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