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f∆
1
J (A) ’P








ν









Λ1 (A)

j1





d

A
Since ∆ is a ¬rst order di¬erential operator, there exists a homomorphism
f ∆ : J 1 (A) ’ P satisfying the equality ∆ = f ∆ —¦ j1 . But ∆ is a derivation,
i.e., ∆(1) = 0, which means that ker(f ∆ ) contains A · j1 (1). Hence, there
exists a unique mapping •∆ such that the above diagram is commutative.

Remark 4.3. From the de¬nition it follows that Λ1 (A), as an A-module,
is generated by the elements da, a ∈ A, with the relations
d(±a + βb) = ±da + βdb, d(ab) = adb + bda,
1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 165

±, β ∈ K, a, b ∈ A, while the de Rham di¬erential takes a to the coset
a mod (A · j1 (1)).
Let us set now
Λi (A) = Λ1 (A) § · · · § Λ1 (A) . (4.16)
i times

The elements of Λi (A) are called di¬erential i-forms of the algebra A. We
def
also formally set Λ0 (A) = A.
Proposition 4.10. The modules Λi (A), i ≥ 0, are representative ob-
jects for the functors Di (•).
Proof. The case i = 0 is trivial while the case i = 1 was proved already
(see Proposition 4.9). Let now i > 1 and a ∈ A. De¬ne the mappings
»a : homA (Λi (A), P ) ’ homA (Λi’1 (A), P ), ia : Di (P ) ’ Di’1 (P )
by setting
def def
(»a •)(ω) = •(da § ω), ia ∆ = ∆(a),
where ω ∈ Λi’1 (A), • ∈ homA (Λi (A), P ), and ∆ ∈ Di (P ).
Using induction on i, let us construct isomorphisms
ψi : homA (Λi (A), P ) ’ Di (P )
in such a way that the diagrams
ψi
homA (Λi (A), P ) ’ Di (P )

»a ia (4.17)
“ “
ψi’1
i’1
’ Di’1 (P )
homA (Λ (A), P )
are commutative for all a ∈ A.
The case i = 1 reduces to Proposition 4.9. Let now i > 1 and assume
that for i ’ 1 the statement is valid. Then from (4.17) we should have
• ∈ homA (Λi (A), P ),
(ψi (•))(a) = ψi’1 (»a (•)),
which completely determines ψi . From the de¬nition of the mapping »a it
follows that
»a —¦ »b = ’»b —¦ »a , a, b ∈ A,
»ab = a»b + b»a ,
i.e., im ψi ∈ Di (P ) (see Proposition 4.5).
Let us now show that ψi constructed in such a way is an isomorphism.
Take ∆ ∈ Di (P ), a1 , . . . , ai and set
def
¯ ’1
ψi (da1 § . . . dai ) = ψi’1 (X(a1 )) (da2 § · · · § dai ).
166 4. BRACKETS

’1
It may be done since ψi’1 exists by the induction assumption. Directly from
¯ ¯
de¬nitions one obtains that ψi —¦ ψi = id, ψi —¦ ψi = id. It is also obvious that
the isomorphisms ψi are natural, i.e., the diagrams
ψi
homA (Λi (A), P ) ’ Di (P )

homA (Λi (A), f ) Di (f )
“ “
ψi
homA (Λi (A), Q) ’ Di (Q)
are commutative for all homomorphisms f ∈ homA (P, Q).
From the result proved we obtain the pairing
·, · : Di (P ) —A Λi (A) ’ P (4.18)
de¬ned by
def
’1
ω ∈ Λi (A), ∆ ∈ Di (P ).
∆, ω = ψi (∆) (ω),
A direct consequence of the proof of Proposition 4.10 is the following
Corollary 4.11. The identity
∆, da § ω = ∆(a), ω (4.19)
holds for any ω ∈ Λi (A), ∆ ∈ Di+1 (A), a ∈ A.
Let us de¬ne the mappings d = di : Λi’1 (A) ’ Λi (A) by taking the ¬rst
de Rham di¬erential for d1 and setting
def
di (a0 da1 § · · · § dai ) = da0 § da1 § · · · § dai
for i > 1. From (4.16) and Remark 4.3 it follows that the mappings d i are
well de¬ned.
Proposition 4.12. The mappings di possess the following properties:
(i) di is a ¬rst order di¬erential operator acting from Λi’1 (A) to Λi (A);
(ii) d(ω § θ) = d(ω) § θ + (’1)i ω § d(θ) for any ω ∈ Λi (A), θ ∈ Λj (A);
(iii) di —¦ di’1 = 0.
The proof is trivial.
In particular, (iii) means that the sequence of mappings
d
d
0 ’ A ’1 Λ1 (A) ’ · · · ’ Λi’1 (A) ’i Λi (A) ’ · · ·
’ ’ (4.20)
is a complex.
Definition 4.6. The mapping di is called the (i-th) de Rham di¬er-
ential. The sequence (4.20) is called the de Rham complex of the algebra
A.
1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 167

Remark 4.4. Before proceeding with further exposition, let us make
some important comments on the relation between algebraic and geometrical
settings. As we saw above, the algebraic de¬nition of a linear di¬erential
operator is in full accordance with the analytical one. The same is true if we
compare algebraic “vector ¬elds” (i.e., elements of the module D(A)) with
vector ¬elds on a smooth manifold M : derivations of the algebra C ∞ (M )
are identical to vector ¬elds on M .
This situation changes, when we pass to representative objects. A simple
example illustrates this e¬ect. Let M = R and A = C ∞ (M ). Consider the
di¬erential one-form ω = dex ’ ex dx ∈ Λ1 (A). This form is nontrivial as an
element of the module Λ1 (A). On the other hand, for any A-module P let
us de¬ne the value of an element p ∈ P at point x ∈ M as follows. Denote
by µx the ideal
def
µx = {f ∈ C ∞ (M ) | f (x) = 0} ‚ C ∞ (M )
def
and set px = p mod µx . In particular, if P = A, thus de¬ned value coincides
with the value of a function f at a point. One can easily see that ωx = 0
for any x ∈ M . Thus, ω is a kind of a “ghost”, not observable at any point
of the manifold. The reader will easily construct similar examples for the
modules J k (A). In other words, we can state that
Λi (M ) = Λi (C ∞ (M )), “(πk ) = J k (“(π))
for an arbitrary smooth manifold M and a vector bundle π : E ’ M .
Let us say that C ∞ (M )-module P is geometrical, if
µx · P = 0.
x∈M
Obviously, all modules of the form “(π) are geometrical. We can introduce
the geometrization functor by setting
def
µx · P.
G(P ) = P/
x∈M
Then the following result is valid:
Proposition 4.13. Let M be a smooth manifold and π : E ’ M be a
smooth vector bundle. Denote by A the algebra C ∞ (M ) and by P the module
“(π). then:
(ii) The functor Di (•) is representable in the category of geometrical A-
modules and one has
Di (Q) = homA (G(Λi (A)), Q)
for any geometrical module Q.
(i) The functor Diff(P, •) is representable in the category of geometrical
A-modules and one has
Diff k (P, Q) = homA (G(J k (P )), Q)
for any geometrical module Q.
168 4. BRACKETS

In particular,
Λi (M ) = G(Λi (C ∞ (M ))), “(πk ) = G(J k (“(π))).
1.5. Smooth algebras. Let us introduce a class of algebras which
plays an important role in geometrical theory.
Definition 4.7. A commutative algebra A is called smooth, if Λ1 (A)
is a projective A-module of ¬nite type while A itself is an algebra over the
¬eld of rational numbers Q.
Denote by S i (P ) the i-th symmetric power of an A-module P .
Lemma 4.14. Let A be a smooth algebra. Then both S i (Λ1 (A)) and
Λi (A) are projective modules of ¬nite type.
def
Proof. Denote by T i = T i (Λ1 (A)) the i-th tensor power of Λ1 (A).
Since the module Λ1 (A) is projective, then it can be represented as a direct
summand in a free module, say P . Consequently, T i is a direct summand
in the free module T i (P ) and thus is projective with ¬nite number of gen-
erators.
On the other hand, since A is a Q-algebra, both S i (Λi (A)) and Λi (A)
are direct summands in T i which ¬nishes the proof.
Proposition 4.15. If A is a smooth algebra, then the following isomor-
phisms are valid :
(i) Di (A) D1 (A) § · · · § D1 (A),
i times
(ii) Di (P ) Di (A) —A P ,
where P is an arbitrary A-module.
Proof. The result follows from Lemma 4.14 combined with Proposition
4.10
For smooth algebras, one can also e¬ciently describe the modules
J k (A). Namely, the following statement is valid:
Proposition 4.16. If A is a smooth algebra, then all the modules J k (A)
are projective of ¬nite type and the isomorphisms
J k (A) S i (Λ1 (A))
i¤k
take place.
Proof. We shall use induction on k. First note that the mapping a ’
aj1 (1) splits the exact sequence
ν1,0
0 ’ ker(ν1,0 ) ’ J 1 (A) ’ ’ J 0 (A) = A ’ 0.

But by de¬nition, ker(ν1,0 ) = Λ1 (A) and thus J 1 (A) = A • Λ1 (A).
Let now k > 1 and assume that for k ’ 1 the statement is true. By
de¬nition, ker(νk,k’1 ) = µk’1 /µk , where µi ‚ A —K A are the submodules
1. DIFFERENTIAL CALCULUS OVER COMMUTATIVE ALGEBRAS 169

introduced in Subsection 1.2. Note that the identity a—b = a(b—1)’aδ b (1—
1) implies the direct sum decomposition µk’1 = µk • (µk’1 /µk ) and thus
the quotient module µk’1 /µk is identi¬ed with the submodule in A —K A
spanned by
(δ a1 —¦ · · · —¦ δ ak ) (1 — 1), a0 , . . . , ak ∈ A.
Consequently, any a ∈ A determines the homomorphism
δ a : µk’2 /µk’1 ’ µk’1 /µk
by
δ a : a — a ’ aa — a ’ a — aa .
But one has δ ab = aδ b + bδ a and hence δ : a ’ δ a is an element of the
module D1 (homA (µk’2 /µk’1 , µk’1 /µk )). Consider the corresponding ho-
momorphism
• = •δ ∈ homA (Λ1 (A), homA (µk’2 /µk’1 , µk’1 /µk )).
Due to the canonical isomorphism
homA (Λ1 (A), homA (µk’2 /µk’1 , µk’1 /µk ))
homA (Λ1 (A) —A µk’2 /µk’1 , µk’1 /µk ),
we obtain the mapping
• : Λ1 (A) —A µk’2 /µk’1 ’ µk’1 /µk ,
and repeating the procedure, get eventually the mapping • : T k ’ µk’1 /µk .
Due to the identity δa —¦δb = δb —¦δa , this mapping induces the homomorphism
•S : S k (Λ1 (A)) ’ µk’1 /µk which, in terms of generators, acts as
•S (da1 · · · · · dak ) = (δ a1 —¦ . . . —¦ δ ak ) (1 — 1)
and thus is epimorphic.
Consider the dual monomorphism
•— : µk’1 /µk = Diff k (A)/ Diff k’1 (A) ’ (S k (Λ1 (A)))— = S k (D1 (A)).
S
Let σ ∈ Diff k (A)/ Diff k’1 (A) and ∆ ∈ Diff k (A) be a representative of the
class σ. Then
(•— (σ))(da1 · · · · · dak ) = (δa1 —¦ · · · —¦ δak ) (∆).
S
But, on the other hand, it is not di¬cult to see that the mapping
1
•— : X1 · . . . Xk ’ [X1 —¦ . . . Xk ],
¯S
k!
•— : S k (D1 (A)) ’ Diff k (A)/ Diff k’1 (A), where [∆] denotes the coset of
¯S
the operator ∆ ∈ Diff k (A) in the quotient module Diff k (A)/ Diff k’1 (A), is
inverse to •— . Thus, •— is an isomorphism. Then the mapping
S S

µk’1 /µk ’ (µk’1 /µk )—— S k (Λ1 (A)),
where the ¬rst arrow is the natural homomorphism, is the inverse to •S .
170 4. BRACKETS

S k (Λ1 (A)) and we have
From the above said it follows that µk’1 /µk
the exact sequence
0 ’ S k (Λ1 (A)) ’ J k (A) ’ J k’1 (A) ’ 0.
But, by the induction assumption, J k’1 (A) is a projective module isomor-
phic to i¤k’1 S i (Λ1 (A)). Hence,
J k (A) S k (Λ1 (A)) • J k’1 (A) S i (Λ1 (A))
i¤k
which ¬nishes the proof.
def
Definition 4.8. Let P be an A-module. The module Smbl— (P ) =
k≥0 Smblk (P ), where
def
Smblk (P ) = Diff k (P )/ Diff k’1 (P ),
is called the module of symbols for P . The coset of ∆ ∈ Diff k (P ) in Smblk (P )
is called the symbol of the operator ∆.
Let σ ∈ Smbli (A) and σ ∈ Smblj (A) and assume that ∆ ∈ Diff i (A) and
∆ ∈ Diff j (A) are representatives of σ, σ respectively. De¬ne the product
σσ as the coset of ∆ —¦ ∆ in Diff i+j (A). It is easily checked that Smbl— (A)
forms a commutative A-algebra with respect to thus de¬ned multiplication.
As a direct consequence of the last proposition and of Proposition 4.4,
we obtain
Corollary 4.17. If A is a smooth algebra, then the following state-
ments are valid :
(i) Diff k (P ) Diff k (A) —A P ,
(ii) Diff — (A), as an associative algebra, is generated by A = Diff 0 (A) and
D1 (A) ‚ Diff 1 (A),
(iii) Smblk (P ) Smblk (A) —A P ,
(iv) Smbl— (A), as a commutative algebra, is isomorphic to the symmetric
tensor algebra of D1 (A).
Remark 4.5. It should be noted that Smbl— A is more than just a com-
mutative algebra. In fact, in the case A = C ∞ (M ), as it can be easily seen,
elements of Smbl— A can be naturally identi¬ed with smooth functions on
T — M polynomial along the ¬bers of the natural projection T — M ’ M . The
manifold T — M is symplectic and, in particular, the algebra C ∞ (T — M ) pos-
sesses a Poisson bracket which induces a bracket in Smbl— A ‚ C ∞ (T — M ).
This bracket, as it happens, is of a purely algebraic nature.
Let us consider two symbols σ1 ∈ Smbli1 A, σ2 ∈ Smbli2 A such that
σr = ∆r mod Diff ir ’1 A, r = 1, 2, and set
def
{σ1 , σ2 } = [∆1 , ∆2 ] mod Diff i1 +i2 ’2 . (4.21)
The operation {·, ·} de¬ned by (4.21) is called the Poisson bracket in the
algebra of symbols and in the case A = C ∞ (M ) coincides with the classical
2. NIJENHUIS BRACKET 171

Poisson bracket on the cotangent space. It possesses the usual properties,
i.e.,
{σ1 , σ2 } + {σ2 , σ1 } = 0,
{σ1 , {σ2 , σ3 }} + {σ2 , {σ3 , σ1 }} + {σ3 , {σ1 , σ2 }} = 0,
{σ1 , σ2 σ3 } = {σ1 , σ2 }σ3 + σ2 {σ1 , σ3 }
and, in particular, Smbl— A becomes a Lie K-algebra with respect to this
bracket. This is a starting point to construct Hamiltonian formalism in a
general algebraic setting. For details and generalizations see [104, 53, 54].

2. Fr¨licher“Nijenhuis bracket
o
We still consider the general algebraic setting of the previous section
and extend standard constructions of calculus to form-valued derivations.
It allows us to de¬ne Fr¨licher“Nijenhuis brackets and introduce a coho-
o
mology theory ( -cohomologies) associated to commutative algebras with
¬‚at connections. In the next chapter, applying this theory to in¬nitely
prolonged partial di¬erential equations, we obtain an algebraic and analyt-
ical description of recursion operators for symmetries and describe e¬cient
tools to compute these operators. These and related results, together with
their generalizations, were ¬rst published in the papers [55, 56, 57] and
[59, 58, 40].

2.1. Calculus in form-valued derivations. Let k be a ¬eld of char-
acteristic zero and A be a commutative unitary k-algebra. Let us recall the
basic notations:
• D(P ) is the module of P -valued derivations A ’ P , where P is an
A-module;
• Di (P ) is the module of P -valued skew-symmetric i-derivations. In
particular, D1 (P ) = D(P );
• Λi (A) is the module of di¬erential i-forms of the algebra A;
• d : Λi (A) ’ Λi+1 (A) is the de Rham di¬erential.
Recall also that the modules Λi (A) are representative objects for the
functors Di : P ’ Di (P ), i.e., Di (P ) = HomA (Λi (A), P ). The isomorphism
D(P ) = HomA (Λ1 (A), P ) can be expressed in more exact terms: for any
derivation X : A ’ P , there exists a uniquely de¬ned A-module homomor-
phism •X : Λ1 (A) ’ P satisfying the equality X = •X —¦ d. Denote by
Z, ω ∈ P the value of the derivation Z ∈ Di (P ) at ω ∈ Λi (A).
Both Λ— (A) = i≥0 Λi (A) and D— (A) = i≥0 Di (A) are endowed with
the structures of superalgebras with respect to the wedge product operations
§ : Λi (A) — Λj (A) ’ Λi+j (A),
§ : Di (A) — Dj (A) ’ Di+j (A),
the de Rham di¬erential d : Λ— (A) ’ Λ— (A) becoming a derivation of Λ— (A).
Note also that D— (P ) = i≥0 Di (P ) is a D— (A)-module.
172 4. BRACKETS

Using the paring ·, · and the wedge product, we de¬ne the inner product
(or contraction) iX ω ∈ Λj’i (A) of X ∈ Di (A) and ω ∈ Λj (A), i ¤ j, by
setting
Y, iX ω = (’1)i(j’i) X § Y, ω , (4.22)
where Y is an arbitrary element of Dj’i (P ), P being an A-module. We
formally set iX ω = 0 for i > j. When i = 1, this de¬nition coincides with
the one given in Section 1. Recall that the following duality is valid:
X, da § ω = X(a), ω , (4.23)
where ω ∈ Λi (A), X ∈ Di+1 (P ), and a ∈ A (see Corollary 4.11). Using the
property (4.23), one can show that
iX (ω § θ) = iX (ω) § θ + (’1)Xω ω § iX (ω)
for any ω, θ ∈ Λ— (A), where (as everywhere below) the symbol of a graded
object used as the exponent of (’1) denotes the degree of that object.
We now de¬ne the Lie derivative of ω ∈ Λ— (A) along X ∈ D— (A) as
LX ω = iX —¦ d ’ (’1)X d —¦ iX ω = [iX , d]ω, (4.24)
where [·, ·] denotes the graded (or super) commutator: if ∆, ∆ : Λ— (A) ’
Λ— (A) are graded derivations, then
[∆, ∆ ] = ∆ —¦ ∆ ’ (’1)∆∆ ∆ —¦ ∆.
For X ∈ D(A) this de¬nition coincides with the ordinary commutator of
derivations.
Consider now the graded module D(Λ— (A)) of Λ— (A)-valued deriva-
tions A ’ Λ— (A) (corresponding to form-valued vector ¬elds ” or, which
is the same ” vector-valued di¬erential forms on a smooth manifold).
Note that the graded structure in D(Λ— (A)) is determined by the splitting
D(Λ— (A)) = i≥0 D(Λi (A)) and thus elements of grading i are derivations
X such that im X ‚ Λi (A). We shall need three algebraic structures asso-
ciated to D(Λ— (A)).
First note that D(Λ— (A)) is a graded Λ— (A)-module: for any X ∈
D(Λ— (A)), ω ∈ Λ— (A) and a ∈ A we set (ω § X)a = ω § X(a). Second,
we can de¬ne the inner product iX ω ∈ Λi+j’1 (A) of X ∈ D(Λi (A)) and
ω ∈ Λj (A) in the following way. If j = 0, we set iX ω = 0. Then, by induc-
tion on j and using the fact that Λ— (A) as a graded A-algebra is generated
by the elements of the form da, a ∈ A, we set
iX (da § ω) = X(a) § ω ’ (’1)X da § iX (ω), a ∈ A. (4.25)
Finally, we can contract elements of D(Λ— (A)) with each other in the fol-
lowing way:
X, Y ∈ D(Λ— (A)), a ∈ A.
(iX Y )a = iX (Y a), (4.26)
Three properties of contractions are essential in the sequel.
2. NIJENHUIS BRACKET 173

Proposition 4.18. Let X, Y ∈ D(Λ— (A)) and ω, θ ∈ Λ— (A). Then
iX (ω § θ) = iX (ω) § θ + (’1)ω(X’1) ω § iX (θ), (4.27)
iX (ω § Y ) = iX (ω) § Y + (’1)ω(X’1) ω § iX (Y ), (4.28)
[iX , iY ] = i[[X,Y ]]rn , (4.29)
where
[[X, Y ]]rn = iX (Y ) ’ (’1)(X’1)(Y ’1) iY (X). (4.30)
Proof. Equality (4.27) is a direct consequence of (4.25). To prove
(4.28), it su¬ces to use the de¬nition and expressions (4.26) and (4.27).
Let us prove (4.29) now. To do this, note ¬rst that due to (4.26), the
equality is su¬cient to be checked for elements ω ∈ Λj (A). Let us use
induction on j. For j = 0 it holds in a trivial way. Let a ∈ A; then one has
[iX , iY ](da § ω) = iX —¦ iY ’ (’1)(X’1)(Y ’1) iY —¦ iX (da § ω)
= iX (iY (da § ω)) ’ (’1)(X’1)(Y ’1) iY (iX (da § ω)).
But
iX (iY (da § ω)) = iX (Y (a) § ω ’ (’1)Y da § iY ω)
= iX (Y (a)) § ω + (’1)(X’1)Y Y (a) § iX ω ’ (’1)Y (X(a) § iY ω
’ (’1)X da § iX (iY ω)),
while
iY (iX (da § ω) = iY (X(a) § ω ’ (’1)X da § iX ω)
= iY (X(a)) § ω + (’1)X(Y ’1) X(a) § iY ω ’ (’1)X (Y (a) § iX ω
’ (’1)Y da § iY (iX ω)).
Hence,
[iX , iY ](da § ω) = iX (Y (a)) ’ (’1)(X’1)(Y ’1) iY (X(a)) § ω
+ (’1)X+Y da § iX (iY ω) ’ (’1)(X’1)(Y ’1) iY (iX ω) .
But, by de¬nition,
iX (Y (a)) ’ (’1)(X’1)(Y ’1) iY (X(a))
= (iX Y ’ (’1)(X’1)(Y ’1) iY X)(a) = [[X, Y ]]rn (a),
whereas
iX (iY ω) ’ (’1)(X’1)(Y ’1) iY (iX ω) = i[[X,Y ]]rn (ω)
by induction hypothesis.
Note also that the following identity is valid for any X, Y, Z ∈ D(Λ— (A)):
Z + (’1)X (X § Y )
X (Y Z) = (X Y) Z. (4.31)
174 4. BRACKETS

Definition 4.9. The element [[X, Y ]]rn de¬ned by (4.30) is called the
Richardson“Nijenhuis bracket of elements X and Y .
Directly from Proposition 4.18 we obtain the following
Proposition 4.19. For any derivations X, Y, Z ∈ D(Λ— (A)) and a form
ω ∈ Λ— (A) one has
[[X, Y ]]rn + (’1)(X+1)(Y +1) [[Y, X]]rn = 0, (4.32)

(’1)(Y +1)(X+Z) [[[[X, Y ]]rn , Z]]rn = 0, (4.33)

[[X, ω § Y ]]rn = iX (ω) § Y + (’1)(X+1)ω ω § [[X, Y ]]rn . (4.34)
Here and below the symbol denotes the sum of cyclic permutations.
Remark 4.6. Note that Proposition 4.19 means that D(Λ— (A))“ is a
Gerstenhaber algebra with respect to the Richardson“Nijenhuis bracket [48].
Here the superscript “ denotes the shift of grading by 1.
Similarly to (4.24), let us de¬ne the Lie derivative of ω ∈ Λ— (A) along
X ∈ D(Λ— (A)) by
LX ω = (iX —¦ d ’ (’1)X’1 d —¦ iX )ω = [iX , d]ω (4.35)
Remark 4.7. Let us clarify the change of sign in (4.35) with respect to
formula (4.24). If A is a commutative algebra, then the module D— (Λ— (A))
is a bigraded module: if ∆ ∈ Di (Λj (A)), then bigrading of this element is
def
(i, j). We can also consider the total grading by setting deg ∆ = i + j. In
this sense, if X ∈ Di (A), then deg X = i, and for X ∈ D1 (Λj (A)), then
deg X = j + 1. This also explains shift of grading in Remark 4.6.
From the properties of iX and d we obtain
Proposition 4.20. For any X ∈ D(Λ— (A)) and ω, θ ∈ Λ— (A), one has
the following identities:
LX (ω § θ) = LX (ω) § θ + (’1)Xω ω § LX (θ), (4.36)
Lω§X = ω § LX + (’1)ω+X d(ω) § iX , (4.37)
[LX , d] = 0. (4.38)
Our main concern now is to analyze the commutator [LX , LY ] of two Lie
derivatives. It may be done e¬ciently for smooth algebras (see De¬nition
4.7).
Proposition 4.21. Let A be a smooth algebra. Then for any derivations
X, Y ∈ D(Λ— (A)) there exists a uniquely determined element [[X, Y ]]fn ∈
D(Λ— (A)) such that
[LX , LY ] = L[[X,Y ]]fn . (4.39)
2. NIJENHUIS BRACKET 175

Proof. To prove existence, recall that for smooth algebras one has
Di (P ) = HomA (Λi (A), P ) = P —A HomA (Λi (A), A) = P —A Di (A)
for any A-module P and integer i ≥ 0. Using this identi¬cation, let us
represent elements X, Y ∈ D(Λ— (A)) in the form
X = ω — X and Y = θ — Y for ω, θ ∈ Λ— (A), X , Y ∈ D(A).
Then it is easily checked that the element
Z = ω § θ — [X , Y ] + ω § LX θ — Y + (’1)ω dω § iX θ — Y
’ (’1)ωθ θ § LY ω — X ’ (’1)(ω+1)θ dθ § iY ω — X
= ω § θ — [X , Y ] + LX θ — Y ’ (’1)ωθ LY ω — X (4.40)
satis¬es (4.39).
Uniqueness follows from the fact that LX (a) = X(a) for any a ∈ A.
Definition 4.10. The element [[X, Y ]]fn ∈ Di+j (Λ— (A)) de¬ned by for-
mula (4.39) (or by (4.40)) is called the Fr¨licher“Nijenhuis bracket of form-
o
i (Λ— (A)) and Y ∈ D j (Λ— (A)).
valued derivations X ∈ D
The basic properties of this bracket are summarized in the following
Proposition 4.22. Let A be a smooth algebra, X, Y, Z ∈ D(Λ— (A)) be
derivations and ω ∈ Λ— (A) be a di¬erential form. Then the following iden-
tities are valid :
[[X, Y ]]fn + (’1)XY [[Y, X]]fn = 0, (4.41)

(’1)Y (X+Z) [[X, [[Y, Z]]fn ]]fn = 0, (4.42)

i[[X,Y ]]fn = [LX , iY ] + (’1)X(Y +1) LiY X , (4.43)
iZ [[X, Y ]]fn = [[iZ X, Y ]]fn + (’1)X(Z+1) [[X, iZ Y ]]fn
(4.44)
+ (’1)X i[[Z,X]]fn Y ’ (’1)(X+1)Y i[[Z,Y ]]fn X,
[[X, ω § Y ]]fn = LX ω § Y ’ (’1)(X+1)(Y +ω) dω § iY X
(4.45)
+ (’1)Xω ω § [[X, Y ]]fn .
Note that the ¬rst two equalities in the previous proposition mean that
the module D(Λ— (A)) is a Lie superalgebra with respect to the Fr¨licher“
o
Nijenhuis bracket.
Remark 4.8. The above exposed algebraic scheme has a geometrical
realization, if one takes A = C ∞ (M ), M being a smooth ¬nite-dimensional
manifold. The algebra A = C ∞ (M ) is smooth in this case. However,
in the geometrical theory of di¬erential equations we have to work with
in¬nite-dimensional manifolds3 of the form N = proj lim{πk+1,k } Nk , where
3
In¬nite jets, in¬nite prolongations of di¬erential equations, total spaces of coverings,
etc.
176 4. BRACKETS

all the mappings πk+1,k : Nk+1 ’ Nk are surjections of ¬nite-dimensional
smooth manifolds. The corresponding algebraic object is a ¬ltered algebra
A = k∈Z Ak , Ak ‚ Ak+1 , where all Ak are subalgebras in A. As it was al-
ready noted, self-contained di¬erential calculus over A is constructed, if one
considers the category of all ¬ltered A-modules with ¬ltered homomorphisms
for morphisms between them. Then all functors of di¬erential calculus in
this category become ¬ltered, as well as their representative objects.
In particular, the A-modules Λi (A) are ¬ltered by Ak -modules Λi (Ak ).
We say that the algebra A is ¬nitely smooth, if Λ1 (Ak ) is a projective Ak -
module of ¬nite type for any k ∈ Z. For ¬nitely smooth algebras, elements
of D(P ) may be represented as formal in¬nite sums k pk — Xk , such that
any ¬nite sum Sn = k¤n pk — Xk is a derivation An ’ Pn+s for some ¬xed
s ∈ Z. Any derivation X is completely determined by the system {Sn } and
Proposition 4.22 obviously remains valid.
Remark 4.9. In fact, the Fr¨licher“Nijenhuis bracket can be de¬ned in
o
a completely general situation, with no additional assumption on the algebra
A. To do this, it su¬ces to de¬ne [[X, Y ]]fn = [X, Y ], when X, Y ∈ D1 (A)
and then use equality (4.44) as inductive de¬nition. Gaining in generality,
we then loose of course in simplicity of proofs.
2.2. Algebras with ¬‚at connections and cohomology. We now
introduce the second object of our interest. Let A be an k-algebra, k being
a ¬eld of zero characteristic, and B be an algebra over A. We shall assume
that the corresponding homomorphism • : A ’ B is an embedding. Let P
be a B-module; then it is an A-module as well and we can consider the B-
module D(A, P ) of P -valued derivations A ’ P .
Definition 4.11. Let • : D(A, •) ’ D(•) be a natural transforma-
tions of the functors D(A, •) : A ’ D(A, P ) and D(•) : P ’ D(P ) in the
category of B-modules, i.e., a system of homomorphisms P : D(A, P ) ’
D(P ) such that the diagram
P
’ D(P )
D(A, P )


D(A, f ) D(f )
“ “
Q
’ D(Q)
D(A, Q)
is commutative for any B-homomorphism f : P ’ Q. We say that • is a
connection in the triad (A, B, •), if P (X) A = X for any X ∈ D(A, P ).
Here and below we use the notation Y |A = Y —¦ • for any derivation
Y ∈ D(P ).
Remark 4.10. When A = C ∞ (M ), B = C ∞ (E), • = π — , where M and
E are smooth manifolds and π : E ’ M is a smooth ¬ber bundle, De¬nition
2. NIJENHUIS BRACKET 177

4.11 reduces to the ordinary de¬nition of a connection in the bundle π. In
fact, if we have a connection • in the sense of De¬nition 4.11, then the
correspondence
B
D(A) ’ D(A, B) ’ ’ D(B)

allows one to lift any vector ¬eld on M up to a π-projectable ¬eld on E.
Conversely, if is such a correspondence, then we can construct a natural
transformation • of the functors D(A, •) and D(•) due to the fact that
for smooth ¬nite-dimensional manifolds one has D(A, P ) = P —A D(A) and
D(P ) = P —B D(P ) for an arbitrary B-module P . We use the notation
= B in the sequel.
Definition 4.12. Let • be a connection in (A, B, •) and consider two
derivations X, Y ∈ D(A, B). The curvature form of the connection • on
the pair X, Y is de¬ned by
R (X, Y ) = [ (X), (Y )] ’ ( (X) —¦ Y ’ (Y ) —¦ X). (4.46)
(X) —¦ Y ’ (Y ) —¦ X is a B-valued
Note that (4.46) makes sense, since
derivation of A.
Consider now the de Rham di¬erential d = dB : B ’ Λ1 (B). Then the
composition dB —¦ • : A ’ B is a derivation. Consequently, we may consider
the derivation (dB —¦ •) ∈ D(Λ1 (B)).
Definition 4.13. The element U ∈ D(Λ1 (B)) de¬ned by
(dB —¦ •) ’ dB
U= (4.47)
is called the connection form of .
Directly from the de¬nition we obtain the following
Lemma 4.23. The equality
iX (U ) = X ’ ( X|A ) (4.48)
holds for any X ∈ D(B).
Using this result, we now prove
Proposition 4.24. If B is a smooth algebra, then
iY iX [[U , U ]]fn = 2R ( X|A , Y |A ) (4.49)
for any X, Y ∈ D(B).
Proof. First note that deg U = 1. Then using (4.44) and (4.41) we
obtain

iX [[U , U ]]fn = [[iX U , U ]]fn + [[U , iX U ]]fn ’ i[[X,U ’ i[[X,U
]]fn U ]]fn U
fn
= 2 [[iX U , U ]] ’ i[[X,U ]]fn U .
178 4. BRACKETS

Applying iY to the last expression and using (4.42) and (4.44), we get now
iY iX [[U , U ]]fn = 2 [[iX U , iY U ]]fn ’ i[[X,Y ]]fn U .

But [[V, W ]]fn = [V, W ] for any V, W ∈ D(Λ0 (A)) = D(A). Hence, by (4.48),
we have
iY iX [[U , U ]]fn = 2 [X ’ ( X|A ), Y ’ ( Y |A )] ’ ([X, Y ] ’ ([X, Y ]|A )) .
( X|A )|A = X|A and [X, Y ]|A = X —¦
It only remains to note now that
Y |A ’ Y —¦ X|A .
Definition 4.14. A connection in (A, B, •) is called ¬‚at, if R = 0.


Fix an algebra A and let us introduce the category FC(A), whose objects
are triples (A, B, •) endowed with a connection • while morphisms are
de¬ned as follows. Let O = (A, B, •, • ) and O = (A, B, •, • ) be two
objects of FC(A). Then a morphism from O to O is a mapping f : B ’ B
such that:
(i) f is an A-algebra homomorphism, i.e., the diagram
f
’B
B
















A
is commutative, and
(ii) for any B-module P (which can be considered as a B-module as well
due to the homomorphism f the diagram
D(B, f )
’ D(B, P )
D(B, P )






P
P




D(A, P )
is commutative, where D(B, f )(X) = X —¦ f for any derivation
X : B ’ P.
Due to Proposition 4.24, for ¬‚at connections we have
[[U , U ]]fn = 0. (4.50)
Let U ∈ D(Λ1 (B)) be an element satisfying equation (4.50). Then from
the graded Jacobi identity (4.42) we obtain
2[[U, [[U, X]]fn ]]fn = [[[[U, U ]]fn , X]]fn = 0
2. NIJENHUIS BRACKET 179

for any X ∈ D(Λ— (A)). Consequently, the operator
‚U = [[U, ·]]fn : D(Λi (B)) ’ D(Λi+1 (B))
de¬ned by the equality ‚U (X) = [[U, X]]fn satis¬es the identity ‚U —¦ ‚U = 0.
Consider now the case U = U , where is a ¬‚at connection.
Definition 4.15. An element X ∈ D(Λ— (B)) is called vertical, if
X(a) = 0 for any a ∈ A. Denote the B-submodule of such elements by
Dv (Λ— (B)).
Lemma 4.25. Let be a connection in (A, B, •). Then
(1) an element X ∈ D(Λ— (B)) is vertical if and only if iX U = X;
(2) the connection form U is vertical, U ∈ Dv (Λ1 (B));
(3) the mapping ‚U preserves verticality, i.e., for all i one has the em-
beddings ‚U (Dv (Λi (B))) ‚ D v (Λi+1 (B)).
Proof. To prove (1), use Lemma 4.23: from (4.48) it follows that
iX U = X if and only if ( X|A ) = 0. But ( X|A )|A = X|A . The second
statements follows from the same lemma and from the ¬rst one:
iU U = U ’ ( U |A ) = U ’ (U ’ ( U |A ))|A = U .
Finally, (3) is a consequence of (4.44).
Definition 4.16. Denote the restriction ‚U |Dv (Λ— (A)) by ‚ and call
the complex
‚ ‚
0 ’ D v (B) ’’ D v (Λ1 (B)) ’ · · · ’ D v (Λi (B)) ’’ D v (Λi+1 (B)) ’ · · ·
(4.51)
the -complex of the triple (A, B, •). The corresponding cohomology is de-
noted by H — (B; A, •) = i≥0 H i (B; A, •) and is called the -cohomology
of the triple (A, B, •).
Introduce the notation
dv = LU : Λi (B) ’ Λi+1 (B). (4.52)
Proposition 4.26. Let be a ¬‚at connection in a triple (A, B, •) and
B be a smooth (or ¬nitely smooth) algebra. Then for any X, Y ∈ D v (Λ— (A))
and ω ∈ Λ— (A) one has
‚ [[X, Y ]]fn = [[‚ X, Y ]]fn + (’1)X [[X, ‚ Y ]]fn , (4.53)
[iX , ‚ ] = (’1)X i‚ X, (4.54)
‚ (ω § X) = (dv ’ d)(ω) § X + (’1)ω ω § ‚ X, (4.55)
[dv , iX ] = i‚ + (’1)X LX . (4.56)
X

Proof. Equality (4.53) is a direct consequence of (4.42). Equality
(4.54) follows from (4.44). Equality (4.55) follows from (4.45) and (4.48).
Finally, (4.56) is obtained from (4.43).
180 4. BRACKETS

Corollary 4.27. The module H — (B; A, •) inherits the graded Lie al-
gebra structure with respect to the Fr¨licher“Nijenhuis bracket [[·, ·]]fn , as well
o
as the contraction operation.
Proof. Note that D v (Λ— (A)) is closed with respect to the Fr¨licher“
o
Nijenhuis bracket: to prove this fact, it su¬ces to apply (4.44). Then the
¬rst statement follows from (4.53). The second one is a consequence of
(4.54).
Remark 4.11. We preserve the same notations for the inherited struc-
tures. Note, in particular, that H 0 (B; A, •) is a Lie algebra with respect to
the Fr¨licher“Nijenhuis bracket (which reduces to the ordinary Lie bracket
o
in this case). Moreover, H 1 (B; A, •) is an associative algebra with respect
to the inherited contraction, while the action
X ∈ H 0 (B; A, •), „¦ ∈ H 1 (B; A, •)
R„¦ : X ’ iX „¦,
is a representation of this algebra as endomorphisms of H 0 (B; A, •).
Consider now the mapping dv : Λ— (B) ’ Λ— (B) de¬ned by (4.52) and
de¬ne dh = dB ’ dv .
Proposition 4.28. Let B be a (¬nitely) smooth algebra and be a ¬‚at
connection in the triple (B; A, •). Then
(1) The pair (dh , dv ) forms a bicomplex, i.e.,
dv —¦ dv = 0, dh —¦ dh = 0, dh —¦ dv + dv —¦ dh = 0. (4.57)
(2) The di¬erential dh possesses the following properties
[dh , iX ] = ’i‚ X, (4.58)
‚ (ω § X) = ’dh (ω) § X + (’1)ω ω § ‚ X, (4.59)
where ω ∈ Λ— (B), X ∈ D v (Λ— (B)).
Proof. (1) Since deg dv = 1, we have
2dv —¦ dv = [dv , dv ] = [LU , LU ] = L[[U = 0.
,U ]]fn
Since dv = LU , the identity [dB , dv ] = 0 holds (see (4.38)), and it concludes
the proof of the ¬rst part.
(2) To prove (4.58), note that
[dh , iX ] = [dB ’ dh , iX ] = (’1)X LX ’ [dv , iX ],
and (4.58) holds due to (4.56). Finally, (4.59) is just the other form of
(4.55).
Definition 4.17. Let be a connection in (A, B, •).
(1) The bicomplex (B, dh , dv ) is called the variational bicomplex associ-
ated to the connection .
(2) The corresponding spectral sequence is called the -spectral sequence
of the triple (A, B, •).
3. STRUCTURE OF SYMMETRY ALGEBRAS 181

Obviously, the -spectral sequence converges to the de Rham cohomology
of B.
To ¬nish this section, note the following. Since the module Λ1 (B) is
generated by the image of the operator dB : B ’ Λ1 (B) while the graded
algebra Λ— (B) is generated by Λ1 (B), we have the direct sum decomposition

Λp (B) — Λq (B),
Λ— (B) = v h
i≥0 p+q=i

where

Λq (B) = Λ1 (B) § · · · § Λ1 (B),
Λp (B) = Λ1 (B) § · · · § Λ1 (B),
v v v h h
h
p times q times

while the submodules Λ1 (B) ‚ Λ1 (B), Λ1 (B) ‚ Λ1 (B) are spanned in
v h
1 (B) by the images of the di¬erentials dv and dh respectively. Obviously,
Λ
we have the following embeddings:

dh Λp (B) — Λq (B) ‚ Λp (B) — Λq+1 (B),
v v
h h
dv Λp (B) — Λq (B) ‚ Λp+1 (B) — Λq (B).
v v
h h

Denote by D p,q (B) the module D v (Λp (B) — Λq (B)). Then, obviously,
v h
v (B) = p,q (B), while from equalities (4.58) and (4.59) we
D D
i≥0 p+q=i
obtain

Dp,q (B) ‚ Dp,q+1 (B).


Consequently, the module H — (B; A, •) is split as

H p,q (B; A, •)
H — (B; A, •) = (4.60)
i≥0 p+q=i

with the obvious meaning of the notation H p,q (B; A, •).

Proposition 4.29. If O = (B, ) is an object of the category FC(A),
then

H p,0 (B) = ker ‚ .
v
D1 (C p Λ(B))



3. Structure of symmetry algebras
Here we expose the theory of symmetries and recursion operators in the
categories FC(A). Detailed motivations for the de¬nition can be found in
previous chapters as well as in Chapter 5. A brief discussion concerning rela-
tions of this algebraic scheme to further applications to di¬erential equations
the reader will ¬nd in concluding remarks below.
182 4. BRACKETS

3.1. Recursion operators and structure of symmetry algebras.
We start with the following
Definition 4.18. Let O = (B, ) be an object of the category FC(A).
(i) The elements of H 0,0 (B) = H 0 (B) are called symmetries of O.
(ii) The elements of H 1,0 (B) are called recursion operators of O.
We use the notations
def
Sym = H 0,0 (B)
and
def
Rec = H 1,0 (B).
From Corollary 4.27 and Proposition 4.29 one obtains
Theorem 4.30. For any object O = (B, ) of the category FC(A) the
following facts take place:
(i) Sym is a Lie algebra with respect to commutator of derivations.
(ii) Rec is an associative algebra with respect to contraction, U being the
unit of this algebra.
(iii) The mapping R : Rec ’ Endk (Sym), where
R„¦ (X) = iX („¦), „¦ ∈ Rec, X ∈ Sym,
is a representation of this algebra and hence
(iv) i(Sym) (Rec) ‚ Sym .
In what follows we shall need a simple consequence of basic de¬nitions:
Proposition 4.31. For any object O = (B, ) of FC(A)
[[ Sym, Rec]] ‚ Rec
and
[[Rec, Rec]] ‚ H 2,0 (B).
Corollary 4.32. If H 2,0 (B) = 0, then all recursion operators of the
object O = (B, ) commute with each other with respect to the Fr¨licher“
o
Nijenhuis bracket.
We call the objects satisfying the conditions of the previous corollary
2-trivial. To simplify notations we denote
R„¦ (X) = „¦(X), „¦ ∈ Rec, X ∈ Sym .
From Proposition 4.31 and equality (4.42) one gets
Proposition 4.33. Consider an object O = (B, ) of FC(A) and let
X, Y ∈ Sym, „¦, θ ∈ Rec. Then
[[„¦, θ]](X, Y ) = [„¦(X), θ(Y )] + [θ(X), „¦(Y )] ’ „¦([θ(X), Y ]
+ [X, θ(Y )]) ’ θ([„¦(X), Y ] + [X, „¦(Y )]) + („¦ —¦ θ + θ —¦ „¦) [X, Y ].
3. STRUCTURE OF SYMMETRY ALGEBRAS 183

In particular, for „¦ = θ one has
1
[[„¦, „¦]](X, Y ) = [„¦(X), „¦(Y )]
2
’ „¦([„¦(X), Y ]) ’ „¦([X, „¦(Y )]) + „¦(„¦([X, Y ])). (4.61)
The proof of this statement is similar to that of Proposition 4.24. The
right-hand side of (4.61) is called the Nijenhuis torsion of „¦ (cf. [49]).
Corollary 4.34. If O is a 2-trivial object, then
[„¦(X), „¦(Y )] = „¦ ([„¦(X), Y ] + [X, „¦(Y )] ’ „¦[X, Y ]) . (4.62)
Choose a recursion operator „¦ ∈ Rec and for any symmetry X ∈ Sym
denote „¦i (X) = Ri (X) by Xi . Then (4.62) can be rewritten as
„¦
[X1 , Y1 ] = [X1 , Y ]1 + [X, Y1 ]1 ’ [X, Y ]2 . (4.63)
Using (4.63) as the induction base, one can prove the following
Proposition 4.35. For any 2-trivial object O and m, n ≥ 1 one has
[Xm , Yn ] = [Xm , Y ]n + [X, Yn ]m ’ [X, Y ]m+n .
Let, as before, X be a symmetry and „¦ be a recursion operator. Then
def
„¦X = [[X, „¦]] is a recursion operator again (Proposition 4.31). Due to
(4.42), its action on Y ∈ Sym can be expressed as
„¦X (Y ) = [X, „¦(Y )] ’ „¦[X, Y ]. (4.64)
From (4.64) one has
Proposition 4.36. For any 2-trivial object O, symmetries X, Y ∈ Sym,
a recursion „¦ ∈ Rec, and integers m, n ≥ 1 one has
n’1
[X, Yn ] = [X, Y ]n + („¦X Yi )n’i’1
i=0
and
m’1
[Xm , Y ] = [X, Y ]m ’ („¦Y Xj )m’j’1 .
j=0

From the last two results one obtains
Theorem 4.37 (the structure of a Lie algebra for Sym). For any 2-
trivial object O, its symmetries X, Y ∈ Sym, a recursion operator „¦ ∈ Rec,
and integers m, n ≥ 1 one has
n’1 m’1
(„¦X Yi )m+n’i’1 ’
[Xm , Yn ] = [X, Y ]m+n + („¦Y Xj )m+n’j’1 .
i=0 j=0

Corollary 4.38. If X, Y ∈ Sym are such that „¦X and „¦Y commute
with „¦ ∈ Rec with respect to the Richardson“Nijenhuis bracket, then
[Xm , Yn ] = [X, Y ]m+n + n(„¦X Y )m+n’1 ’ m(„¦Y X)m+n’1 .
184 4. BRACKETS

We say that a recursion operator „¦ ∈ Rec is X-invariant, if „¦X = 0.
Corollary 4.39 (on in¬nite series of commuting symmetries). If O is
a 2-trivial object and if a recursion operator „¦ ∈ Rec is X-invariant,
X ∈ Sym, then a hierarchy {Xn }, n = 0, 1, . . . , generated by X and „¦
is commutative:
[Xm , Xn ] = 0
for all m, n.
3.2. Concluding remarks. Here we brie¬‚y discuss relations of the
above exposed algebraic scheme to geometry of partial di¬erential equations
exposed in the previous chapters and the theory of recursion operators dis-
cussed in Chapters 5“7.
First recall that correspondence between algebraic approach and geo-
metrical picture is established by identifying the category of vector bun-
dles over a smooth manifold M with the category of geometrical mod-
ules over A = C ∞ (M ), see [60]. In the case of di¬erential equations, M
plays the role of the manifold of independent variables while B = ± B±
is the function algebra on the in¬nite prolongation of the equation E and
B± = C ∞ (E ± ), where E ± , ± = 0, 1, . . . , ∞, is the ±-prolongation of E. The
mapping • : A ’ B is dual to the natural projection π∞ : E ∞ ’ M and
thus in applications to di¬erential equations it su¬ces to consider the case
A = ± B± .
If E is a formally integrable equation, the bundle π∞ : E ∞ ’ M pos-
sesses a natural connection (the Cartan connection C) which takes a vector
¬eld X on M to corresponding total derivative on E ∞ . Consequently, the
category of di¬erential equations [100] is embedded to the category of alge-
bras with ¬‚at connections FC(C ∞ (M )). Under this identi¬cation the spec-
tral sequence de¬ned in De¬nition 4.17 coincides with A. Vinogradov™s C-
spectral sequence [102] (or variational bicomplex), the module Sym, where
O = (C ∞ (M ), C ∞ (E ∞ ), C), is the Lie algebra of higher symmetries for the
equation E and, in principle, Rec consists of recursion operators for these
symmetries. This last statement should be clari¬ed.
In fact, as we shall see later, if one tries to compute the algebra Rec
straightforwardly, the results will be trivial usually ” even for equations
which really possess recursion operators. The reason lies in nonlocal char-
acter of recursion operators for majority of interesting equations [1, 31, 4].
Thus extension of the algebra C ∞ (E ∞ ) with nonlocal variables (see 3) is the
way to obtain nontrivial solutions ” and actual computation show that all
known (as well as new ones!) recursion operators can be obtained in such
a way (see examples below and in [58, 40]). In practice, it usually su¬ces
to extend C ∞ (E ∞ ) by integrals of conservation laws (of a su¬ciently high
order).
The algorithm of computations becomes rather simple due to the follow-
ing fact. It will shown that for non-overdetermined equations all cohomology
3. STRUCTURE OF SYMMETRY ALGEBRAS 185

p,q
groups HC (E) are trivial except for the cases q = 0, 1 while the di¬erential
‚C : D1 (C p (E)) ’ D1 (C p (E) § Λh (E)) coincides with the universal lineariza-
v v
1
tion operator E of the equation E extended to the module of Cartan forms.
p,0
Therefore, the modules HC (E) coincide with ker( E ) (see 4.29)
p,0
HC (E) = ker( E ) (4.65)
and thus can be computed e¬ciently.
In particular, it will shown that for scalar evolution equations all coho-
p,0
mologies HC (E), p ≥ 2, vanish and consequently equations of this type are
2-trivial and satisfy the conditions of Theorem 4.37 which explains commu-
tativity of some series of higher symmetries (e.g., for the KdV equation).
186 4. BRACKETS
CHAPTER 5


Deformations and recursion operators

In this chapter, we apply the algebraic formalism of Chapter 4 to the
speci¬c case of partial di¬erential equations. Namely, we consider a formally
integrable equation E ‚ J k (π), π : E ’ M , taking the associated triple
(C ∞ (M ), F(E), π∞ ) for the algebra with ¬‚at connection, where F(E) =

∞ ∞ ’ M is the
k Fk (E) is the algebra of smooth functions on E , π∞ : E
natural projection and the Cartan connection C plays the role of .
We compute the corresponding cohomology groups for the case E ∞ =
J ∞ (π) and deduce de¬ning equations for a general E. We also establish
relations between in¬nitesimal deformations of the equation structure and
recursion operators for symmetries and consider several illustrative exam-
ples.
We start with repeating some de¬nitions and proofs of the previous
chapter in the geometrical situation.


1. C-cohomologies of partial di¬erential equations
Here we introduce cohomological invariants of partial di¬erential equa-
tions based on the results of Sections 1, 2 of Chapter 4. We call these in-
variants C-cohomologies since they are determined by the Cartan connection
C on E ∞ . We follow the scheme from the classical paper by Nijenhuis and
Richardson [78], especially in interpretation of the cohomology in question.
Let ξ : P ’ M be a ¬ber bundle with a connection , which is considered
as a C ∞ (M )-homomorphism : D(M ) ’ D(P ) taking a ¬eld X ∈ D(M ) to
the ¬eld (X) = X ∈ D(P ) and satisfying the condition X (ξ — f ) = X(f )
for any f ∈ C ∞ (M ).
Let y ∈ P , ξ(y) = x ∈ M , and denote by Py = ξ ’1 (x) the ¬ber of
the projection ξ passing through y. Then determines a linear mapping
y : Tx (M ) ’ Ty (P ) such that ξ—,y ( y (v)) = v for any v ∈ Tx (M ). Thus
with any point y ∈ P a linear subspace y (Tx (M )) ‚ Ty (P ) is associated. It
determines a distribution D on P which is called the horizontal distribution
of the connection . If is ¬‚at, then D is integrable.
As it is well known (see, for example, [46, 47]), the connection form
U = U ∈ Λ1 (P ) — D(P ) can be de¬ned as follows. Let y ∈ P , Y ∈ D(P ),
Yy ∈ Ty (P ) and v = ξ—,y (Yy ). Then we set

U )y = Y y ’
(Y y (v). (5.1)

187
188 5. DEFORMATIONS AND RECURSION OPERATORS

In other words, the value of U at the vector Yy ∈ Ty (P ) is the projection of
Yy onto the tangent plane Ty (P ) along the horizontal plane1 passing through
y ∈ P.
If (x1 , . . . , xn ) are local coordinates in M and (y 1 , . . . , y s ) are coordinates
along the ¬ber of ξ (the case s = ∞ is included), we can de¬ne by the
following equalities
s
‚ ‚ ‚
j
= + = i. (5.2)
i
‚y j
‚xi ‚xi
j=1

Then U is of the form
s n

j
j
dy ’ dxi —
U= , (5.3)
i
‚y j
j=1 i=1

From equality (4.40) on p. 175 it follows that
‚k ‚k ‚
j j
fn ±
dxi § dxj — k .
[[U , U ]] = 2 + (5.4)
i
‚y ±
‚xi ‚y
±
i,j,k

Recall that the curvature form R of the connection is de¬ned by the
equality
’ X, Y ∈ D(M ).
R (X, Y ) = [ X, Y] [X,Y ] ,

We shall express the element [[U , U ]]fn in terms of the form R now
(cf. Proposition 4.24). Let us consider a ¬eld X ∈ D(P ) and represent it in
the form
X = X v + X h, (5.5)
where, by de¬nition,
Xv = X Xh = X ’ Xv
U,
are vertical and horizontal components of X respectively. In the same
manner one can de¬ne vertical and horizontal components of any element
„¦ ∈ Λ— (P ) — D(P ).
Obviously, X v ∈ Dv (P ), where
Dv (P ) = {X ∈ D(P ) | Xξ — (f ) = 0, f ∈ C ∞ (M )},
while the component X h is of the form

fi ∈ C ∞ (P ), Xi ∈ D(M ),
Xh = fi Xi ,
i

and lies in the distribution D .

1
With respect to .
1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 189

Proposition 5.1. Let : D(M ) ’ D(P ) be a connection in the ¬ber
bundle ξ : P ’ M . Then for any ξ-vertical vector ¬eld X v one has
[[U , U ]]fn = 0.
Xv
fi , gj ∈ C ∞ (P ), Xi , Yj ∈ D(M ),
If X h = i fi Xi , Y h = gj Yj ,
j
are horizontal vector ¬elds, then
[[U , U ]]fn = 2
Yh Xh fi gj R (Xi , Yj ), (5.6)
i,j

or, to be short,
[[U , U ]]fn = 2R .
Proof. Let X ∈ D(P ). Then from equality (4.45) on p. 175 it follows
that
[[U, U ]]fn = 2([[U, X]]fn U ]]fn ),
U ’ [[U, X
X
where U = U . Hence, if X = X v is a vertical ¬eld, then
[[U, U ]]fn = 2([[U, X v ]]fn U ]]fn ) = ’2([[U, X v ]]fn )h .
Xv U ’ [[U, X v
But the left-hand side of this equality is vertical (see (5.4)) and thus vanishes.
This proves the ¬rst part of the proposition.
Let now X = X h be a horizontal vector ¬eld. Then
[[U, U ]]fn = 2[[U, X h ]]fn U = 2([[U, X h ]]fn )v .
Xh
Hence, if Y h is another horizontal ¬eld, then, by (4.31) on p. 173, one has
[[U, U ]]fn ) = 2Y h ([[U, X h ]]fn [[U, X h ]]fn )
Yh (X h U ) = 2(Y h U.
But from (4.45) (see p. 175) it follows that
[[U, X h ]]fn = [[X h , Y h ]]fn
Yh U = [X h , Y h ] U.
Therefore,

[[U, U ]]fn = 2[X h , Y h ]
Yh Xh U = 2([X h , Y h ] ’ [X h , Y h ]h )
h
’[
=2 fi gi ([ Xi , Yj ] Xi , Yj ] )
i,j

But obviously, for any f ∈ C ∞ (M )
for X = i fi and Y = gj Yj .
Xi j
one has
[ Xi , Yj ](f ) = [Xi , Yj ](f )
and, consequently,
h
[ Xi , Yj ] = [Xi ,Yj ] ,

which ¬nishes the proof.
190 5. DEFORMATIONS AND RECURSION OPERATORS

From equality (5.6) and from the considerations in the end of Section 2
of Chapter 4 it follows that if the connection in question is ¬‚at, i.e. R = 0,
then the element U determines a complex
‚0
0 ’ D(P ) ’’ Λ1 (P ) — D(P ) ’ · · ·
‚i
’ Λ (P ) — D(P ) ’’ Λi+1 (P ) — D(P ) ’ · · · ,
i
(5.7)

where ‚ = ‚ i = [[U , · ]]fn .
Remark 5.1. Horizontal vector ¬elds X h are de¬ned by the condition
X h U = 0. Denote the module of such ¬elds by D h (P ):
Dh (P ) = {X ∈ D(P ) | X U = 0}.
Then, by setting ˜ = U = U in (4.31) on p. 173, one can see that
‚ („¦ U ) = ‚ („¦) U
for any „¦ ∈ Λ— (P ) — D(P ). Hence,
‚ (Λ— (P ) — D v (P )) ‚ Λ— (P ) — D v (P )
and
‚ (Λ— (P ) — D h (P )) ‚ Λ— (P ) — D h (P ).
Considering a direct sum decomposition
Λ— (P ) — D(P ) = Λ— (P ) — D v (P ) • Λ— (P ) — D h (P )
one can see that
‚ = ‚v • ‚h ,
where
‚ v = ‚ |Λ— (P )—Dv (P ) , ‚ h = ‚ |Λ— (P )—Dh (P ) .

To proceed further let us compute 0-cohomology of the complex (5.7).
From equality (4.31) on p. 173 it follows that for any two vector ¬elds
Y, Z ∈ D(P ) the equality
‚ 0 Y + [Z, Y ]
Z U = [Z U ,Y ]
holds. Thus Y ∈ ker(‚ 0 ) if and only if
[Z, Y ] U = [Z U ,Y ]
for any Z ∈ D(P ). Using decomposition (5.5) for the ¬elds Y and Z and
substituting it into the last equation, we get that the condition Y ∈ ker(‚ 0 )
is equivalent to the system of equations
[Z v , Y h ] U = [Z v , Y h ], [Z h , Y v ] U = 0. (5.8)
1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 191

Let Y h = i fi (see above). Then from the ¬rst equality of (5.8) it
Xi
follows that
Z v (fi ) v
= fi [ Xi , Z ].
Xi
i i

But the left-hand side of this equation is a horizontal vector ¬eld while the
right-hand side is always vertical. Hence,
Z v (fi ) =0
Xi
i

for any vertical ¬eld Z v . Choosing locally independent vector ¬elds Xi , we
see that the functions fi actually lie in C ∞ (M ) (or, strictly speaking, in
ξ — (C ∞ (M )) ‚ C ∞ (P )). It means that, at least locally, Y h is of the form
Yh = X ∈ D(M ).
X,

But since X = X if and only if X = X , the ¬eld X is well de¬ned on
the whole manifold M .
On the other hand, from the second equality of (5.8) we see that Y v ∈
ker(‚ 0 ) if and only if the commutator [Z h , Y v ] is a horizontal ¬eld for any
horizontal Z h . Thus we get the following result:
Proposition 5.2. A direct sum decomposition
ker(‚ 0 ) = Dv (P ) • (D(M ))
: D(M ) ’ D(P )
(D(M )) is the image of the mapping
takes place, where
and
Dv (P ) = {Y ∈ D v (P ) | [Y, D h (P )] ‚ D h (P )}.
One can see now that D v (P ) consists of nontrivial in¬nitesimal sym-
metries of the distribution D while the elements of (D(M )) are trivial
symmetries (in the sense that the corresponding transformations slide inte-
gral manifolds of D along themselves). To skip this trivial part of ker(‚ 0 ),
note that
(i) U ∈ Λ1 (P ) — D v (P ),
and (see Remark 5.1)
(ii) ‚ i Λi (P ) — D v (P ) ‚ Λi+1 (P ) — D v (P ).
Thus we have a vertical complex
‚0
0 ’ D (P ) ’’ Λ1 (P ) — D v (P ) ’ · · ·
v

‚i
’ Λi (P ) — D v (P ) ’’ Λi+1 (P ) — D v (P ) ’ · · · ,
the i-th cohomology of which is denoted by H i (P ). From the above said it
follows that H 0 (P ) coincides with the Lie algebra of nontrivial in¬nitesimal
symmetries for the distribution D .
192 5. DEFORMATIONS AND RECURSION OPERATORS

Consider now an in¬nitely prolonged equation E ∞ ‚ J ∞ (π) and the
Cartan connection C = CE in the ¬ber bundle π∞ : E ∞ ’ M . The corre-
sponding connection form U , where = C, will be denoted by UE in this
case. Knowing the form UE , one can reconstruct the Cartan distribution on
E ∞ . Since this distribution contains all essential information about solutions
of E, one can state that UE determines the equation structure on E ∞ (see
De¬nition 2.4 in Chapter 2).
By rewriting the vertical complex de¬ned above in the case ξ = π∞ , we
get a complex
0
‚C
0 ’ D (E) ’ Λ1 (E) — D v (E) ’ · · ·
v

i
‚C
’ Λ (E) — D (E) ’ Λi+1 (E) — D v (E) ’ · · · ,
i v
’ (5.9)
where, for the sake of simplicity, Λi (E) stands for Λi (E ∞ ). The cohomologies
i
of (5.9) are denoted by HC (E) and are called C-cohomologies of the equation
E.
From the de¬nition of the Lie algebra sym(E) and from the previous
considerations we get the following
Theorem 5.3. For any formally integrable equation E one has the iso-
morphism
0
HC (E) = sym(E).
1
To obtain an interpretation of the group HC (E), consider the element
U = UE ∈ Λ1 (E) — D v (E) and its deformation U (µ), U (0) = U , where µ ∈ R
is a small parameter. It is natural to expect this deformation to satisfy the
following conditions:
(i)
U (µ) ∈ Λ1 (E) — D v (E) (verticality)
and
(ii)
[[U (µ), U (µ)]]fn = 0 (integrability). (5.10)
Let us expand U (µ) into a formal series in µ,
U (µ) = U0 + U1 µ + · · · + Ui µi + · · · , (5.11)
and substitute (5.11) into (i) and (ii). Then one can see that U1 ∈ Λ1 (E) —
Dv (E) and
[[U0 , U1 ]]fn = 0.
1 1
Since U0 = U (0) = U , it follows that U1 ∈ ker(‚E ). Thus ker(‚E ) con-
sists of all (vertical) in¬nitesimal deformations of U preserving the natural
conditions (i) and (ii).
1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 193

0 0
On the other hand, im(‚E ) consists of elements of the form ‚E (X) =
[[U, X]]fn , X ∈ Dv (E). Such elements can be viewed as in¬nitesimal defor-
mations of U originating from transformations of E ∞ which are trivial on
M (i.e., ¬ber-wise transformations of the bundle π∞ : E ∞ ’ M ). In fact,
let P be a manifold and At : P ’ P , t ∈ R, A0 = id, be a one-parameter
group of di¬eomorphisms with
d
(At ) = X ∈ D(P ).
dt t=0

Then for any ˜ ∈ Λ— (P ) — D(P ) one can consider the element At,— (L˜ )
de¬ned by means of the commutative diagram

Λ— (P ) ’ Λ— (P )

A— A— (5.12)
t t
“ “
At,— (L˜ ) —

’ Λ (P )
Λ (P )
Then, obviously, for any homogeneous element ˜ = θ — Y ∈ Λ— (P ) — D(P )
and a form ω ∈ Λ— (P ) we have

d
At,— (L˜ )(ω)
dt t=0
d
A— (θ) § A— Y A— ω + (’1)θ dA— θ § A— (Y A— ω)
= ’t ’t
t t t t
dt t=0
= X(θ) § Y (ω) + θ § [X, Y ](ω) + (’1)θ dX(θ) § (Y ω)
+ (’1)θ dθ § [X, Y ] ω = L([[X,θ—Y ]]fn ) (ω).

— Yi ∈ Λ— (P ) — D(P ) and sets
Thus, if one takes ˜ = i θi

A— (˜) = ˜(t) = A— (θi ) — A— Yi A— , (5.13)
’t
t t t
i

then
˜(t) = ˜ + [[X, ˜]]fn t + o(t).

In other words, [[X, ˜]]fn is the velocity of the transformation of ˜ with re-
spect to At . Taking P = E ∞ and ˜ = U , one can see that the elements
V = [[U, X]]fn are in¬nitesimal transformations of U arising from transfor-
mations At : E ∞ ’ E ∞ . If π∞ —¦ At = π∞ , then X ∈ D v (E) and V ∈ im(‚E ).
0

It is natural to call such deformations of U trivial.
Since, as it was pointed out above, the element U determines the struc-
ture of di¬erential equation on the manifold E ∞ we obtain the following
result.
194 5. DEFORMATIONS AND RECURSION OPERATORS

1
Theorem 5.4. The elements of HC (E) are in one-to-one correspondence
with the classes of nontrivial in¬nitesimal vertical deformations of the equa-
tion E.
Remark 5.2. One can consider deformations of UE not preserving the
verticality condition. Then classes of the corresponding in¬nitesimal defor-
mations are identi¬ed with the elements of the ¬rst cohomology module of
the complex (5.7) (for P = E ∞ and = C). The theory of such deformations
is quite interesting but lies beyond the scope of the present book.
Remark 5.3. Since the operation [[·, ·]]fn de¬ned on HC (E ∞ ) takes its
1
2 2
values in HC (E) the elements of the module HC (E) (or a part of them at least)
can be interpreted as the obstructions for the deformations of E (cf. [78]).
Local coordinate expressions for the element UE and for the di¬erentials
‚C = ‚E in the case E ∞ = J ∞ (π) look as follows.
i

Let (x1 , . . . , xn , u1 , . . . , um ) be local coordinates in J 0 (π) and pj , j =
σ
1, . . . , m, |σ| ≥ 0, be the corresponding canonical coordinates in J ∞ (π).
Then from equality (1.35) on p. 26 and (5.3) it follows that

j
ωσ —
U= , (5.14)
‚pj
σ
j,σ
j
where ωσ are the Cartan forms on J ∞ (π) given by (1.27) (see p. 18).
Consider an element ˜ = j,σ θσ — ‚/‚pj ∈ Λ— (π) — D v (π). Then, due
j
σ
to (5.14) and (4.40), p. 175, we have
n m

j j
dxi § θσ+1i ’ Di (θσ ) —
‚π (˜) = , (5.15)
‚pj
σ
i=1 j=1 |σ|≥0

where Di (θ) is the Lie derivative of the form θ ∈ Λ— (E) along the vector ¬eld
Di ∈ D(E).

As it follows from the above said, the cohomology module HC (E) inherits
from Λ— — D1 the structure of the graded Lie algebra with respect to the
Fr¨licher“Nijenhuis bracket. In the case when U = U is the connection
o
form of a connection : D(M ) ’ D(P ), additional algebraic structures
arise in the cohomology modules H — (P ) = i H i (P ) of the corresponding
vertical complex.
First of all note that for any element „¦ ∈ Λ— (P ) — D v (P ) the identity
„¦ U =„¦ (5.16)
holds. Hence, if ˜ ∈ Λ— (P ) — D v (P ) is a vertical element too, then equality
(4.31) on p. 173 acquires the form
‚ ˜ + (’1)„¦ ‚ („¦
„¦ ˜) = ‚ („¦) ˜. (5.17)
From (5.17) it follows that
ker(‚ ) ‚ ker(‚ ),
ker(‚ )
1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 195

im(‚ ) ‚ im(‚ ),
ker(‚ )
ker(‚ ) ‚ im(‚ ).
im(‚ )
Therefore, the contraction operation
: Λi (P ) — D v (P ) — Λj (P ) — D v (P ) ’ Λi+j’1 (P ) — D v (P )
induces an operation
: H i (P ) —R H j (P ) ’ H i+j’1 (P ),
which is de¬ned by posing
[„¦] [˜] = [„¦ ˜],
where [·] denotes the cohomological class of the corresponding element.
In particular, H 1 (P ) is closed with respect to the contraction operation,
and due to (4.31) this operation determines in H 1 (P ) an associative algebra
structure. Consider elements φ ∈ H 0 (P ) and ˜ ∈ H 1 (P ). Then one can
de¬ne an action of ˜ on φ by posing
˜ ∈ H 0 (P ).
R˜ (φ) = φ (5.18)
Thus we have a mapping
R : H 1 (P ) ’ EndR (H 0 (P ))
which is a homomorphism of associative algebras due to (4.31) on p. 173.
In particular, taking P = E ∞ and ξ = π∞ , we obtain the following
Proposition 5.5. For any formally integrable equation E ‚ J k (π) the
1
module HC (E) is an associative algebra with respect to the contraction opera-
0
tion . This algebra acts on HC (E) = sym(E) by means of the representation
R de¬ned by (5.18).
When (5.16) takes place, equality (4.55), see p. 179, acquires the form
‚ (ρ § „¦) = (LU ’ dρ) § „¦ + (’1)ρ ρ § ‚ („¦). (5.19)
Let us set
dh = d ’ L U (5.20)
and note that
(dh )2 = (LU )2 ’ LU —¦ d ’ d —¦ LU + d2 = ’LU —¦ d ’ d —¦ LU .
But
L„¦ —¦ d = (’1)„¦ d —¦ L„¦ (5.21)
and, therefore, (dh )2 = 0. Thus we have the di¬erential
dh : Λi (P ) ’ Λi+1 (P ), i = 0, 1, . . . ,
and the corresponding cohomologies
H h,i (P ) = ker(dh,i+1 )/im(dh,i ).
196 5. DEFORMATIONS AND RECURSION OPERATORS

From (5.19) it follows that
ker(dh ) § ker(‚ ) ‚ ker(‚ ),
im(dh ) § ker(‚ ) ‚ im(‚ ),
ker(dh ) § im(‚ ) ‚ im(‚ ),
and hence a well-de¬ned wedge product
§ : H h,i (P ) —R H j (P ) ’ H i+j (P ).
Moreover, from (5.20) and (5.21) it follows that
L„¦ —¦ dh = L[[„¦,U + (’1)„¦ dh —¦ L„¦
]]fn

for any „¦ ∈ Λ— (P ) — D v (P ). It means that by posing
ω ∈ Λ— (P )
L[„¦] [ω] = [L„¦ ω],
we get a well-de¬ned homomorphism of graded Lie algebras
L : H — (P ) ’ D gr (H h,— (P )),
where H h,— (P ) = i H h,i (P ).
If (x1 , . . . , xn , y 1 , . . . , y s ) are local coordinates in P , then an easy com-
putation shows that
dh (f ) = i (f )dxi ,
i
dh (dxi ) = 0,
j
dh (dy j ) = § dxi ,
d (5.22)
i
i

where f ∈ C ∞ (P ), i = 1, . . . , n, j = 1, . . . , s, while the coe¬cients j
i
and vector ¬elds i are given by (5.2). Obviously, the di¬erential dH is
completely de¬ned by (5.22).

2. Spectral sequences and graded evolutionary derivations
In this section, we construct three spectral sequences associated with
C-cohomologies of in¬nitely prolonged equations. One of them is used to
compute the algebra HC (π) = HC (J ∞ (π)) of the “empty” equation. The
— —

result obtained leads naturally to the notion of graded evolutionary deriva-
tions which seem to play an important role in the geometry of di¬erential
equations.
The ¬rst of spectral sequences to be de¬ned originates from a ¬ltration
in Λ— (E) — D v (E) associated with the notion of the degree of horizontality.
Namely, an element ˜ ∈ Λp (E) — D v (E) is said to be i-horizontal if
X1 (X2 . . . (Xp’i+1 ˜) . . . ) = 0
p
for any X1 , . . . , Xp’i+1 ∈ Dv (E). Denote by Hi (E) the set of all such ele-

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