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We study some properties of first order differential operators from an algebraic viewpoint. We show this last can be decomposed in sum of an element of a module and a derivation. From a geometric viewpoint, we give some properties on the algebra of smooth functions. The Dirac mass at a point is the best example of first order differential operators at this point. This allows to construct a basis of this set and its dual basis.

When L is a commutative graded algebra with unit 1 L over a commutative field K with characteristic zero, a K -linear map

σ : L → L

is a first order differential operator and of degree r ( r ∈ ℤ ), if σ fulfills

σ ( x ⋅ y ) = σ ( x ) ⋅ y + ( − 1 ) r ⋅ | x | x ⋅ σ ( y ) − ( − 1 ) r ⋅ | x ⋅ y | x ⋅ y ⋅ σ ( 1 L ) (1)

for homogeneous elements x , y ∈ L , where | x | denotes the degree of x . Let σ be a first order differential operator and of degree r, let σ ′ be another first order differential operator and of degree r ′ , then the bracket

[ σ , σ ′ ] : L → L

such that

[ σ , σ ′ ] = σ ∘ σ ′ − ( − 1 ) r ⋅ r ′ σ ′ ∘ σ (2)

is a first order differential operator and of degree r + r ′ .

Thus the pair ( L , σ ) is a differential algebra if L is a commutative graded algebra with unit 1 L and

σ : L → L

is a first order differential operator and of degree +1 such that σ ∘ σ = 0 .

We consider a commutative algebra A with unit 1 A over a commutative field K with characteristic zero.

We denote E an A -module. For any x ∈ E ,

L x : A → E , a ↦ a ⋅ x

denotes the multiplication by x .

The set, D K ( A , E ) , of all first order differential operators from A into E is an A -module and admits a K -Lie algebra structure.

We denote, D e r K ( A , E ) , the set of K -derivations from A into E which is an A -submodule and a K -Lie subalgebra of D K ( A , E ) . The main goal of this paper is to study some properties of first order differential operators from an algebraic viewpoint and geometric viewpoint. We also give an example of these applications in the last case at a point and we construct the basis of the set of all these maps and its dual basis.

A differential operator of order ≤ r , r ∈ ℕ , from A into E is a K -linear map

D : A → E

such that for any a ∈ A , the map

A → E , b ↦ D ( a ⋅ b ) − a ⋅ D ( b ) ,

is a differential operator of order ≤ ( r − 1 ) from A into E .

A differential operator of order zero is an A -linear map from A into E .

Proposition 1 A K -linear map

D : A → E

is a first order differential operator from A into E if and only if

D ( a ⋅ b ) = D ( a ) ⋅ b + a ⋅ D ( b ) − a ⋅ b ⋅ D ( 1 A ) (3)

for all a and b elements of A .

Proof. Assume that D : A → E is a first order differential operator from A into E , then for any a ∈ A , the map

D a : A → E , b ↦ D ( a ⋅ b ) − a ⋅ D (b)

is A -linear. Thus for any a , b ∈ A , D a ( b ) = b ⋅ D a ( 1 A ) . Therefore we have the following

D a ( b ) = b ⋅ D a (1A)

D ( a ⋅ b ) − a ⋅ D ( b ) = b ⋅ ( D ( a ⋅ 1 A ) − a ⋅ D ( 1 A ) ) = b ⋅ ( D ( a ) − a ⋅ D ( 1 A ) ) = b ⋅ D ( a ) − a ⋅ b ⋅ D (1A)

We deduce that

D ( a ⋅ b ) = D ( a ) ⋅ b + a ⋅ D ( b ) − a ⋅ b ⋅ D ( 1 A ) .

Conversely if

D ( a ⋅ b ) = D ( a ) ⋅ b + a ⋅ D ( b ) − a ⋅ b ⋅ D (1A)

for any a , b ∈ A , then we obtain

D a ( b ) = b ⋅ D a (1A)

That ends the proof.

For any x ∈ E , we can easily see that the map

L x : A → E , a ↦ a ⋅ x ,

is a first order differential operator from A into E .

Proposition 2 [

D − L D ( 1 A ) : A → E , a ↦ D ( a ) − a ⋅ D ( 1 A ) ,

is a derivation.

Proof. As D is a first order differential operator from A into E , for any a , b ∈ A , we verify that

( D − L D ( 1 A ) ) ( a ⋅ b ) − ( D − L D ( 1 A ) ) ( a ) ⋅ b − a ⋅ ( D − L D ( 1 A ) ) ( b ) = 0.

Conversely, let

D − L D ( 1 A ) : A → E , a ↦ D ( a ) − a ⋅ D (1A)

be a derivation, then by straightforward calculation do that D is a first order differential operator.

Theorem 3 The map

τ : D K ( A , E ) → E × D e r K ( A , E ) , D ↦ ( D ( 1 A ) , D − L D ( 1 A ) )

is an isomorphism of A -modules.

Proof. The map

τ : D K ( A , E ) → E × D e r K ( A , E ) , D ↦ ( D ( 1 A ) , D − L D ( 1 A ) )

is A -linear.

For any x ∈ E , d ∈ D e r K ( A , E ) , the map

τ ′ ( x , d ) : A → E , a ↦ a ⋅ x + d ( a ) ,

is K -linear and is a first order differential operator. Indeed for any a , b ∈ A ,

τ ′ ( x , d ) ( a b ) − τ ′ ( x , d ) ( a ) ⋅ b − a ⋅ τ ′ ( x , d ) ( b ) + a ⋅ b ⋅ τ ′ ( x , d ) ( 1 A ) = 0.

Then

τ ′ ( x , d ) ( a b ) = τ ′ ( x , d ) ( a ) ⋅ b + a ⋅ τ ′ ( x , d ) ( b ) − a ⋅ b ⋅ τ ′ ( x , d ) ( 1 A ) .

We also verify that the map

τ ′ : E × D e r K ( A , E ) → D K ( A , E ) , ( x , d ) ↦ τ ′ ( x , d ) ,

is A -linear. For any pair ( x , d ) , ( x ′ , d ′ ) ∈ E × D e r K ( A , E ) and for a , b ∈ A

τ ′ ( ( x , d ) + ( x ′ , d ′ ) ) ( a ) = ( τ ′ ( x , d ) + ( x ′ , d ′ ) ) ( a ) = ( τ ′ ( x , d ) + τ ′ ( x ′ , d ′ ) ) ( a ) .

We deduce that

τ ′ ( ( x , d ) + ( x ′ , d ′ ) ) = τ ′ ( x , d ) + τ ′ ( x ′ , d ′ )

and

τ ′ ( a ⋅ ( x , d ) ) ( b ) = b ⋅ ( a ⋅ x ) + a ⋅ d ( b ) = [ a ⋅ τ ′ ( ( x , d ) ) ] ( b ) .

Thus

τ ′ ( a ⋅ ( x , d ) ) = a ⋅ τ ′ ( ( x , d ) ) .

We have

( τ ′ ∘ τ ) ( D ) ( a ) = τ ′ ( D ( 1 A ) , D − L D ( 1 A ) ) ( a ) = a ⋅ D ( 1 A ) + ( D − L D ( 1 A ) ) ( a ) = D ( a ) .

So τ ′ ∘ τ = i d D K ( A , E ) for any a ∈ A and D ∈ D K ( A , E ) .

And

( τ ∘ τ ′ ) ( ( x , d ) ) ( a ) = τ [ τ ′ ( x , d ) ( a ) ] = a ⋅ x + d ( a ) = ( x , d ) ( a ) .

So τ ∘ τ ′ = i d E × D e r K ( A , E ) .

Thus the map τ is an isomorphism of A -modules.

In the following, we shall consider the A -module A × Ω K ( A ) where Ω K ( A ) is the A -module of Kähler K -differential of A and

d A / K : A → Ω K ( A ) , a ↦ d A / K (a)

is the canonical derivation [

Theorem 4 [

D A / K : A → A × Ω K ( A ) , a ↦ ( a , d A / K ( a ) ) ,

is a differential operator and the image of D A / K generates the A -module A × Ω K ( A ) . Furthermore the pair ( A × Ω K ( A ) , D A / K ) possesses the following universal property: for any A -module E and for any differential operator

φ : A → E ,

then there exists one and only one A -linear map

φ ˜ : A × Ω K ( A ) → E

such that

φ ˜ ∘ D A / K = φ . (4)

The map

H o m A ( A × Ω K ( A ) , E ) → D K ( A , E ) , ψ ↦ ψ ∘ D A / K ,

is an isomorphism of A -modules [

We recall that an alternating p-differential operator from A into E is an alternating K -multilinear map

φ : A p → E

such that for all p elements a 1 , a 2 , ⋯ , a p ∈ A , the map

φ ( a 1 , a 2 , ⋯ , a i ^ , ⋯ , a p ) : A → E , a i ↦ φ ( a 1 , a 2 , ⋯ , a i , ⋯ , a p ) ,

is a differential operator for any i = 1 , 2 , ⋯ , p .

Theorem 5 [

φ : A p → E

be an alternating p-differential operator, then there exists an unique alternating A -multilinear map

φ ˜ : [ A × Ω K ( A ) ] p → E

such that

φ ˜ ( D A / K ( a 1 ) , D A / K ( a 2 ) , ⋯ , D A / K ( a p ) ) = φ ( a 1 , a 2 , ⋯ , a p ) (5)

for all a 1 , a 2 , ⋯ , a p ∈ A .

We note Λ A p [ A × Ω K ( A ) ] , the p-exterior power of the A -module A × Ω K ( A ) .

Theorem 6 [

D A / K ( p ) : A p → Λ A p [ A × Ω K ( A ) ] , ( a 1 , a 2 , ⋯ , a p ) ↦ D A / K ( a 1 ) ∧ D A / K ( a 2 ) ∧ ⋯ ∧ D A / K ( a p ) ,

is a p-alternating differential operator and the image of D A / K ( p ) generates the A -module Λ A p [ A × Ω K ( A ) ] . Moreover, the pair ( Λ A p [ A × Ω K ( A ) ] , D A / K ( p ) ) possesses the following universal property: for any A -module E and for any alternating p-differential operator

φ : A p → E ,

there exists an unique A -linear map

φ ˜ : Λ A p [ A × Ω K ( A ) ] → E

such that

φ ˜ ∘ D A / K ( p ) = φ . (6)

We deduce that

H o m A ( Λ A p [ A × Ω K ( A ) ] , E ) → D K ( p ) ( A , E ) , ψ ↦ ψ ∘ D A / K ( p ) ,

is an isomorphism of A -modules.

Let Λ A [ A × Ω K ( A ) ] = ⊕ n ∈ ℕ Λ A n [ A × Ω K ( A ) ] be the exterior A -algebra of A -module A × Ω K ( A ) [

Theorem 7 The differential operator D A / K : A → A × Ω K ( A ) extends along a differential operator, denoted once again D A / K , from Λ A [ A × Ω K ( A ) ] into Λ A [ A × Ω K ( A ) ] of degree +1 and of square zero such that the pair ( Λ A [ A × Ω K ( A ) ] , D A / K ) is the differential algebra in the sense of Okassa [

Proposition 8 For any integer p ( p ≥ 1 ) , the map

[ A × Ω K ( A ) ] p → Λ A p − 1 [ A × Ω K ( A ) ] , ( x 1 , x 2 , ⋯ , x p ) ↦ ∑ i = 1 p ( − 1 ) i − 1 φ ˜ ( x i ) ⋅ x 1 ∧ x 2 ∧ ⋯ ∧ x i ^ ∧ ⋯ ∧ x p

is alternating A -multilinear and induces an A -linear map

i φ : Λ A p [ A × Ω K ( A ) ] → Λ A p − 1 [ A × Ω K ( A ) ]

which extends along an A -endomorphism of Λ A [ A × Ω K ( A ) ] , noticed once again i φ , of degree −1.

Theorem 9 For any φ ∈ D K ( A ) , the inner product

i φ : Λ A [ A × Ω K ( A ) ] → Λ A [ A × Ω K ( A ) ]

satisfies

i φ ( η 1 ∧ η 2 ) = [ i φ ( η 1 ) ] ∧ η 2 + ( − 1 ) | η 1 | η 1 ∧ [ i φ ( η 2 ) ] (7)

for all η 1 , η 2 ∈ Λ A [ A × Ω K ( A ) ] and where | η 1 | denotes the degree of η 1 . Moreover, the bracket

θ φ = i φ ∘ D A / K + D A / K ∘ i φ : Λ A [ A × Ω K ( A ) ] → Λ A [ A × Ω K ( A ) ]

is a differential operator of Λ A [ A × Ω K ( A ) ] and of degree zero.

Proof. It is obvious.

Proposition 10 For any φ ∈ D K ( A ) , we get

θ φ ∘ D A / K = D A / K ∘ θ φ . (8)

Proof. As D A / K ∘ D A / K = 0 and θ φ = i φ ∘ D A / K + D A / K ∘ i φ , then we have

θ φ ∘ D A / K = D A / K ∘ i φ ∘ D A / K = D A / K ∘ θ φ .

That ends the proof.

Theorem 11 For all φ ∈ D K ( A ) , a ∈ A and η ∈ Λ A [ A × Ω K ( A ) ] , we have

θ φ ( a ⋅ η ) = [ φ ( a ) − a ⋅ φ ( 1 A ) ] ⋅ η + a ⋅ θ φ ( η ) (9)

θ a ⋅ φ ( η ) = a ⋅ θ φ ( η ) + [ D A / K ( a ) − a ⋅ D A / K ( 1 A ) ] ∧ i φ ( η ) ; (10)

and

θ φ [ D A / K ( a ) ] = D A / K [ φ ( a ) ] . (11)

Proof. The result is immediate.

In what follows, M denotes a paracompact and connected smooth manifold, C ∞ ( M ) the algebra of numerical functions of class C ∞ on M , X ( M ) the C ∞ ( M ) -module of vector fields on M ,1 the unit of C ∞ ( M ) , D ( M ) the C ∞ ( M ) -module of differential operators on C ∞ ( M ) and δ the cohomology operator associated with the identity map

i d : D ( M ) → D ( M ) .

We recall that if [ , ] is the usual Lie bracket on D ( M ) then for all D , D ′ ∈ D ( M ) and f , g ∈ C ∞ ( M ) , we have

[ D , f ⋅ D ′ ] = [ D ( f ) − f ⋅ D ( 1 ) ] ⋅ D ′ + f ⋅ [ D , D ′ ] . (12)

In particular

[ D , f ] = D ( f ) − f ⋅ D ( 1 ) ; (13)

[ D ,1 ] = 0 ; (14)

[ f , g ] = 0. (15)

Proposition 12 [

[ D , D ′ ] = X ( g ) − Y ( f ) + [ X , Y ] . (16)

Proof. By straightforward calculation, one has

[ D , D ′ ] = [ f + X , g + Y ] = [ f , g ] + [ f , Y ] + [ X , g ] + [ X , Y ] = X ( g ) − Y ( f ) + [ X , Y ] .

That ends the proof.

Let N be a smooth manifold and let f : M → N be a differential map.

The map

f ∗ : C ∞ ( N ) → C ∞ ( M ) , g ↦ g ∘ f

is an homomorphism of real algebras.

If

f : M → N

is a diffeomorphism and

f − 1 : N → M

its inverse, then

f ∗ : C ∞ ( N ) → C ∞ (M)

is an isomorphism of real algebras such that

( f ∗ ) − 1 = ( f − 1 ) ∗ . (17)

Proposition 13 If f is a diffeomorphism and D is a differential operator of C ∞ ( M ) , then the map

f ∗ D = ( f − 1 ) ∗ ∘ D ∘ f ∗ : C ∞ ( N ) → C ∞ (N)

is a differential operator of C ∞ ( N ) called image of D by f.

Theorem 14 Let D , D ′ be two differential operators of C ∞ ( M ) , if f is a diffeomorphism, then

[ f ∗ D , f ∗ D ′ ] = f ∗ [ D , D ′ ] . (18)

Proof. Indeed, we verify that

[ f ∗ D , f ∗ D ′ ] = f ∗ D ∘ f ∗ D ′ − f ∗ D ′ ∘ f ∗ D = f ∗ [ D , D ′ ] .

That ends the proof.

Proposition 15 Let D be a differential operator, when f is a diffeomorphism then for any g element of C ∞ ( M ) , we have

D ( g ∘ f ) = [ ( f ∗ D ) ( g ) ] ∘ f . (19)

Proof. Let x ∈ M , we have

[ D ( g ∘ f ) ] ( x ) = [ D [ f ∗ ( g ) ] ] ( x ) = ( f ∗ D ) [ f ( x ) ] ( g ) = [ [ ( f ∗ D ) ( g ) ] ∘ f ] ( x ) .

We deduce that D ( g ∘ f ) = [ ( f ∗ D ) ( g ) ] ∘ f .

Proposition 16 When f is a diffeomorphism and ξ be a multilinear p-form on D ( N ) , for all D 1 , D 2 , ⋯ , D p differential operators of C ∞ ( M ) , we have

( f ∗ ξ ) ( D 1 , D 2 , ⋯ , D p ) = [ ξ ( f ∗ D 1 , f ∗ D 2 , ⋯ , f ∗ D p ) ] ∘ f . (20)

Theorem 17 If f is a diffeomorphism then δ and f ∗ commute.

Proof. We have

[ δ ( f ∗ ξ ) ] ( D 1 , D 2 , ⋯ , D p , D p + 1 ) = ∑ i = 1 p + 1 ( − 1 ) i + 1 D i [ ( f ∗ ξ ) ( D 1 , D 2 , ⋯ , D i ^ , ⋯ , D p , D p + 1 ) ] + ∑ i < j ( − 1 ) i + j ( f ∗ ξ ) ( [ D i , D j ] , D 1 , D 2 , ⋯ , D i ^ , ⋯ , D j ^ , ⋯ , D p , D p + 1 ) = [ ∑ i = 1 p + 1 ( − 1 ) i + 1 ( f ∗ D i ) [ ξ ( f ∗ D 1 , f ∗ D 2 , ⋯ , f ∗ D i ^ , ⋯ , f ∗ D p + 1 ) ] ] ∘ f + [ ∑ i < j ( − 1 ) i + j ξ ( [ f ∗ D i , f ∗ D j ] , f ∗ D 1 , ⋯ , f ∗ D i ^ , ⋯ , f ∗ D j ^ , ⋯ , f ∗ D p , f ∗ D p + 1 ) ] ∘ f .

On the other hand, we get

[ f ∗ ( δ ξ ) ] ( D 1 , D 2 , ⋯ , D p , D p + 1 ) = [ ( δ ξ ) ( f ∗ D 1 , f ∗ D 2 , ⋯ , f ∗ D p + 1 ) ] ∘ f = [ ∑ i = 1 p + 1 ( − 1 ) i + 1 ( f ∗ D i ) [ ξ ( f ∗ D 1 , f ∗ D 2 , ⋯ , f ∗ D i ^ , ⋯ , f ∗ D p + 1 ) ] ] ∘ f + [ ∑ i < j ( − 1 ) i + j ξ ( [ f ∗ D i , f ∗ D j ] , f ∗ D 1 , ⋯ , f ∗ D i ^ , ⋯ , f ∗ D j ^ , ⋯ , f ∗ D p + 1 ) ] ∘ f .

That ends the proof.

Let C p ∞ ( M ) be the algebra of smooth functions defined in the neighborhood of p ∈ M ,1 the unit of C p ∞ ( M ) .

A differential operator at p ∈ M is a linear map

D p : C p ∞ ( M ) → ℝ

such that

for all functions f and g in

Example 18 The Dirac mass at

is a differential operator at p.

We verify that the set,

is an isomorphism of vector spaces. Moreover, for

is a linear form on

Theorem 19 Let U be an open neighborhood of p in M of coordinate functions

is the dual basis of the basis of

Conclusion 20 In this paper, after having given the definition of differential operator by a relation of recurrence, one decomposes it and one states some properties from the algebraic viewpoint. We define the Lie derivative with respect to a differential operator. From a geometric viewpoint, some properties on the algebra of smooth functions were given. The Dirac mass at a point was the best example of differential operators at this point.

The author declares no conflicts of interest regarding the publication of this paper.

Gatse, S.C. (2019) Some Properties of First Order Differential Operators. Advances in Pure Mathematics, 9, 934-943. https://doi.org/10.4236/apm.2019.911046