚ ‚
˜i
Dx (θj ) — j + ρs —
= , (6.41)
θ,ρ
‚ws
‚ui s
i=0 j=1

where θ = (θ 1 , . . . , θm ) and ρ = (ρ1 , . . . , ρs , . . . ), θj , ρs ∈ C p Λ(E• ), are
vector-valued forms satisfying the equations
˜(p) (θ) = 0, (6.42)
E
and
∞ m
‚Xs ‚Xs
˜β ˜
Dx (θ± ) ± + ρj = Dx (ρs ),
‚uβ ‚wj
β=0 ±=1 j
∞m
‚Ts ‚Ts
˜β ˜
Dx (θ± ) ± + ρj = Dt (ρs ), (6.43)
‚uβ ‚wj
β=0 ±=1 j
(p) (p)
s = 1, 2, . . . , where ˜E is the natural extension of with Dx and Dt
E
˜ ˜
replaced by Dx and Dt in (6.37).
Similar to Chapter 5, we call (6.42) shadow equations and (6.42) relation
equations for the element (θ, ρ); solutions of (6.42) are called shadow solu-
tions, or simply shadows. Our main concern lies in reconstruction elements
p,0
of the module HC (E• ) from their shadows. Denote the set of such shadows
p,0
by SHC (E• ).
Remark 6.9. Let • be a covering. Consider horizontal one-forms
s
ω• = dh ws = Xs dx + Ts dt, s = 1, 2, . . . ,
3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 267

where dh is the horizontal de Rham di¬erential associated to •. Then (6.42)
can be rewritten as
s
θ,ρ (ω• ) = d h ρs , s = 1, 2, . . . (6.44)
Remark 6.10. When Xs and Ts do not depend on nonlocal variables,
the conditions of • being a covering is equivalent to
s
dh ω• = 0, s = 1, 2, . . . ,
dh being the horizontal di¬erential on E. In particular, one-dimensional
coverings are identi¬ed with elements of ker(dh ). We say a one-dimensional
covering • to be trivial if corresponding form ω• is exact (for motivations see
Chapter 3). Thus, the set of classes of nontrivial one-dimensional coverings •
with ω• independent of nonlocal variables is identi¬ed with the cohomology
1
group Hh (E), or with the group of nontrivial conservation laws for E.
3.5. The functors K and T . Keeping in mind the problem of recon-
structing recursion operators from their shadows, we introduce two functors
in the category GDE(M ). One of them is known from the classical (non-
graded) theory (cf Chapter 3), the other is speci¬c to graded equations and
is a super counterpart of the Cartan even covering constructed in Chapter
3 (see also [97]).
1
Let (F, F ) be an object of the category GDE(M ) and Hh (F) be the
R-module of its ¬rst horizontal cohomology. Let {w± } be a set of generators
for Hh (F), each w± being the cohomology class of a form ω± ∈ Λ1 (F),
1
h
m i dx . We de¬ne the functor K : GDE(M ) ’ GDE(M ) of
ω± = i=1 X± i
1 (F) as follows.
killing Hh
The algebra KF is a graded commutative algebra freely generated by
i
{w± } over F with gr(w± ) = gr(X± ). The connection KF looks as
‚ ‚ ‚
i
= + X± .
F
KF
‚xi ‚xi ‚w±
±
1
From the fact that Hh is a covariant functor from GDE(M ) into the category
of R-modules it easily follows that K is a functor as well.
To de¬ne the functor T : GDE(M ) ’ GDE(M ), let us set T F = C — Λ(F),
where C — Λ— (F) = p≥0 C p Λ(F) is the module of all Cartan forms on F (see
Subsection 2.7). If F is n-graded, then T F carries an obvious structure of
(n + 1)-graded algebra. The action of vector ¬elds F (X), X ∈ D(M ), on
Λ— (F) by Lie derivatives preserves the submodule C — Λ(F). Since C — Λ(F),
as a graded algebra, is generated by the elements χ and dC ψ, χ, ψ ∈ F, this
action can be written down as
L χ= F (X)χ,
F (X)

L dC ψ = d C F (X)(ψ),
F (X)

(χ) · dC ψ + χL
L (χdC ψ) = L dC ψ.
F (X) F (X)
F
268 6. SUPER AND GRADED THEORIES

Moreover, for any X ∈ D(M ) and ω ∈ C — Λ(F) one has
F (X) ω = 0;
hence, for any θ ∈ C — Λ(F)
(θ § L )(ω)
F (X)

(ω) + (’1)θ1 dθ § (
=θ§L ω) = θ § L
F (X) , (ω),
F (X) F (X)

which means that we have a natural extension of the connection F in F
up to a connection T F in T F. It is easy to see that the correspondence
T : (F, F ) ’ (T F, T F ) is functorial. We call (T F, ∆T F ) the (odd ) Car-
tan covering of (F, F ).
In the case when (F, F ) is an evolution equation E of the form (6.31),
T (F, F ) is again an evolution equation T E with additional dependent vari-
ables v 1 , . . . , v m and additional relations
±
‚f 1 j
1
v = vi ,
t

‚uj

 i,j i
(6.45)
...............

 m
‚f j
m
v =
t vi .

‚uj
i,j i
Note that if a variable uj is of grading (i1 , . . . , in ), then the grading of v j is
(i1 , . . . , in , 1).
3.6. Reconstructing shadows. Computerized computations on non-
local objects, such as symmetries and recursion operators, can be e¬ectively
realized for shadows of these objects (see examples below). Here we describe
a setting which guarantees the existence of symmetries and, in general, el-
p,0
ements of HC (E) corresponding to the shadows computed. Below we still
consider evolution equations only.
Proposition 6.28. Let E be an evolution equation and • be its covering.
p,0
Let θ ∈ SHC (E• ). Then, if the coe¬cients Xs and Ts for the extensions of
total derivatives do not depend on nonlocal variables for all s, then
(i) for any extension θ,ρ of θ up to a vector ¬eld on E• the forms
s def
„¦s
θ,ρ (ω• ) =

(see Remark 6.9 in Subsection 3.4) are dh -closed on E• ;
p,0
(ii) the element θ is extendable up to an element of HC (E• ) if and only
if all „¦s are dh -exact forms.
Proof. To prove the ¬rst statement, note that using Proposition
6.13 (i) one has


2 s
+ d h ρs ) —
0 = ‚• ( θ,ρ ) = ‚• ( θ,ρ (ω• )
‚ωs
s
3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 269


dh „¦ s —
=’ . (6.46)
‚ωs
s

The second statement immediately follows from (6.43).
Remark 6.11. If Xs , Ts depend on w1 , w2 , . . . , then (6.46) transforms
into
‚Xs ‚Ts ‚
dh „¦s ’ („¦s + dh ρs ) § —
dx + dt = 0. (6.47)
‚ws ‚ws ‚ws
s
p,0 q,0
Let now θ ∈ SHC (E• ) and ¦ ∈ HC (E• ). Then from Proposition
6.13 (iii) it follows that
[i¦ , ‚• ]θ = (’1)q (‚• ¦) θ = 0.
Hence, since by the de¬nition of shadows ‚• θ is a •-vertical element,
i¦ ‚• θ is vertical too. It means that ‚• i¦ θ is a •-vertical element, i.e.,
p+q’1
i• θ ∈ SHC (E• ). It proves the following result (cf. similar results of
Chapter 5):
p,0 q,0
Proposition 6.29. For any θ ∈ SHC (E• ) and ¦ ∈ HC (E• ) the ele-
p+q’1
ment ¦ θ lies in SHC (E• ). In particular, when applying a shadow of
a recursion operator to a symmetry, one gets a shadow of a symmetry.
The next result follows directly from the previous ones.
Theorem 6.30. Let E be an evolution equation of the form (6.31) and
E• be its covering constructed by in¬nite application of the functor K : E • =
K (∞) E, where
K (∞) E = inj lim(K n E), K n E = (K —¦ · · · —¦ K) E.
n’∞
n times
Then for any shadow R of a recursion operator in E• and a symmetry ¦ ∈
sym E• the shadow R(¦) can be extended up to a symmetry of E• . Thus, an
1,0
action of SHC (E• ) on sym(E• ) is de¬ned modulo “shadowless” symmetries.
1,0
To be sure that elements of SHC (E• ) can be extended up to recursion
operators in an appropriate setting, we prove the following two results.
Proposition 6.31. Let E be an equation and E• be its covering by means
of T E. Then there exists a natural embedding
—,0
Tsym : HC (E) ’ sym(T E)
of graded Lie algebras.
—,0
Proof. Let ¦ ∈ HC (E). Then L¦ acts on Λ— (E) and this action pre-
serves the submodule C — Λ(E) ‚ Λ— (E), since
[L¦ , dC ] = L[[¦,UE ]]fn = 0.
270 6. SUPER AND GRADED THEORIES


Let X ∈ CD(E). Then, due to (6.11), [[X, ¦]]fn UE = 0. But, using
(6.11) again, one can see that [[X, ¦]]fn is a vertical element. Hence,
[[X, ¦]]fn UE = [[X, ¦]]fn = 0.


—,0
Proposition 6.31 allows one to compute elements of HC (E) as nonlocal
symmetries in E• = T E. This is the base of computational technology used
in applications below.
The last result of this subsection follows from the previous ones.
Theorem 6.32. Let E be an evolution equation and E• be its covering
constructed by in¬nite application of the functor K —¦ T . Then any shadow
—,0 —,0
¦ ∈ SHC (E• ) can be extended up to an element of HC (E• ). In particular,
1,0
to any shadow SHC (E• ) a recursion operator corresponds in E• .
Remark 6.12. For “¬ne obstructions” to shadows reconstruction one
should use corresponding term of A.M. Vinogradov™s C-spectral sequence
([102], cf. [58]).

4. The Kupershmidt super KdV equation
As a ¬rst application of the graded calculus for symmetries of graded par-
tial di¬erential equations we discuss the symmetry structure of the so-called
Kupershmidt super KdV equation, which is an extension of the classical
KdV equation to the graded setting [24].
At this point we have already to make a remark. The equation under
consideration will be a super equation but not a supersymmetric equation
in the sense of Mathieu, Manin“Radul, where a supersymmetric equation
is an equation admitting and odd, or supersymmetry [74], [72]. The super
KdV equation is given as the following system of graded partial di¬erential
equations E for an even function u and an odd function • in J 3 (π; •), where
J 3 (π; •) is the space J 3 (π) for the bundle π : R — R2 ’ R2 , (u, x, t) ’ (x, t),
extended by the odd variable •:
ut = 6uux ’ uxxx + 3••xx ,
•t = 3ux • + 6u•x ’ 4•xxx , (6.48)
where subscripts denote partial derivatives with respect to x and t. As
usual, t is the time variable and x is the space variable. Here u, x, t, u, ux ,
ut , uxx , uxxx are even (commuting) variables, while •, •x , •xx , •xxx are
odd (anticommuting) variables. In the sequel we shall often use the term
“graded” instead of “super”.
We introduce the total derivative operators Dx and Dt on the space
∞ (π; •), by
J
‚ ‚ ‚ ‚ ‚
+ ··· ,
Dx = + ux + •x + uxx + •xx
‚x ‚u ‚• ‚ux ‚•x
4. THE KUPERSHMIDT SUPER KDV EQUATION 271

‚ ‚ ‚ ‚ ‚
+ ···
Dt = + ut + •t + utx + •tx (6.49)
‚t ‚u ‚• ‚ux ‚•x
The in¬nite prolongation E ∞ is the submanifold of J ∞ (π; •) de¬ned by
the graded system of partial di¬erential equations
nm
Dx Dt (ut ’ 6uux + uxxx ’ 3••xx ) = 0,
nm
Dx Dt (•t ’ 3ux • ’ 6u•x + 4•xxx ) = 0, (6.50)
where n, m ∈ N.
We choose internal coordinates on E ∞ as x, t, u, •, u1 , •1 , . . . , where we
introduced a further notation
ux = u 1 , •x = • 1 , uxx = u2 , •xx = •2 , . . . (6.51)
The restriction of the total derivative operators Dx and Dt to E ∞ , again
denoted by the same symbols, are then given by
‚ ‚ ‚
Dx = + (un+1 + •n+1 ),
‚x ‚un ‚•n
n≥0
‚ ‚ ‚
Dt = + ((un )t + (•n )t ). (6.52)
‚t ‚un ‚•n
n≥0

We note that (6.48) admits a scaling symmetry, which leads to the in-
troduction of a degree to each variable,
deg(x) = ’1, deg(t) = ’3,
deg(u) = 2, deg(u1 ) = 3, . . . ,
3 5
deg(•) = , deg(•1 ) = , . . . (6.53)
2 2
From this we see that each term in (6.48) is of degree 5 and 4 1 respectively.
2

4.1. Higher symmetries. We start the discussion of searching for
(higher) symmetries at the representation of vertical vector ¬elds,
‚ ‚ ‚ ‚
= ¦u + ¦• Dx (¦u )
n
+ Dx (¦• )
n
+ , (6.54)
¦
‚u ‚• ‚un ‚•n
n>0
where ¦ = (¦u , ¦• ) is the generating function of the vertical vector ¬eld ¦ .
We restrict our search for higher symmetries to even vector ¬elds, meaning
that ¦u is even, while ¦• is odd.
Moreover we restrict our search for higher symmetries to vector ¬elds
u •
¦ whose generating function ¦ = (¦ , ¦ ) depends on the variables x, t,
u, •, . . . , u5 , •5 . These requirements lead to a representation of the func-
tion ¦ = (¦u , ¦• ), ¦u , ¦• ∈ C ∞ (x, t, u, u1 , . . . , u5 ) — Λ(•, . . . , •5 ) in the
following form
¦u = f0 + f1 ••1 + f2 ••2 + f3 ••3 + f4 ••4 + f5 ••5 + f6 •1 •2
+ f7 •1 •3 + f8 •1 •4 + f9 •1 •5 + f10 •2 •3 + f11 •2 •4 + f12 •2 •5
+ f13 •3 •4 + f14 •3 •5 + f15 •4 •5 + f16 ••1 •2 •3 + f17 ••1 •2 •4
272 6. SUPER AND GRADED THEORIES

+ f18 ••1 •2 •5 + f19 ••1 •3 •4 + f20 ••1 •3 •5 + f21 ••1 •4 •5
+ f22 ••2 •3 •4 + f23 ••2 •3 •5 + f24 ••2 •4 •5 + f25 ••3 •4 •5
+ f26 •1 •2 •3 •4 + f27 •1 •2 •3 •5 + f28 •2 •3 •4 •5
+ f29 ••1 •2 •3 •4 •5 ,

¦ • = g 1 • + g 2 •1 + g 3 •2 + g 4 •3 + g 5 •4 + g 6 •5
+ g7 ••1 •2 + g8 ••1 •3 + g9 ••1 •4 + g10 ••1 •5 + g11 ••2 •3
+ g12 ••2 •4 + g13 ••2 •5 + g14 ••3 •4 + g15 ••3 •5 + g16 ••4 •5
+ g17 •1 •2 •3 + g18 •1 •2 •4 + g19 •1 •2 •5 + g20 •1 •3 •4 + g21 •1 •3 •5
+ g22 •1 •4 •5 + g23 •2 •3 •4 + g24 •2 •3 •5 + g25 •2 •4 •5 + g26 •3 •4 •5
+ g27 ••1 •2 •3 •4 + g28 ••1 •2 •3 •5 + g29 ••1 •2 •4 •5 + g30 ••1 •3 •4 •5
+ g31 ••2 •3 •4 •5 + g32 •1 •2 •3 •4 •5 , (6.55)
where f0 , . . . , f29 , g1 , . . . , g32 are functions depending on the even variables
x, t, u, u1 , . . . , u5 . We have to mention here that we are constructing
generic elements, even and odd explicitly, of the following exterior alge-
bra C ∞ (x, t, u, . . . , u5 ) — Λ(•, . . . , •5 ), where Λ(•, . . . , •5 ) is the (exterior)
algebra generated by •, . . . , •5 The symmetry condition (6.37) for p = 0
reads in this case to the system
Dt (¦u ) = ’ u3 + 3••2 ),
¦ (6uu1
Dt (¦• ) = + 6u•1 ’ 4•3 ),
¦ (3u1 • (6.56)
which results in equations
Dt (¦u ) ’ 6¦u u1 ’ 6uDx (¦u ) + Dx (¦u ) ’ 3¦• •2 ’ 3•Dx (¦• ) = 0,
3 2

Dt (¦• ) ’ 3Dx (¦u )• ’ 3u1 ¦• ’ 6¦u •1 ’ 6uDx (¦• ) + 4Dx (¦• ) = 0.
3

(6.57)
Substitution of the representation(6.55) of ¦ = (¦u , ¦• )), leads to an
overdetermined system of classical partial di¬erential equations for the co-
e¬cients f0 , . . . , f26 , g1 , . . . , g32 , which are, as mentioned above, functions
depending on the variables x, t, u, u1 , . . . , u5 .
The general solution of equations (6.57) and (6.55) is generated by the
functions
¦1 = (u1 , •1 );
¦2 = (6uu1 ’ u3 + 3••2 , 3u1 • + 6u•1 ’ 4•3 );
¦3 = (6tu1 + 1, 6t•1 );
¦4 = (3t(6uu1 ’ u3 + 3••2 ) + x(u1 ) + 2u,
3
3t(3u1 • + 6u•1 ’ 4•3 ) + x•1 + •);
2
¦5 = (u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 ’ 15••4 ’ 10•1 •3
+ 30u1 ••1 + 30u••2 ,
4. THE KUPERSHMIDT SUPER KDV EQUATION 273

16•5 ’ 40u•3 ’ 60u1 •2 ’ 50u2 •1 + 30u2 •1 + 30u1 u• ’ 15u3 •).
(6.58)
We note that the vector ¬elds ¦1 , ¦2 , ¦3 , ¦4 are equivalent to the
classical symmetries

S1 = ,
‚x

S2 = ,
‚t
‚ 1‚

S3 = t ,
‚x 6 ‚u
‚ ‚ ‚ 3‚
S4 = ’x ’ 3t + 2u +• . (6.59)
‚x ‚t ‚u 2 ‚•
In (6.59) S1 , S2 re¬‚ect space and time translation, S3 re¬‚ects Galilean
invariance, while S4 re¬‚ects the scaling as mentioned already. In (6.50), the
evolutionary vector ¬eld ¦5 is the ¬rst higher symmetry of the super KdV
equation and reduces to

(u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 ) + ..., (6.60)
‚u
in the absence of odd variables •, •1 , . . . , being then just the classical ¬rst
higher symmetry of the KdV equation
ut = 6uu1 ’ u3 . (6.61)
4.2. A nonlocal symmetry. In this subsection we demonstrate the
existence and construction of nonlocal higher symmetries for the super KdV
equation (6.48). The construction runs exactly along the same lines as it is
for the classical equations.
So we start at the construction of conservation laws, conserved densities
and conserved quantities as discussed in Section 2. According to this con-
struction we arrive, amongst others, at the following two conservation laws,
i.e.,
Dt (u) = Dx (3u2 ’ u2 + 3••1 ),
Dt (u2 + 3••1 ) = Dx (4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 ), (6.62)
1
from which we obtain the nonlocal variables
x
p1 = u dx,
’∞
x
(u2 + 3••1 ) dx.
p3 = (6.63)
’∞
Now using these new nonlocal variables p1 , p3 , we de¬ne the augmented
system E of partial di¬erential equations for the variables u, p1 , p3 , •,
where u, p1 , p3 are even and • is odd,
ut = 6uux ’ uxxx + 3••xx ,
274 6. SUPER AND GRADED THEORIES

•t = 3ux • + 6u•x ’ 4•xxx ,
(p1 )x = u,
(p1 )t = 3u2 ’ u2 + 3••1 ,
(p3 )x = u2 + 3••1 ,
(p3 )t = 4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 . (6.64)
1

Internal coordinates for the in¬nite prolongation E ∞ of this augmented
system (6.64) are given as x, t, u, p1 , p3 , •, u1 , •1 , . . . . The total derivative
operators Dx and Dt on E ∞ are given by
‚ ‚
+ (u2 + 3••1 )
Dx = D x + u ,
‚p1 ‚p3

Dt = Dt + (3u2 ’ u2 + 3••1 )
‚p1

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