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[Y + (2, 0), Y1+ ] = 2Z1 ,
+

[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

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