+

[Y ’ (2, 0), Y0’ ] = 0,

[Y + (2, 0), Y0+ ] = 0,

’ ’

[Y ’ (2, 0), Y’1 ] = 2Z’1 ,

+ +

[Y + (2, 0), Y’1 ] = 2Z’1 ,

[Y + (2, 0), Yi’ ] = 0, [Y ’ (2, 0), Yi+ ] = 0, (3.125)

where i = ’1, 0, 1. These results suggest to set

Y ± (1, i) = Zi± , Y ± (0, i) = Yi± , i ∈ Z. (3.126)

The complete Lie algebra structure is obtained in Subection 7.5.3.

7.5.2. Hamiltonian structures. We shall now discuss Hamiltonians (or

conserved functionals) for the Federbush model described by (3.84),

u1,t + u1,x ’ m1 v2 = »(u2 + v4 )v1 ,

2

4

’v1,t ’ v1,x ’ m1 u2 = »(u2 + v4 )u1 ,

2

4

u2,t ’ u2,x ’ m1 v1 = ’»(u2 + v3 )v2 ,

2

3

’v2,t + v2,x ’ m1 u1 = ’»(u2 + v3 )u2 ,

2

3

u3,t + u3,x ’ m2 v4 = ’»(u2 + v2 )v3 ,

2

2

’v3,t ’ v3,x ’ m2 u4 = ’»(u2 + v2 )u3 ,

2

2

u4,t ’ u4,x ’ m2 v3 = »(u2 + v2 )v4 ,

2

2

’v4,t + v4,x ’ m2 u3 = »(u2 + v2 )u4 .

2

(3.127)

2

We introduce functions R1 , . . . , R4 by

R1 = u2 + v1 ,

2

R2 = u 2 + v 2 ,

2

1 2

R3 = u2 + v3 ,

2

R4 = u 2 + v 4 .

2

3 4

We ¬rst rewite the Federbush model as a Hamiltonian system, i.e.,

du

= „¦’1 δH, (3.128)

dt

7. SYMMETRIES OF THE FEDERBUSH MODEL 141

where „¦ is a symplectic operator, H is the Hamiltonian and δH is the

Fr´chet derivative1 of H, u = (u1 , v1 , . . . , u4 , v4 ). In (3.128) we have

e

«

J000

¬0 J 0 0· 01

„¦=¬ ·, J= ,

0 0 J 0 ’1 0

000J

and

∞

1

u1x v1 ’u1 v1x ’u2x v2 +u2 v2x +u3x v3 ’u3 v3x ’u4x v4 +u4 v4x dx

H=

’∞ 2

» »

’ m1 (u1 u2 + v1 v2 ) ’ m2 (u3 u4 + v3 v4 ) ’ R1 R4 + R2 R3 .

2 2

By de¬nition, to each Hamiltonian symmetry Y (also called canonical sym-

metry) there corresponds a Hamiltonian F (Y ), where

∞

F(Y ) dx,

F (Y ) = (3.129)

’∞

F(Y ) being the Hamiltonian density, such that

Y = „¦’1 δF (Y ), (3.130)

and the Poisson bracket of F (Y ) and H vanishes.

Suppose that Y1 , Y2 are two Hamiltonian symmetries. Then [Y1 , Y2 ] is a

Hamiltonian symmetry and

F ([Y1 , Y2 ]) = {F (Y1 ), F (Y2 )}, (3.131)

where {·, ·} is the Poisson bracket de¬ned by

{F (Y1 ), F (Y2 )} = δF (Y1 ), Y2 , (3.132)

·, · denoting the contraction of a 1-form and a vector ¬eld:

d

H(x + y)| =0 = δH, y . (3.133)

d

The Hamiltonians F (X) associated to the Hamiltonian densities F(X) are

de¬ned by (3.134):

∞

F(X) dx.

F (X) = (3.134)

’∞

From these de¬nitions it is a straightforward computation that the symme-

tries Y0+ , Y1+ , Y’1 , Y0’ , Y1’ , Y’1 obtained sofar are all Hamiltonian, where

’

+

the Hamiltonian densities are given by

1

F(Y0+ ) = (R1 + R2 ),

2

1 » 1

F(Y1+ ) = ’ (u2x v2 ’ u2 v2x ) + R34 R2 ’ m1 (u1 u2 + v1 v2 ),

2 4 2

1

By the Fr´chet derivative the components of the Euler“Lagrange operator are

e

understood.

142 3. NONLOCAL THEORY

1 » 1

+

F(Y’1 ) = ’ (u1x v1 ’ u1 v1x ) + R34 R1 + m1 (u1 u2 + v1 v2 ),

2 4 2

1

F(Y0’ ) = (R3 + R4 ),

2

1 » 1

F(Y1’ ) = ’ (u4x v4 ’ u4 v4x ) ’ R12 R4 ’ m2 (u3 u4 + v3 v4 ),

2 4 2

1 » 1

’

F(Y’1 ) = ’ (u3x v3 ’ u3 v3x ) ’ R12 R3 + m2 (u3 u4 + v3 v4 ), (3.135)

2 4 2

whereas the densities F(Yi± ), i = ’2, 2, are given by

1 » 1

F(Y2+ ) = ’ (u2 + v2x ) + R34 (u2x v2 ’ u2 v2x ) ’ m1 (u2x v1 ’ u1 v2x )

2

2 2x 2 2

122 1 12

’ » R34 R2 + m1 »R34 (u1 u2 + v1 v2 ) ’ m1 R12 ,

8 4 8

1 » 1

+

F(Y’2 ) = ’ (u2 + v1x ) + R34 (u1x v1 ’ u1 v1x ) + m1 (u1x v2 ’ u2 v1x )

2

1x

2 2 2

1 1 1

’ »2 R34 R1 ’ m1 »R34 (u1 u2 + v1 v2 ) ’ m2 R12 ,

2

81

8 4

1 » 1

F(Y2’ ) = ’ (u2 + v4x ) ’ R12 (u4x v4 ’ u4 v4x ) ’ m2 (u4x v3 ’ u3 v4x )

2

2 4x 2 2

1 1 1

’ »2 R12 R4 ’ m2 »R12 (u3 u4 + v3 v4 ) ’ m2 R34 ,

2

82

8 4

1 » 1

’

F(Y’2 ) = ’ (u2 + v3x ) ’ R12 (u3x v3 ’ u3 v3x ) + m2 (u3x v4 ’ u4 v3x )

2

2 3x 2 2

122 1 12

’ » R12 R3 + m2 »R12 (u3 u4 + v3 v4 ) ’ m2 R34 , (3.136)

8 4 8

and the densities associated to Y3+ , Y’3 are given by

+

F(Y3+ ) = ’(u2xx v2x ’ v2xx u2x ) ’ »R34 (u2xx u2 + v2xx v2 )

»

+ R34 (u2 + v2x )

2

2x

2

3

’ m1 (u1x u2x + v1x v2x ) ’ »2 R34 (u2x v2 ’ u2 v2x )

2

4

1

+ m1 »R34 (u1x v2 ’ u1 v2x + u2x v1 ’ u2 v1x )

2

1 1 1

’ m2 (u1x v1 ’ u1 v1x ) ’ m2 (u2x v2 ’ u2 v2x ) ’ m3 (u1 u2 + v1 v2 )

1 1

41

4 2

1 1 1

+ »3 R34 R2 ’ m1 »2 R34 (u1 u2 + v1 v2 ) + m2 »R34 (R1 + 2R2 ),

3 2

81

8 4

»

+

F(Y’3 ) = u1xx v1x ’ v1xx u1x + »R34 (u1xx u1 + v1xx v1 ) + R34 (u2 + v1x )

2

1x

2

3

’ m1 (u1x u2x + v1x v2x ) + »2 R34 (u1x v1 ’ u1 v1x )

2

4

1

+ m1 »R34 (u1x v2 ’ u1 v2x + u2x v1 ’ u2 v1x )

2

7. SYMMETRIES OF THE FEDERBUSH MODEL 143

1 1 1

+ m2 (u1x v1 ’ u1 v1x ) + m2 (u2x v2 ’ u2 v2x ) ’ m3 (u1 u2 + v1 v2 )

21 41 41

1 1 1

’ »3 R34 R1 ’ m1 »2 R34 (u1 u2 + v1 v2 ) ’ m2 »R34 (2R1 + R2 ).

3 2

81

8 4

’ ’

+ +

The vector ¬elds Z’1 , Z1 , Z’1 , Z1 are Hamiltonian vector ¬elds too,

and the associated densities are given by

F(Z0 ) = x F(Y1+ ) ’ F(Y’1 ) + t F(Y1+ ) + F(Y’1 ) ,

+ + +

1 1

F(Z1 ) = x F(Y2+ ) ’ m2 F(Y0+ ) + t F(Y2+ ) + m2 F(Y0+ ) ,

+

41 41

1 1

F(Z’1 ) = x ’ F(Y’2 ) + m2 F(Y0+ ) + t F(Y’2 ) + m2 F(Y0+ ) ,

+ + +

1

41

4

F(Z0 ) = x F(Y1’ ) ’ F(Y’1 ) + t F(Y1’ ) + F(Y’1 ) ,

’ ’ ’

1 1

F(Z1 ) = x F(Y2’ ) ’ m2 F(Y0’ ) + t F(Y2’ ) + m2 F(Y0’ ) ,

’

2

42

4

1 1

F(Z’1 ) = x(’F(Y’2 ) + m2 F(Y0’ ) + t F(Y’2 ) + m2 F(Y0’ ) .

’ ’ ’

2

42

4

We now arrive at the following remarkable fact: The vector ¬elds Y + (2, 0)

and Y ’ (2, 0) are again Hamiltonian vector ¬elds, the corresponding Hamil-

tonian densities being given by

1

F(Y ’ (2, 0)) = x2 F(Y2’ ) ’ m2 F(Y0’ ) + F(Y’2 )’

2

2

’ ’

+ 2xt F(Y2 ) ’ F(Y’2 )

1

+ t2 F(Y2’ ) + m2 F(Y0’ ) + F(Y’2 )

’

2

2

1

= (x + t)2 F(Y2’ ) ’ m2 (x + t)(x ’ t)F(Y0’ )

22

’

+ (x ’ t)2 F(Y’2 ), (3.137)

and similarly

1

F(Y + (2, 0)) = (x + t)2 F(Y2+ ) ’ m2 (x + t)(x ’ t)F(Y0+ )

21

+

+ (x ’ t)2 F(Y’2 ), (3.138)

’ ’

+ +

Now the Hamiltonians F (Z1 ), F (Z’1 ), F (Z1 ), F (Z’1 ) act as cre-

ating and annihilating operators on the (x, t)-independent Hamiltonians

F (Y’3 ), . . . , F (Y3+ ) and F (Y’3 ), . . . , F (Y3’ ), by the action of the Poisson

’

+

bracket: for example

{F (Z1 ), F (Y0+ )} = 0,

+

∞

1 1

+ +

(R1 + R2 ) = m2 F (Y0+ ),

{F (Z1 ), F (Y’1 )} = m2

41 41

’∞

{F (Z1 ), F (Y1+ )} = ’F (Y2+ ).

+

144 3. NONLOCAL THEORY

In the next subsection we give a formal proof for the existence of in-

¬nite number of hierarchies of higher symmetries by proving existence of

in¬nite number of hierarchies of Hamiltonians, thus leading to those for the

symmetries.

7.5.3. The in¬nity of the hierarchies. We shall prove here a lemma con-

cerning the in¬niteness of the hierarchies of Hamiltonians for the Federbush

model. From this we obtain a similar result for the associated hierarchies of

Hamiltonian vector ¬elds.

r r

Lemma 3.12. Let Hn (u, v) and Kn (u, v) be de¬ned by

∞

r

xr (u2 + vn ),

2

Hn (u, v) = n

’∞

∞

r

xr (un+1 vn ’ vn+1 un ),

Kn (u, v) = (3.139)

’∞

whereas in (3.139) r, n = 0, 1, . . . , and r, n are such that the degrees of

r r

Hn (u, v) and Kn (u, v) are positive.

Let the Poisson bracket of F and L, denoted by {F, L}, be de¬ned as

∞

δF δL δF δL

{F, L} = ’ . (3.140)

’∞ δv δu δu δv

Then the following results hold

1 r r

{H1 , Hn } = 4(n ’ r)Kn ,

1 r r r’2

{H1 , Kn } = (4(n ’ r) + 2)Hn+1 + r(r ’ 1)(r ’ n ’ 1)Hn ,

2 r r+1

{H1 , Hn } = 4(2n ’ r)Kn ,

r+1

{H1 , Kn } = (2n + 1 ’ r)(4Hn+1 ’ r2 Hn ),

2 r r’1

(3.141)

r, n = 0, 1, . . .

Proof. We shall now prove the third and fourth relation in (3.141), the

proofs of the other two statements running along similar lines.

r r

Calculation of the Fr´chet derivatives of Hn , Kn yields

e

r

δHn

= (’Dx )n (2xr un ),

δu

r

δHn

= (’Dx )n (2xr vn ),

δv

r

δKn

= (’Dx )n+1 (xr vn ) ’ (’Dx )n (xr vn+1 ),

δu

r

δKn

= ’(’Dx )n+1 (xr un ) + (’Dx )n (xr un+1 ). (3.142)

δv

Substitution of (3.142) into the third relation of (3.141) yields

∞

2 r

’Dx (2x2 v1 ) · (’1)n Dx (2xr un )

n

{H1 , Hn } =

’∞

+ Dx (2x2 u1 ) · (’1)n Dx (2xr vn )

n

7. SYMMETRIES OF THE FEDERBUSH MODEL 145

∞

2n’1

Dx (2x2 u1 )Dx (2xr vn ) ’ Dx (2x2 v1 )Dx (2xr un )

n n

= (’1)

’∞

∞

(x2 un+1 + 2nxun + n(n ’ 1)un’1 )(xr vn+1 + rxr’1 vn )

= ’4

’∞

’ (x vn+1 + 2nxvn + n(n ’ 1)vn’1 )(xr un+1 + rxr’1 un )

2

∞

rxr+1 (un+1 vn ’ vn+1 un ) ’ 2nxr+1 (un+1 vn ’ vn+1 un )

= ’4

’∞

+ n(n ’ 1)x (vn+1 un’1 ’ un+1 vn’1 ) + n(n ’ 1)rxr’1 (vn un’1 ’ un vn’1 )

r

r+1

= 4(2n ’ r)Kn , (3.143)

which proves the third relation in (3.141).

The last equality in (3.143) results from the fact that the last two terms

are just constituting a total derivative of

n(n ’ 1)xr (vn un’1 ’ un vn’1 ). (3.144)

In order to prove the fourth relation in (3.141), we substitute (3.142),

which leads to

∞

2 r

’Dx (2x2 v1 ) · (’1)n+1 Dx (xr vn ) ’ (’1)n Dx (xr vn+1 )

n+1 n

{H1 , Hn } =

’∞

+ Dx (2x2 u1 ) · (’1)n+1 Dx (xr un ) ’ (’1)n Dx (xr un+1 ) . (3.145)

n+1 n

Integration, n times, of the terms in brackets leads to

∞

2 r

Dx (x2 v1 ) · (Dx (xr vn ) + xr vn+1 )

n+1

{H1 , Hn } =2

’∞

+ Dx (x2 u1 ) · (Dx (xr un ) + xr un+1 )

n+1

∞

(x2 vn+2 + 2(n + 1)xvn+1 + n(n + 1)vn )(2xr vn+1 + rxr’1 vn )

=2

’∞

+ (x2 un+2 + 2(n + 1)xun+1 + n(n + 1)un )(2xr un+1 + rxr’1 un ). (3.146)

By expanding the expressions in (3.146), we arrive, after a short calculation,

at

r+1

{H1 , Kn } = (2n + 1 ’ r)(4Hn+1 ’ r2 Hn ),

2 r r’1

(3.147)

which proves the fourth relation in (3.141).

We are now in a position to formulate and prove the main theorem of this

subsection.

Theorem 3.13. The conserved functionals F (Y ± (2, 0)) associated to

the symmetries Y ± (2, 0) generate in¬nite number of hierarchies of Hamilto-

nians, starting at the hierarchies F (Yi+ ), F (Yi’ ), where i ∈ Z, by repeated

’

+

action of the Poisson bracket (3.140). The hierarchies F (Zj ), F (Zj ),

j ∈ Z, are obtained by the ¬rst step of this procedure.

146 3. NONLOCAL THEORY

Moreover, the hierarchies F (Yj+ ), F (Yj’ ), j ∈ Z, are obtained from

± ±

F (Y±1 ) by repeated action of the conserved functionals F (Z±1 )

1

± ±

F (Z±1 ) = ± F [Y ± (2, 0), Y±1 ] . (3.148)

2

Proof. The proof of this theorem is a straightforward application of

the previous lemma, and the observation that the (», m1 , m2 )-independent

±

parts of the conserved densities Y±1 , Y + (2, 0), Y ’ (2, 0) are just given by

1

F(Y1+ ) ’’ ’ (u2x v2 ’ v2x u2 ),

2

1

+

F(Y’1 ) ’’ ’ (u1x v1 ’ v1x u1 ),

2

1

F(Y1’ ) ’’ ’ (u4x v4 ’ v4x u4 ),

2

1

’

F(Y’1 ) ’’ ’ (u3x v3 ’ v3x u3 ),

2

1 1

F(Y + (2, 0)) ’’ ’ (x + t)2 (u2 + v2x ) ’ (x ’ t)2 (u2 + v1x ),

2 2

2x 1x

2 2

1 1

F(Y ’ (2, 0)) ’’ ’ (x + t)2 (u2 + v4x ) ’ (x ’ t)2 (u2 + v3x ).

2 2

4x 3x

2 2

Note that in applying the lemma we have to choose (u, v) = (u1 , v1 ), etc.

7.6. Nonlocal symmetries. In this last subsection concerning the

Federbush model, we discuss existence of nonlocal symmetries. We start

from the conservation laws, conserved quantities and the associated nonlo-

cal variables p1 , p2 :

p1t = ’R1 + R2 ,

p1x = R1 + R2 ,

p2t = ’R3 + R4 .

p2x = R3 + R4 , (3.149)

Including these two nonlocal variables, we ¬nd two new nonlocal symmetries

‚ ‚ ‚ ‚ ‚ ‚

Z + (0, 0) = u1 ’ »p1 v3 ’ u3

+ v1 + u2 + v2

‚u1 ‚v1 ‚u2 ‚v2 ‚u3 ‚v3

‚ ‚ ‚

’ u4

+ v4 + 2p1 ,

‚u4 ‚v4 ‚p1

‚ ‚ ‚ ‚ ‚ ‚

Z ’ (0, 0) = u3 ’ u1

+ v3 + u4 + v4 + »p2 v1

‚u3 ‚v3 ‚u4 ‚v4 ‚u1 ‚v1

‚ ‚ ‚

’ u2

+ v2 + 2p2 . (3.150)

‚u2 ‚v2 ‚p2

Analogously to the construction of conservation laws and nonlocal variables

in previous sections, we obtained nonlocal variables p3 , p4 , p5 , p6 de¬ned by

1

p3x = »(R1 + R2 )R4 + m2 (u3 u4 + v3 v4 ) ’ u4 v4x + v4 u4x ,

2

1

p3t = »(R1 + R2 )R4 ’ u4 v4x + v4 u4x ,

2

7. SYMMETRIES OF THE FEDERBUSH MODEL 147

1

p4x = »(R1 + R2 )R3 + m2 (u3 u4 + v3 v4 ) + u3 v3x ’ v3 u3x ,

2

1

p4t = »(R1 + R2 )R3 ’ u3 v3x + v3 u3x ,

2

1

p5x = »(R3 + R4 )R2 ’ m1 (u1 u2 + v1 v1 ) + u2 v2x ’ v2 u2x ,

2

1

p5t = »(R3 + R4 )R2 + u2 v2x ’ v2 u2x ,

2

1

p6x = »(R3 + R4 )R1 + m1 (u1 u2 + v1 v1 ) + u1 v1x ’ v1 u1x ,

2

1

p6t = ’ »(R3 + R4 )R1 ’ u1 v1x + v1 u1x . (3.151)

2

Using these nonlocal variables we ¬nd four additional nonlocal symmetries

Z + (0, ’1), Z + (0, +1), Z ’ (0, ’1), Z ’ (0, +1):

1 ‚

Z + (0, ’1) = ’ »u1 (R3 + R4 ) ’ m1 u2 ’ 2v1x

2 ‚u1

1 ‚

’ »v1 (R3 + R4 ) ’ m1 v2 + 2u1x

+

2 ‚v1

1 ‚ 1 ‚

’ m 1 u1 ’ m 1 v1

2 ‚u2 2 ‚v2

‚ ‚ ‚ ‚

’ u3 ’ u4

+ »p6 v3 + v4 ,

‚u3 ‚v3 ‚u4 ‚v4

1 ‚ 1 ‚

Z + (0, +1) = m1 u2 + m 1 v2

2 ‚u1 2 ‚v1

1 ‚

’ »u2 (R3 + R4 ) + m1 u1 ’ 2v2x

+

2 ‚u2

1 ‚

’ »v2 (R3 + R4 ) + m1 v1 + 2u2x

+

2 ‚v2

‚ ‚ ‚ ‚

’ u3 ’ u4

+ »p5 v3 + v4 ,

‚u3 ‚v3 ‚u4 ‚v4

‚ ‚ ‚ ‚

Z ’ (0, ’1) = ’»p4 (v1 ’ u1 ’ u2

+ v2 )

‚u1 ‚v1 ‚u2 ‚v2

1 ‚

’ »u3 (R1 + R2 ) ’ m2 u4 ’ 2v3x

+

2 ‚u3

1 ‚

+ »v3 (R1 + R2 ) ’ m2 v4 + 2u3x

+

2 ‚v3

1 ‚ 1 ‚

’ m 2 u3 ’ m 2 v3 ,

2 ‚u4 2 ‚v4

‚ ‚ ‚ ‚

Z ’ (0, +1) = »p3 v1 ’ u1 ’ u2

+ v2

‚u1 ‚v1 ‚u2 ‚v2

1 ‚ 1 ‚

+ m 2 u4 + m 2 v4

2 ‚u3 2 ‚v3

148 3. NONLOCAL THEORY

1 ‚

»u4 (R1 + R2 ) + m2 u3 ’ 2v4x

+

2 ‚u4

1 ‚

+ »v4 (R1 + R2 ) + m2 v3 + 2u4x .

2 ‚v4

According to standard lines of computations, including prolongation towards

nonlocal variables as explained in previous sections, we arrive at the follow-

ing commutators:

[Y ± (1, ±1), Z ± (0, 0)] = 0,

[Y + (1, ’1), Z + (0, ’1)] = Z + (0, ’2),

1

[Y + (1, ’1), Z + (0, +1)] = ’ m2 Z + (0, 0),

41

1

[Y + (1, +1), Z + (0, ’1)] = m2 Z + (0, 0),

41

[Y + (1, +1), Z + (0, +1)] = Z + (0, +2)

and

[Y ’ (1, ’1), Z ’ (0, ’1)] = Z ’ (0, ’2),

1

[Y ’ (1, ’1), Z ’ (0, +1)] = ’ m2 Z ’ (0, 0),

42

1

[Y ’ (1, +1), Z ’ (0, ’1)] = m2 Z ’ (0, 0),

42

[Y ’ (1, +1), Z ’ (0, +1)] = Z ’ (0, +2), (3.152)

the vector ¬elds Z + (0, ’2), Z + (0, +2), Z ’ (0, ’2), Z ’ (0, +2) just being new

nonlocal symmetries.

Summarising these results, we conclude that the action of the symmetries

± (1, ±1) on Z ± (0, ±1) constitute hierarchies of nonlocal symmetries.

Y

Finally we compute the Lie brackets of Y + (2, 0), (3.121), and Z + (0, ±1)

which results in

[Y + (2, 0), Z + (0, ’1)] = Z + (1, ’1),

[Y + (2, 0), Z + (0, +1)] = Z + (1, +1), (3.153)

whereas in (3.153) Z + (0, ±1) are de¬ned by

1

Z + (1, ’1) = 2(’x + t)Z + (0, ’2) + m2 (x + t)Z + (0, 0)

21

‚ ‚

+ »v1 R34 + m1 v2 ’ 2u1x ’ »u1 R34 + m1 u2 + 2v1x

‚u1 ‚v1

» ‚ ‚ ‚ ‚ +

’ ’ u3 ’ u4

v3 + v4 K’1 ,

2 ‚u3 ‚v3 ‚u4 ‚v4

1

Z + (1, +1) = 2(x + t)Z + (0, ’2) + m2 (x ’ t)Z + (0, 0)

21

‚

+ »v2 R34 ’ m1 v1 ’ 2u2x

‚u2

¨

8. BACKLUND TRANSFORMATIONS AND RECURSION OPERATORS 149

‚

’ »u2 R34 + m1 u1 + 2v2x

‚v2

» ‚ ‚ ‚ ‚ +

’ ’ u3 ’ u4

v3 + v4 K+1 , (3.154)

2 ‚u3 ‚v3 ‚u4 ‚v4

+

while K±1 are given by

x x x x

+ +

m2 F(Y + (0, 0)),

F(Y (0, ’2)) ’

K’1 =8 1

’∞ ’∞ ’∞ ’∞

x x x x

+

F(Y + (0, +2)) ’ m2 F(Y + (0, 0)).

K+1 = 8 (3.155)

1

’∞ ’∞ ’∞ ’∞

The previous formulas re¬‚ect the fact that Y + (2, 0) constructs an (x, t)-

dependent hierarchy Z + (1, —) from Z + (0, —) by action of the Lie bracket. We

expect similar results for the action of Y + (2, 0) on the hierarchy Z + (1, —).

Results conserning the action of Y ’ (2, 0) on Z ’ (0, —) and from this, on

Z ’ (1, —) will be similar.

8. B¨cklund transformations and recursion operators

a

In this section, we mainly follow the results by M. Marvan exposed in

[73]. Our aim here is to show that recursion opeartors for higher symmetries

may be unberstood as B¨cklund transformations of a special type.

a

Let E1 and E2 be two di¬erential equations in unknown functions u1 and

u2 respectively. Informally speaking, a B¨cklund transformation between E1

a

and E2 is a third equation E containing both independent variables u1 and

u2 and possessing the following property:

1. If u1 is a solution of E1 , then solving the equation E[u1 ] with respect

0 0

to u2 , we obtain a family of solutions to E2 .

2. Vice versa, if u2 is a solution of E2 , then solving the equation E[u2 ]

0 0

with respect to u1 , we obtain a family of solutions to E1 .

Geometrically this construction is expressed in a quite simple manner.

Definition 3.10. Let N1 and N2 be objects of the category DM∞ . A

B¨klund transformation between N1 and N2 is a pair of coverings

a

N

•2

1

•

’

←

N1 N2

where N is a third object of DM∞ . A B¨cklund transformation is called a

a

B¨cklund auto-transformation, if N1 = N2 .

a

In fact, let Ni = Ei∞ , i = 1, 2, and s ‚ E1 be a solution. Then the set

∞

•’1 s ‚ N is ¬bered by solutions of N and they are projected by •2 (at

1

∞

nonsingular points) to a family of solutions of E2 .

150 3. NONLOCAL THEORY

We are now interested in B¨cklund auto-transformations of the total

a

v : V E ∞ ’ E ∞ (see Example 3.2). The reason

space of the Cartan covering „

to this is the following

Proposition 3.14. A section X : E ∞ ’ V E ∞ of the projection „ v is a

symmetry of the equation E if and only if it is a morphism in the category

DM∞ , i.e., if it preserves Cartan distributions.

The proof is straightforward and is based on the de¬nition of the Cartan

distribution on V E ∞ . The result is in full agreement with equalities (3.2)

on p. 101: the equations for V E ∞ are just linearization of E and symmetries

are solutions of the linearized equation.

Thus, we can hope that B¨cklund auto-transformations of V E ∞ will

a

relate symmetries of E to each other. This motivates the following

Definition 3.11. Let E ∞ be an in¬nitely prolonged equation. A recur-

sion operator for symmetries of E is a pair of coverings K, L : R ’ V E ∞

such that the diagram

R

K

L

←

’

V E∞ V E∞

v

„

v

„

’

←

E∞

is commutative. A recursion operator is called linear, if both K and L are

linear coverings.

Example 3.4. Consider the KdV equation E = {ut = uux + uxxx }.

Then V E ∞ is described by additional equation

vt = uvx + ux v + vxxx .

Let us take for R the system of equations

wx = v,

wt = vxx + uv,

vt = vxxx uvx + ux v,

ut = uxxx + uux ,

while the mappings K and L are given by

K : v = wx ,

2 1

L : v = vxx + uv + ux w.

3 3

¨

8. BACKLUND TRANSFORMATIONS AND RECURSION OPERATORS 151

Obviously, K and L determine covering structures over V E ∞ (the ¬rst being

one-dimensional and the second three-dimensional) while the triple (R, K, L)

corresponds to the classical Lenard operator Dx + 3 u + 1 ux Dx .

2 ’1

2

3

Let us now study action of recursion operators on symmetries in more

details. Let X be a symmetry of an equation E. Then, due to Proposition

3.14, it can be considered as a section X : E ∞ ’ V E ∞ which is a morphism

in DM∞ . Thus we obtain the following commutative diagram

X— L

—

’ V E∞

R ’R

P = X — (K) „v

K

“ “ “

„v

X

∞ ∞

’ E∞

E ’VE

where the composition of the arrows below is the identity while P = X — (K)

is the pull-back. As a consequence, we obtain the following morphism of

coverings

L —¦ X—

—

’ V E∞

R

v

P

„

’

←

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 }

which is called the module of in¬nite jets for P . Denote by ν∞,k : J ∞ (P ) ’

J k (P ) the corresponding projections. Since νk+1,k —¦ jk = jk+1 for any k ≥ 0,

the system of operators jk induces the mapping j∞ : P ’ J ∞ (P ) satisfying

the condition ν∞,k —¦ j∞ = jk . Obviously, J ∞ (P ) is the representative object

for the functor Diff — (P, •) while the mapping j∞ possesses the universal

property similar to that of jk : for any ∆ ∈ Diff — (P, Q) there exists a unique

1

It makes no di¬erence whether we span µk by the left or the right multiplication due

to the identity la δ a (b — p) = ra δ a (b — p) + δ a δ a (b — p).

160 4. BRACKETS

homomorphism f ∆ : J ∞ (P ) ’ Q such that ∆ = f ∆ —¦ j∞ . Note that j∞ is

not a di¬erential operator in the sense of De¬nition 4.1.2

The functors J k (•) possess the properties dual to those of Diff + (•).

k

Namely, we can de¬ne the l-th Jet-prolongation of ∆ ∈ Diff k (P, Q) by

setting

def

∆(l) = jl —¦ ∆ : P ’ J l (Q)

and consider the commutative diagram

jk

’ J k (P )

P

jk (l)

jk+l jl

“ ’“

cl,k

J k+l (P ) ’ J l J k (P )

(l)

where cl,k = f jk is called the cogluing transformation. Similar to Diagram

(4.6), for any operators ∆ : P ’ Q, : Q ’ R of orders k and l respectively,

we have the commutative diagram

—¦∆

f

k+l

J ’R

(P )

‘

cl,k f

“

J l (f ∆ ) l

lk

J J (P ) ’ J (Q)

and call cl,k the universal cocompositon operation. This operation is coasso-

ciative, i.e., the diagram

ck+l,s

k+l+s

’ J k+l J s (P )

J (P )

ck,l+s ck,l

“ “

J k (cl,s ) k l s

J k J l+s (P ) ’ J J J (P )

is commutative for all k, l, s ≥ 0.

1.3. Derivations. We shall now deal with special di¬erential operators

of order 1.

Definition 4.4. Let P be an A-module. A P -valued derivation is a

¬rst order operator ∆ : A ’ P satisfying ∆(1) = 0.

2

One might say that j∞ is a di¬erential operator of “in¬nite order”, but this concept

needs to be more clari¬ed. Some remarks concerning a concept of in¬nite order di¬erential

operators were made in Chapter 1, see also [51] for more details.

1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 161

The set of such derivations will be denoted by D(P ). From the above

de¬nition and from De¬nition 4.1 it follows that ∆ ∈ D(P ) if and only if

a, b ∈ A.

∆(ab) = a∆(b) + b∆(a), (4.7)

It should be noted that the set D(P ) is a submodule in Diff 1 (P ) but not in

Diff + (P ).

1

Remark 4.2. In the case A = C ∞ (M ), M being a smooth manifold,

and P = A the module D(A) coincides with the module D(M ) of vector

¬elds on the manifold M .

For any A-homomorphism f : P ’ Q and a derivation ∆ ∈ D(P ), the

def

composition D(f ) = f —¦ ∆ lies in D(Q) and thus P ’ D(P ) is a functor

from the category of A-modules into itself. This functor can be generalized

as follows.

Let P be an A-module and N ‚ P be a subset in P . Let us de¬ne

def

D(N ) = {∆ ∈ D(P ) | ∆(A) ‚ N }.

def

Let us also set (Diff + )i = Diff + —¦ · · ·—¦Diff + , where the composition is taken

1 1 1

i times. We now de¬ne a series of functors Di , i ≥ 0, together with natural

embeddings Di (P ) ’ (Diff + )i (P ) by setting D0 (P ) = P , D1 (P ) = D(P )

1

and, assuming that all Dj (P ), j < i, were de¬ned,

Di (P ) = D(Di’1 (P ) ‚ (Diff + )i’1 (P )).

1

Since

D(Di’1 (P ) ‚ (Diff + )i’1 (P )) ‚ D((Diff + )i’1 (P )) ‚ (Diff + )i (P ), (4.8)

1 1 1

the modules Di (P ) are well de¬ned.

Let us show now that the correspondences P ’ Di (P ) are functors for

all i ≥ 0. In fact, the case i = 0 is obvious while i = 1 was considered

above. We use induction on i and assume that i > 1 and that for j < i all

j

Dj are functors. We shall also assume that the embeddings ±P : Dj (P ) ’

(Diff + )i (P ) are natural, i.e., the diagrams

1

j

±P

’ (Diff + )j (P )

Dj (P ) 1

(Diff + )j (f )

Dj (f ) (4.9)

1

“

“ j

±Q

’ (Diff + )j (Q)

Dj (Q) 1

are commutative for any homomorphism f : P ’ Q (in the cases j = 0, 1,

def

this is obvious). Then, if ∆ ∈ Di (P ) and a ∈ A, we set (Di (f ))(∆) =

Di’1 (∆(a)). Then from commutativity of diagram (4.9) it follows that Di (f )

takes Di (P ) to Di (Q) while (4.8) implies that ±P : Di (P ) ’ (Diff + )i (P ) is

i

1

a natural embedding.

162 4. BRACKETS

Note now that, by de¬nition, elements of Di (P ) may be understood as

K-linear mappings A ’ Di’1 (P ) possessing “special properties”. Given an

element a ∈ A and an operator ∆ ∈ Di (P ), we have ∆(a) ∈ Di’1 (P ), i.e.,

∆ : A ’ Di’1 (P ), etc. Thus ∆ is a polylinear mapping

∆ : A —K · · · —K A ’ P. (4.10)

i times

Let us describe the module Di (P ) in these terms.

Proposition 4.5. A polylinear mapping of the form (4.10) is an ele-

ment of Di (P ) if and only if

∆(a1 , . . . , a±’1 , ab, a±+1 , . . . , ai )

= a∆(. . . , a±’1 , b, a±+1 , . . . ) + b∆(. . . , a±’1 , a, a±+1 , . . . ) (4.11)

and

∆(. . . , a± , . . . , aβ , . . . ) = (’1)±β ∆(. . . , aβ , . . . , a± , . . . ) (4.12)

for all a, b, a1 , . . . , ai ∈ A, 1 ¤ ± < β ¤ i. In other words, Di (P ) consists of

skew-symmetric polyderivations (of degree i) of the algebra A with the values

in P .

Proof. Note ¬rst that to prove the result it su¬ces to consider the

case i = 2. In fact, the general case is proved by induction on i whose step

literally repeats the proof for i = 2.

Let now ∆ ∈ D2 (P ). Then, since ∆ is a derivation with the values in

Diff + (P ), one has

1

∆(ab) = a+ ∆(b) + b+ ∆(a), a, b ∈ A.

Consequently,

∆(ab, c) = ∆(b, ac) + ∆(a, bc) (4.13)

for any c ∈ A. But ∆(ab) ∈ D(P ) and thus ∆(ab, 1) = 0. Therefore, (4.13)

implies ∆(a, b) + ∆(b, a) = 0 which proves (4.12). On the other hand, from

the result proved we obtain that ∆(ab, c) = ’∆(c, ab) while, by de¬nition,

one has ∆(c) ∈ D(P ) for any c ∈ A. Hence,

∆(ab, c) = ’∆(c, ab) = ’a∆(c, b) ’ b∆(c, a) = a∆(b, c) + b∆(a, c)

which ¬nishes the proof.

To ¬nish this subsection, we establish an additional algebraic structure

in the modules Di (P ). Namely, we de¬ne by induction the wedge product

§ : Di (A) —K Dj (P ) ’ Di+j (P ) by setting

def

a § p = ap, a ∈ D0 (A) = A, p ∈ D0 (P ) = P, (4.14)

and

def

(∆ § )(a) = ∆ § (a) + (’1)j ∆(a) § (4.15)

for any ∆ ∈ Di (A), ∈ Dj (P ), i + j > 0.

1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 163

Proposition 4.6. The wedge product of polyderivations is a well-

de¬ned operation.

Proof. It needs to prove that ∆ § de¬ned by (4.14) and (4.15) lies

in Di+j (P ). To do this, we shall use Proposition 4.5 and induction on i + j.

The case i + j < 2 is trivial.

Let now i + j ≥ 2 and assume that the result was proved for all k < i + j.

Then from (4.15) it follows that (∆ § )(a) ∈ Di+j’1 (P ). Let us prove that

∆ § satis¬es identities (4.11) and (4.12) of Proposition 4.5. In fact, we

have

(∆ § )(a, b) = (∆ § (a))(b) + (’1)j (∆(a) § )(b)

= ∆ § (a, b) + (’1)j’1 ∆(b) § (a) + (’1)j (∆(a) § (b)

+ (’1)j ∆(a, b) § ) = ’ ∆ § (b, a) + (’1)j’1 ∆(a) § (b)

+ (’1)j ∆(b) § (a) + ∆(b, a) § = ’(∆ § )(b, a),

where a and b are arbitrary elements of A.

On the other hand,

(∆ § )(ab) = ∆ § (ab) + (’1)j ∆(ab) §

= ∆ § a (b) + b (a) + (’1)j a∆(b) + b∆(a) §

= a ∆ § (b) + (’1)j ∆(b) § + b ∆ § (a) + (’1)j ∆(a) §

= a ∆ § )(b) + b(∆ § (a).

We used here the fact that ∆ § (a ) = a(∆ § ) which is proved by trivial

induction.

Proposition 4.7. For any derivations ∆, ∆1 , ∆2 ∈ D— (A) and , 1,

2 ∈ D— (P ), one has

(i) (∆1 + ∆2 ) § = ∆1 § + ∆2 § ,

(ii) ∆ § ( 1 + 2 ) = ∆ § 1 + ∆ § 2 ,

(iii) ∆1 § (∆2 § ) = (∆1 § ∆2 ) § ,

(iv) ∆1 § ∆2 = (’1)i1 i2 ∆2 § ∆1 ,

where ∆1 ∈ Di1 (A), ∆2 ∈ Di2 (A).

Proof. All statements are proved in a similar way. As an example, let

us prove equality (iv). We use induction on i1 + i2 . The case i1 + i2 = 0 is

obvious (see (4.14)). Let now i1 + i2 > 0 and assume that (iv) is valid for

all k < i1 + i2 . Then

(∆1 § ∆2 )(a) = ∆1 § ∆2 (a) + (’1)i2 ∆1 (a) § ∆2

= (’1)i1 (i2 ’1) ∆2 (a) § ∆1 + (’1)i2 (’1)(i1 ’1)i2 ∆2 § ∆1 (a)

= (’1)i1 i2 (∆2 § ∆1 (a) + (’1)i1 ∆2 (a) § ∆1 ) = (’1)i1 i2 (∆2 § ∆1 )(a)

for any a ∈ A.

164 4. BRACKETS

Corollary 4.8. The correspondence P ’ D— (P ) is a functor from the

category of A-modules to the category of graded modules over the graded

commutative algebra D— (A).

1.4. Forms. Consider the module J 1 (A) and the submodule in it gen-

erated by j1 (1), i.e., by the class of the element 1 — 1 ∈ A —K A. Denote by

ν : J 1 (A) ’ J 1 (A)/(A · j1 (1)) the natural projection of modules.

def

Definition 4.5. The quotient module Λ1 (A) = J 1 (A)/(A · j1 (1)) is

called the module of di¬erential 1-forms of the algebra A. The composition

def

d = d1 = ν —¦ j1 : A ’ Λ1 (A) is called the (¬rst) de Rham di¬erential of A.

Proposition 4.9. For any derivation ∆ : A ’ P , a uniquely de¬ned

A-homomorphism •∆ : Λ1 (A) ’ P exists such that the diagram

d

’ Λ1 (A)

A

∆

∆

•

’

←

P

Λ1 (A)

is commutative. In particular, is the representative object for the

functor D(•).

Proof. The mapping d, being the composition of j1 with a homomor-

phism, is a ¬rst order di¬erential operator and it is a tautology that f d (see

Proposition 4.4) coincides with the projection ν : J 1 (A) ’ Λ1 (A). On the

other hand, consider the diagram