play an essential role. According to this, the associated nonlocal components

of the symmetries play a signi¬cant role too. These components have to be

constructed from the invariance of the associated di¬erential equations for

p and q. Since the system at hand is not of evolutionary type, we have a

choice to compute these components from the invariance of either px or py

and similar for the qx and qy .

Due to the discrete symmetry (5.166), we choose the invariance of the

following equations

py =wx wy ,

q x = ’ w x wy .

242 5. DEFORMATIONS AND RECURSION OPERATORS

From these invariances, we obtain for the generating function of a symmetry

¦ = (¦u , ¦v , ¦w ), terms like

’1

¦p = Dy (wx Dy (¦w ) + wy Dx (¦w )),

’1

¦q = Dx (wx Dy (¦w ) + wy Dx (¦w )). (5.178)

From the above considerations we expect the recursion operator to contain

’1 ’1

terms like Dy (wx Dy (·) + wy Dx (·)), Dx (wx Dy (·) + wy Dx (·)).

Moreover from the expected degree of the operator, which probably will

be equal to 2, due to the degrees of the symmetries of the previous sub-

section, we arrive at the ansatz for the recursion operator for symmetries.

From this ansatz we arrive at the following expression for R:

«2

Dy + u2 + wx ’ wy ’wx Dx + uv + wxx uwx Dx ’ 2uwy Dy

2 2

R= ’Dx ’ v 2 + wx ’ wy 2vwx Dx ’ vwy Dy

2 2 2

wy Dy + wxx

2

0 u Dy

«

’1 ’1

0 0 vwx Dy (wy Dx + wx Dy ) ’ uy Dx (wy Dx + wx Dy )

+ 0 0 vx Dy (wy Dx + wx Dy ) ’ uwy Dx (wy Dx + wx Dy ) (5.179)

’1 ’1

’1 ’1

0 0 wx Dy (wy Dx + wx Dy ) ’ wy Dx (wy Dx + wx Dy )

It is a straightforward check that the operator R is a recursion operator for

higher symmetries since

—¦R=S —¦ F, (5.180)

F

where the matrix operator S is given by

«2

Dy + u2 + wx ’ wy ’Dx ’ v 2 + wx ’ wy ’uDy ’ vwy

2 2 2 2 2

S = ’wy Dx ’ 2wxy (5.181)

wx Dy + 2wxy uwx

S31 S32 S33

where S31 , S32 , S33 are given by

’1 ’1

S31 = 2(uv + wxx )Dx u ’ wy Dx Dy u,

’1 ’1

S32 = 2wxx Dy v + wx Dy Dx v,

’1 ’1 ’1

S33 = 2wxx Dy wy + wx Dy Dx wy + 2(uv + wxx )Dx wx

’1 2 2 2

’ w y D x D y wx + D y + w x ’ w y . (5.182)

It would have been possible not to start from the invariance of py , qx , but

from the invariance of for instance px , qx , but in that case we had to in-

’1 ’1 ’1

corporate terms like Dx v, Dx wx Dx , Dx wy Dy into the matrix recursion

operator R.

CHAPTER 6

Super and graded theories

We shall now generalize the material of the previous chapters to the case

of super (or graded ) partial di¬erential equations. We con¬ne ourselves to

the case when only dependent variables admit odd gradings and develop a

theory closely parallel to that exposed in Chapters 1“5.

We also show here that the cohomological theory of recursion operators

may be considered as a particular case of the symmetry theory for graded

equations, which, in a sense, explains the main result of Chapter 5, i.e.,

(p)

p,0

HC (E) = ker E . It is interesting to note that this reduction is accom-

plished using an odd analog of the Cartan covering introduced in Example

3.3 of Chapter 3.

Our main computational object is a graded extension of a classical partial

di¬erential equation. We discuss the principles of constructing nontrivial

extensions of such a kind and illustrate them in a series of examples. Other

applications are considered in Chapter 7.

1. Graded calculus

Here we rede¬ne the Fr¨licher“Nijenhuis bracket for the case of n-graded

o

commutative algebras. All de¬nitions below are obvious generalizations of

those from 4. Proofs also follow the same lines and are usually omitted.

1.1. Graded polyderivations and forms. Let R be a commutative

ring with a unit 1 ∈ R and A be a commutative n-graded unitary algebra

over R, i.e.,

A= Ai , Ai Aj ‚ Ai+j

i∈Zn

and

ab = (’1)a·b ba

for any homogeneous elements a, b ∈ A. Here and below the notation (’1) a·b

means (’1)i1 j1 +...in jn , where i = (i1 , . . . , in ), j = (j1 , . . . , jn ) ∈ Zn are the

gradings of the elements a and b respectively. We also use the notation a · b

for the scalar product of the gradings of elements a and b. In what follows,

one can consider Zn -graded objects as well. We consider the category of n-

2

graded (left) A-modules Mod = Mod(A) and introduce the functors

Di : Mod(A) ’ Mod(A)

243

244 6. SUPER AND GRADED THEORIES

as follows (cf. [54, 58]):

D0 (P ) = P

for any P ∈ Ob(Mod), P = Pi , and

i∈Zn

D1,j (P ) = {∆ ∈ homR (A, P ) | ∆(Ai ) ‚ Pi+j , ∆(ab)

= ∆(a)b + (’1)∆·a a∆(b)},

where j = (j1 , . . . , jn ) = gr(∆) ∈ Zn is the grading of ∆; we set

D1 (P ) = D1,j (P ).

i∈Zn

Remark 6.1. We can also consider objects of Mod(A) as right A-mod-

ules by setting pa = (’1)a·p ap for any homogeneous a ∈ A, p ∈ P . In

a similar way, for any graded homomorphism • ∈ homR (P, Q), the right

action of • can be introduced by (p)• = (’1)p• •(p).

Further, if D0 , . . . , Ds are de¬ned, we set

Ds+1,j (P ) = {∆ ∈ homR (A, Ds (P )) | ∆(Ai ) ‚ Ds,i+j (P ),

∆(ab) = ∆(a)b + (’1)∆·a a∆(b), ∆(a, b) + (’1)a·b ∆(b, a) = 0}

and

Ds+1 (P ) = Ds+1,j (P ).

j∈Zn

Elements of Ds (P ) a called graded P -valued s-derivations of A and elements

of D— (P ) = s≥0 Ds (P ) are called graded P -valued polyderivations of A.

Proposition 6.1. The functors Ds , s = 0, 1, 2, . . . , are representable in

the category Mod(A), i.e., there exist n-graded modules Λ0 , Λ1 , . . . , Λs , . . . ,

such that

Ds (P ) = homA (Λs , P )

for all P ∈ Ob(Mod).

Elements of the module Λs = Λs (A) are called graded di¬erential forms

of degree s.

Our local target is the construction of graded calculus in the limits needed

for what follows. By calculus we mean the set of basic operations related to

the functors Ds and to modules Λs as well as most important identities con-

necting these operations. In further applications, we shall need the following

particular case:

(i) A0 = C ∞ (M ) for some smooth manifold M , where 0 = (0, . . . , 0);

(ii) All homogeneous components Pi of the modules under consideration

are projective A0 -modules of ¬nite type.

1. GRADED CALCULUS 245

Remark 6.2. In fact, the entire scheme of calculus over commutative

algebras is carried over to the graded case. For example, to de¬ne graded

linear di¬erential operators, we introduce the action δa : homR (P, Q) ’

homR (P, Q), a ∈ A, by setting δa • = a• ’ (’1)a• • · a, • ∈ homR (P, Q),

and say that • is an operator of order ¤ k, if

(δa0 —¦ · · · —¦ δak )• = 0

for all a0 , . . . , ak ∈ A, etc. A detailed exposition of graded calculus can be

found in [106, 52].

1.2. Wedge products. Let us now consider some essential algebraic

structures in the above introduced objects.

Proposition 6.2. Let A be an n-graded commutative algebra. Then:

(i) There exists a derivation d : A ’ Λ1 of grading 0 such that for any

A-module P and any graded derivation ∆ : A ’ P there exists a

uniquely de¬ned morphism f∆ : Λ1 ’ P such that f∆ —¦ d = ∆.

(ii) The module Λ1 is generated over A by the elements da = d(a), a ∈ A,

with the relations

a, b ∈ A.

d(±a + βb) = ±da + βdb, d(ab) = (da)b + adb,

The j-th homogeneous component of Λ1 is of the form

Λ1 = { adb | a, b ∈ A, gr(a) + gr(b) = j},

j

(iii) The modules Λs are generated over A by the elements of the form

ω1 , . . . , ω s ∈ Λ 1 ,

ω1 § · · · § ω s ,

with the relations

ω § θ + (’1)ωθ θ § ω = 0, ω, θ ∈ Λ1 , a ∈ A.

ω § aθ = ωa § θ,

The j-th homogeneous component of Λs is of the form

Λs = { ω1 § · · · § ωs | ωi ∈ Λ1 , gr(ω1 ) + · · · + gr(ωs ) = j}.

j

(iv) Let ω ∈ Λs , j = (j1 , . . . , jn ). Set gr1 (ω) = (j1 , . . . , jn , s). Then

j

Λ— = Λs = Λs

j

s≥0 j∈Zn

s≥0

is an (n + 1)-graded commutative algebra with respect to the wedge

product

ω ∈ Λs , θ ∈ Λr , ω± , θ β ∈ Λ 1 ,

ω § θ = ω § · · · § ω s § θ1 § · · · § θ r ,

i.e.,

ω § θ = (’1)ω·θ+sr θ § ω,

where ω · θ in the power of (’1) denotes scalar product of gradings

inherited by ω and θ from A.

246 6. SUPER AND GRADED THEORIES

Remark 6.3. When working with the algebraic de¬nition of di¬erential

forms in the graded situation, one encounters the same problems as in a pure

commutative setting, i.e., the problem of ghost elements. To kill ghosts, the

same procedures as in Chapter 4 (see Remark 4.4) are to be used.

A similar wedge product can be de¬ned in D— (A). Namely for a, b ∈

D0 (A) = A we set

a § b = ab

and then by induction de¬ne

def ·a+r

(∆ § )(a) = ∆ § ∆(a) §

(a) + (’1) , (6.1)

where a ∈ A, ∆ ∈ Ds (A), ∈ Dr (A) and in the power of (’1) denotes

the grading of in the sense of the previous subsection.

Proposition 6.3. For any n-graded commutative algebra A the follow-

ing statements are valid :

(i) De¬nition (6.1) determines a mapping

§ : Ds (A) —A Dr (A) ’ Ds+r (A),

which is in agreement with the graded structure of polyderivations:

Ds,i (A) § Dr,j (A) ‚ Ds+r,i+j (A).

(ii) The module D— (A) = s≥0 j∈Zn Ds,j is an (n + 1)-graded commu-

tative algebra with respect to the wedge product:

= (’1)∆· +rs

∆§ §∆

for any ∆ ∈ Ds (A), ∈ Dr (A).1

(iii) If A satis¬es conditions (i), (ii) on page 244, then the module D— (A)

is generated by D0 (A) = A and D1 (A), i.e., any ∆ ∈ Ds (A) is a sum

of the elements of the form

a∆1 § · · · § ∆s , ∆i ∈ D1 (A), a ∈ A.

Remark 6.4. One can de¬ne a wedge product § : Di (A) —A Dj (P ) ’

Di+j (P ) with respect to which D— (P ) acquires the structure of an (n + 1)-

graded D— (A)-module (see [54]), but it will not be needed below.

1.3. Contractions and graded Richardson“Nijenhuis bracket.

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