Differential Operators of Higher Order

Posted on 2025-06-19 In Math

Outline

Section 7: Differential Operators of Higher Order

7 Dierential Operators of Higher Order

We will now turn to dierential operators of higher order and set up the machinery

of a symbol calculus. This will be systematically used in the subsequent chapters.

We have remarked that interesting operators on a dierential manifold are often

dened as maps of a tensor bundle into another, which are, in terms of their local

expressions, dierential operators in the usual sense. Many such operators are of rst

order, but there are also interesting operators of higher orders, for example the Laplacian

on a Riemannian manifold. We will therefore proceed to dene higher order operators,

rst on functions and later, from one vector bundle to another.

7.1. Denition. A (linear) dierential operator (of nite order) on functions is an R-

linear sheaf homomorphism A

→ A

which is in the algebra generated by dierential

operators of order ≤ 1. It is said to be of order ≤ k if it can be expressed as a linear

combination of composites of k operators of order ≤ 1. The set of all such operators

will be denoted by Di

or sometimes for shortness D

7.2. Remarks.

1) Since we have seen that dierential operators of order ≤ 1 have local expressions

of the form



∂

∂x

+ φ, it is clear that a dierential operator of order ≤ k is

locally of the form



|≤

∂

∂x

2) If M is not compact, it is possible to construct dierential operators of ‘innite

order’ by constructing operators on compact sets with orders that increase at

innity. For example, consider the open covering of R

by open intervals U

(n, n+2), n ∈ Z

. If {φ

} is a partition of unity with respect to U

, then



∂

∂x

is such an operator. But this phenomenon cannot happen if M is compact. In any

case, we will only consider operators of nite order.

3) Any sheaf homomorphism of A into itself can be shown to be locally a dierential

operator. This is a theorem of Peetre.

Structure of the algebra of dierential operators.

We have seen that the set of homogeneous rst order operators form a Lie algebra,

besides being a module over the ring of functions. We would like now to construct an

analogue of the universal enveloping algebra in this case, although it is not a Lie algebra

over functions. Recall rst the denition of the enveloping algebra.

7.3. Denition. Let g be a Lie algebra over a eld k. Then the enveloping algebra of

g is the quotient of the tensor algebra of the vector space g by the 2-sided ideal generated

by elements of the form X ⊗ Y − Y ⊗ X − [X, Y ], with X, Y ∈ g.

Chapter II: Differential Operators

In order to imitate this construction, we would like rst to construct the analogue of

the tensor algebra. As a starting point, it is better to take D

than T , since the former

is well adapted to the construction of the tensor algebra. We will now explain why.

As a sheaf of R-vector spaces, D

is simply A⊕T . For any f ∈ A(M ) it is convenient

to denote the corresponding element of D

(M) by m(f ) rather than f itself (although

when we judge that no confusion is likely, we may use f instead of m(f )). Then D

an A-bimodule. In fact, if f ∈ A(U ), and D = m(g) + X ∈ D

(U), then we may dene

f ◦ D = m(f) ◦ (m(g) + X) = m( f g) + fX,

D ◦ f = (m(g) + X) ◦ m(f ) = m(fg + Xf ) + f X.

The reason for the latter denition is that

(D ◦ m(f))(φ) = (m(g) + X)(fφ)

= gfφ + (Xf )φ + f Xφ

= m(gf + Xf)φ + fXφ.

It is easy to check therefore that this denition makes D

an A-bimodule.

It will turn out that the structure of the algebra of dierential operators can be

determined from the structure of D

as an A-bimodule on the one hand and as a Lie

algebra over R on the other. Indeed, the construction is akin to that of the enveloping

algebra of a Lie algebra.

In the case of a Lie algebra g over a eld k, the situation is like starting with g ⊕ k.

Let us denote inclusion of k in g ⊕ k by λ 7→ m(λ). We leave the following as a simple

exercise to the reader.

7.4. Exercise. Consider the tensor algebra of W = V ⊕ k. Pass to the quotient by

the 2-sided ideal generated by

(1)

−

. Show that it is canonically isomorphic to the

tensor algebra of V .

Consider D

⊗ D

where the tensor product is with respect to the right A-module

structure of the rst factor and the left one of the second factor. Also using the bimodule

structure of both we see that the tensor product is also a bimodule. We can iterate this

construction to obtain a tensor algebra. Now we will take the quotient of this ring by

the two-sided ideal spanned by m(1) − 1 to obtain an R-algebra. This is the algebra

which we sought to construct. We will denote it by C(M ).

It is easy to describe this algebra by a universal property.

7.5. Proposition. Let B be any R-algebra containing A. Then B has a natural

bimodule structure over A. Any bimodule homomorphism P of D

into B which is the

identity on A gives rise to a unique R-algebra homomorphism of C into B which is the

identity on A and coincides with P on D

Section 7: Differential Operators of Higher Order

Proof. In fact, we will dene inductively, for each r, a bimodule homomorphism P

(r)

from



) into B. Taking P

(0)

to be the identity map on A, and assuming that

(r)

has been dened, we notice that the map D



→ B given by (Y, Z) 7→

P (Y )P

(r)

(Z) is balanced. Hence it goes down to a map P

(r+1)



r+1

→ B as

claimed. It is evidently a homomorphism from the R-algebra T (D

) into B. From the

assumption, we see that m(1) − 1 is in the kernel of this map and so induces a map of

C into B. It is unique since D

and A together generate T ( D

7.6. Remark. We could have restated the assumption on P by saying that it is a left

A-linear map from T into B which satises

P (X)f = Xf + f P (X).

Thus we also have a characterisation of C-modules. It is an A-module E together

with an R-bilinear map T (M ) × E → E denoted (X, s) 7→ ∇

m such that

∇

(s) = f∇

s, and ∇

(fs) = (Xf ).s + f∇

We have constructed, for any open set U, such an algebra C(U) (although for no-

tational convenience we dropped the parenthetical U). It is easy to dene compatible

restriction maps and obtain a presheaf C. We will denote the image of any vector eld

X in C by ∇

and the image of any function f by m(f).

7.7

∇

(

) =

∇

and

∇

(

) = (

)

∇

7.8. Denition. We call C the connection algebra of M.

Notice that this is not an algebra over A, since the right and left multiplications by

functions are not the same. The R-subsheaves C

dened above as images of



are

all submodules for the A-bimodule structure. Moreover, it is clear from the denition

that these satisfy the condition

⊂ C

r+s

This is paraphrased by saying that the algebra C is actually a ltered algebra, of which

we recall the denition below. It is easily veried that C

are sheaves.

7.9. Denition. An algebra A together with subspaces F

(A), r ∈ Z

is said to be

a ltered algebra if F

(A) ⊂ F

r+1

(A) for all r ∈ Z

and F

(A).F

(A) ⊂ F

r+s

(A) for all

r, s ∈ Z

Every ltered algebra gives rise to a graded algebra as follows. Consider the direct

sum



(A)/F

r−1

(A)). It is an algebra under the natural denition which follows.

Suppose v ∈ Gr

(A) = F

(A)/F

r−1

(A) and w ∈ Gr

(A). Then taking representatives

Chapter II: Differential Operators

for v, w in F

(A), F

(A), and multiplying them in A, we obtain an element of F

r+s

(A).

Its image in Gr

r+s

(A) is independent of the choices made and this multiplication denes

an algebra operation. This is called the associated graded algebra and is denoted by

Gr(A).

Let us now consider the graded R-algebra associated to the ltered algebra C. Its

0th graded component, A, is a subalgebra. Indeed, we claim it actually commutes with

all elements of Gr(C). Let us prove this assertion by induction. Since A is itself commu-

tative, we can start the induction. Assume that for every function f, the corresponding

element m(f) commutes with all elements of C

modulo C

r−1

. Any element α of C

r+1

is a linear combination of images of elements of the form β ⊗ (X + m(g)), with β ∈ C

Then we have

m(f)(β ⊗ (X + m(g))) −(β ⊗ (X + m(g)))m(f)

≡ m(f)(β ⊗ X) − (β ⊗ X)m(f) (mod C

)

= m(f)(β ⊗ X) − β ⊗ (m(f )X + m(Xf ))

≡ ((m(f)β − βm(f )) ⊗ X (mod C

but by the induction assumption m(f)β −βm(f) ∈ C

r−1

. This proves our claim. Notice

also that the rst graded component of Gr(C) is canonically D

/A = T . Hence we

have a canonical algebra homomorphism of the tensor algebra over A of T into Gr(C).

We claim that this homomorphism is actually an isomorphism. It is obvious from the

denition that C is generated as an R-algebra by C

and A so that Gr(C) is generated

as an A-algebra by its rst graded component, namely T . This implies that the above

homomorphism is surjective.

Since C

are all sheaves and all these maps are compatible with restrictions to open

sets, it is enough to check the isomorphism statement over open sets in R

We will dene a C-module structure on a specic A-module, namely the tensor

algebra Q of T . We have shown that we need only to dene an action ∇

on Q subject

to equation 7.7. We dene ∇

(



) to be X ⊗



(Xf

. Here α is a

multiindex (i

, . . . , i

), X

denotes

∂

∂x

and X

denotes the element X

⊗···⊗X

, in Q.

That this satises the two conditions 7.7 is obvious. It is also clear that if F

Q denotes

the ltration on Q associated to the natural gradation on the tensor algebra, then we

have C

Q ⊂ F

r+s

Q. Hence we have an induced action of the associated graded

algebra Gr(C) on the tensor algebra Q. Since we have a natural homomorphism of the

tensor algebra Q into Gr, we therefore have an action of Q on itself. This action is by

denition the juxtaposition action, namely the multiplication on the left in the algebra

structure of Q. Since this action is faithful, we conclude that the map Q → Gr(C(M ))

is injective. We have therefore proved the following.

7.10. Theorem. The ltered R-algebra C has as its associated graded algebra an

A-module which is in fact the tensor algebra of T .

Section 7: Differential Operators of Higher Order

We shall go on to the denition of the algebra D of dierential operators. Consider

the two-sided ideal I in C generated by elements of the form R(X, Y ) = ∇

∇

−

∇

− ∇

[X,Y ]

with X, Y vector elds. The quotient of C by I gives rise to an

R-algebra. Again, since this can be done for all U, it follows easily that we have a

presheaf. The resulting sheaf of algebras D comes with a ltration F

(D) making it a

ltered algebra. Now the sheaf A may be made into a C-module by dening X.f to be

Xf , for all vector elds X. It is clear that this passes down to a D-module structure on

A. Under this action, elements of F

(D) act as R-homomorphisms of A which are linear

combinations of composites of k or less vector elds. In other words, we have given a

sheaf homomorphism F

(D) → Di

. In order to check that this is an isomorphism it

is enough to verify it on sections over any open domain U in R

. It is surjective by

denition, and in order to prove injectivity, we note that modulo the relations we have

imposed, any element of F

(D)(U) can be written as



,...,i



∂

∂x



···



∂

∂x



, i

+ ··· + i

≤ k.

The image in Di

(U) is the dierential operator on functions given by the above expres-

sion. Thus we have only to check that it is zero if and only if the action of the operator

on all functions is 0. But this is easily seen by looking at its action on functions of the

form Π(x

− a

)

, where a = (a

, . . . , a

) is some point in U at which the coecient

functions are nonzero. Thus we have proved the following theorem.

7.11. Theorem. The algebra D is naturally isomorphic to the quotient of the con-

nection algebra C by the two-sided ideal generated by elements of the form R(X, Y ) =

∇

− ∇

∇

− ∇

[X,Y ]

with X, Y vector elds.

7.12. Exercise. Show that left invariant dierential operators on a Lie group form

an algebra and that it is canonically isomorphic to the universal enveloping algebra of

its Lie algebra.

7.13. The symbol sequence. Consider the graded algebra associated to the ltered

algebra D. As in the case of Gr(C), we see that the left and right A-module structures

on D pass down to identical structures on Gr(D), making it actually an A-algebra.

Moreover, in view of the fact that R(X, Y ) are all 0 in this algebra, one also sees

that it is a commutative algebra. We have a natural inclusion of T in D as the space

of homogeneous rst order operators and hence we also get a homomorphism of T

into Gr(D). By the universal property of symmetric algebras, this induces an algebra

homomorphism of the symmetric algebra S(T ) into Gr(D), which is obviously onto.

7.14. Theorem. The algebra D of dierential operators on a dierential manifold

M is a ltered algebra whose associated graded algebra is canonically isomorphic to the

Chapter II: Differential Operators

symmetric algebra of T .

Proof. As in the proof of Theorem 7.11, we have only to check that the natural map

(T ) → D

k−1

is an isomorphism on a domain U in R

. But then we have proved

that D

(U) has an A(U)-basis consisting of



∂

∂x



···



∂

∂x



, i

+ ···+ i

≤ k, and

hence the images of



∂

∂x



···



∂

∂x



, i

+ ··· + i

= k, in D

k−1

form a basis.

But clearly these are images of a basis in S

(T ) under the natural map we have given

above. This completes the proof.

7.15. Denition. In particular, we have shown that there is an exact sequence of

vector bundles:

0 → D

k−1

→ D

→ S

(T ) → 0.

It is called the symbol sequence. If D is a dierential operator on an open set U of order

≤ k, then its image in S

(T )(U) is called its symbol or kth order symbol.

7.16. Remarks.

1) If we consider a dierential operator of order ≤ k, and take its image modulo lower

order operators, it is essentially the same as considering its kth order terms. The

above sequence says in eect that the ‘highest order term’ of a dierential operator

makes invariant sense only as a section of S

(T ).

2) A dierential operator D satisfying D(1) = 0 may be said to be an operator

without constant term. In particular an operator of order 1 without constant

term is a homogeneous operator. For higher orders, of course these two notions

are dierent, and while an operator without constant term makes sense in any

manifold, there is no invariant way of dening a homogeneous operator in general.

3) An operator of order at most 1 is of the form X + m(f ) where X is a vector eld

and f is a function. Its symbol is X.

7.17. Dierential operators on vector bundles. We have seen a few examples of

dierential operators on manifolds, but they are rarely on functions, or even on systems

of functions. They are often dened on sheaves of tensors on a dierential manifold.

We are therefore obliged to generalise the notion of a dierential operator. In fact,

we will dene a dierential operator (of order ≤ k) from one locally free sheaf E into

another, say F. This is done by considering an (R-linear) sheaf homomorphism, which

has the following property. Whenever s is a section of E and A a section of F

∗

, the map

f 7→ hD(fs), Ai = A(D(f s)) from functions to functions is a dierential operator (of

order ≤ k). While this is a practical denition, the theoretically satisfactory denition

would be that it is a section of F ⊗D⊗E

∗

, where the rst tensor product is with respect

Section 7: Differential Operators of Higher Order

to the left, and the second with respect to the right A-module structure on D. A slightly

dierent (but obviously equivalent) formulation is the following.

7.18. Denition. A right A-linear homomorphism of E into F ⊗ D

is called a

dierential operator from E to F of order ≤ k.

Using the left A-linear map D

→ A which associates to any D its constant term D(1)

we can associate to any dierential operator from E to F an R-linear homomorphism

E → F. We will denote the sheaf of such operators by D(E, F ). In the case when

E and F are trivial, we refer to such an operator as a system. If D is a dierential

operator from E to F of order ≤ k, its kth order symbol is the A-linear homomorphism

E → F ⊗ S

(T ) obtained by using the symbol map D

→ S

(T ).

7.19. Remark. One can also dene inductively an R-linear map D : E → F to be

a dierential operator of order at most k, by requiring that for every function f , the

operator f 7→ [D, f ] = D ◦ m(f) − m(f) ◦ D be an operator of order at most k − 1. If

}, 1 ≤ i ≤ k, are functions, then

[. . . [[D, f

], f

], . . . , f

]

is a 0th order operator, namely an A-linear homomorphism E → F. This func-

tion depends symmetrically on the f

’s. For example, this follows from the equality

[[D, f

], f

] + [[f

, D], f

] = 0 by the Jacobi identity. Moreover, assume that all the f

’s

vanish at a point m ∈ M. Then evaluation of this bracketed function at m goes down

to a map S

∗

) → Hom(E

, F

). In other words we get a homomorphism ˜σ(D) of

∗

) into Hom(E, F ). Associated to this, we get also a section of S

(T )⊗Hom(E, F ).

This can be easily seen to be the symbol σ as we have dened. We will call either of

these the symbol, but usually reserve the notation ˜σ for one and σ for the other.

7.20. Exercise. Show that the above denition coincides with the symbol of a dif-

ferential operator as dened in 7.16 on functions and hence also on bundles as in 7.19.

Computation of symbols. We will now give some examples of computation of sym-

bols of rst order operators.

7.21. Examples.

1) Consider the exterior derivative d : Λ

∗

) → Λ

i+1

∗

). From its local expression

it is clear that it is a rst order operator. In any case, let α be an i-form and

, . . . , X

i+1

be vector elds providing a linear form, namely ω 7→ ω(X

, . . . , X

i+1

)

on Λ

i+1

∗

). Then in order to see that d is a dierential operator we have only to

show that the map f 7→ d(fα)(X

, . . . , X

i+1

) is a dierential operator. But since

Chapter II: Differential Operators

d(fα) = df ∧ α + f dα, we have

d(fα)(X

, . . . , X

i+1

) =



(−1)

r+1

df(X

)α(X

, . . . ,

, . . . , X

i+1

)

+ f(dα)(X

, . . . , X

i+1

)



(−1)

r+1

f)α(X

, . . . ,

, . . . , X

i+1

)

+ fdα(X

, . . . , X

i+1

Hence this is a dierential operator of order 1 on functions whose symbol is

the vector eld



(−1)

r+1

α(X

, . . . ,

, . . . , X

i+1

and whose constant term

is dα(X

, . . . , X

i+1

). Now the symbol of d is a homomorphism σ(d) : Λ

∗

) →

i+1

∗

) ⊗ T , and so can equally well be interpreted to be a bilinear homomor-

phism Λ

∗

)×T

∗

→ Λ

i+1

∗

). Then the above computation gives the following:

σ(d)(α, β)( X

, . . . , X

i+1

) = β





(−1)

r+1

α(X

, . . . ,

, . . . , X

i+1





(−1)

r+1

α(X

, . . . ,

, . . . , X

i+1

)β(X

)

= (β ∧ α)(X

, . . . , X

i+1

In other words, σ(d)(α, β) = β ∧ α.

2) Consider the Lie derivative with respect to a vector eld X, say of an i-form α.

Then for f ∈ A(M ) and vector elds X

∈ T (M ), r ≤ i, we see that the operator

f 7→ L(X)(fα)(X

, . . . , X

) is given by

f 7→ X( f α(X

, . . . , X

)) −



r=1

(fα)(X

, . . . , [X, X

], . . . , X

)

= (Xf )α(X

, . . . , X

) + f(Xα(X

, . . . , X

))

−



r=1

fα(X

, . . . , [X, X

], . . . , X

This map is obviously a dierential operator of order 1 whose symbol is α(X

, . . . , X

)X.

This shows that the symbol of L(X), which is a homomorphism Λ

∗

) → Λ

∗

)⊗

T , is given by σ(L(X))(α) = α ⊗ X.

7.22. Symbol of a composite of dierential operators. Let D

: E → F and

: F → G be operators of order ≤ k, l, respectively. By composition we obtain an

operator of order ≤ k + l from E to G. We wish to compute the (k + l) th symbol of

◦D

. Firstly, we remark that in the case when E = F = G = A, this is easily done.

For the symbol is the natural homomorphism D

→ S

(T ), and this is multiplicative

in the sense that σ

k+l

◦ D

) = σ

).σ

) where the multiplication on the right

Section 7: Differential Operators of Higher Order

side is intended in the sense of symmetric algebras. In general, the symbols of D

, D

are homomorphisms σ : E → F ⊗ S

(T ), τ : F → G ⊗ S

(T ), respectively. Any two

maps σ, τ as above give rise to a map (τ ⊗ 1

(T )

) ◦ σ : E → G ⊗ S

(T ) ⊗ S

(T ), and

using the multiplication in the symmetric algebra, we get a map E → G ⊗ S

k+l

(T ).

This is called the composite of the symbols σ and τ . Thus our computation above can

be restated as

7.23. Proposition. If D

, D

are dierential operators E → F and F → G of orders

k, l, respectively, then the (k + l)th order symbol of the composite dierential operator

◦ D

: E → G is the composite of the two symbols σ

) and σ

7.24. Tensorisation with local systems. If D is a dierential operator from a

vector bundle E to F , and L is any other vector bundle, there is no natural way of

dening a dierential operator L ⊗ E → L ⊗ F . The reason is that the tensorisation

is over A, while D is not A-linear. For the same reason, it is clear that if L is a local

system of R- or C-vector spaces, then one can dene such an operator. It is moreover

clear that its symbol, considered as a map L ⊗ E → L ⊗ F ⊗ S

(T ), is obtained as the

tensor product of the symbol of D with 1

In particular, if L is a local system, then one can talk of the de Rham complex with

coecients in L, namely

0 → L → L ⊗ A → ··· → L ⊗ Λ

∗

) → ··· ,

and clearly the de Rham sequence of sheaves remains exact after tensorisation with the

local system L. Notice that if L is a local system of R- (resp. C-) vector spaces and E

is a locally free A-sheaf, then L ⊗ A is a locally free A-module.

Exercises

1) Determine the space of derivations of A = R[x

, . . . , x

2) Show that the group B of triangular matrices, namely matrices (A

) such that

= 0 whenever i is greater than j, is a Lie subalgebra of gl(n, R). What is the

corresponding Lie subgroup of GL(n, R)?

3) Let X be a connected topological space, x a point of X and L a local system of

vector spaces over X. Show that the natural map L(X) → L

is injective.

4) If E is a complex vector bundle of rank n, treated as a real vector bundle of rank

2n, show that E ⊗C is isomorphic to E ⊕ E.

5) Show that the Plücker imbedding of the Grassmannian has nonzero dierential at

all points.

Chapter : Differential Operators

6) If a Lie group G acts on a dierential manifold M , then there is a natural homo-

morphism of the Lie algebra g of G into the Lie algebra of dierentiable vector

elds on M. When is this map injective? Compute this map for the action of

GL(n, R) on R

7) Show that the universal covering space of any connected Lie group is also a Lie

group. What is its Lie algebra?

8) Show that if Y is a (dierentiable) vector eld and (φ

) is its ow, then

lim

t→0



∗

(Y ) − Y

− L

(Y )



exists uniformly, where Y is a vector eld with compact support.

9) Let ω be a 2-form on R

with coordinates ( x, y, z, t). Write ω in terms of coordi-

nates and note that it gives rise to a 2-form α and a 1-form β on R

, depending

on t. Compute dω in terms of α and β, giving rise to a 1-form γ and a function φ

on R

(again depending on t).

10) Let G be a compact connected Lie group. Consider the dierentiable map g → g

of G into itself. Compute its dierential at any x ∈ G and determine at what

points it is an isomorphism.