Differential operators

Definition 2.21. If U is an F-module, and if h ∈ _Fw,w⁻¹

, then we shall write L_h for the mapU

w,w⁻¹

→U

w,w⁻¹

, a 7→ha. (Notice that U is implicit in this notation; therefore it sometimes leads to several different maps being denoted byL_h. But this rarely makes any troubles, since these maps can be distinguished by their domains.)

Also, if U is an F-module, and if h ∈ U

w,w⁻¹

, then we shall write L_h for the mapF

w,w⁻¹

→U

w,w⁻¹

, a 7→ha. (Again, U is implicit, and we rely on context to clarify what we mean by L_h.)

When U is an F-module, and when h belongs to either F

w,w⁻¹ or U

w,w⁻¹

, we are often going to abbreviate the map L_h by h (by abuse of notation). Thus, h(a) = ha for every h. This is a generalization of the abuse of notation we mentioned in Remark 2.4.

We next define a (purely algebraic) notion of differential operators – one of many reasonable such notions.

If U is an F-module, and if c_j(w)_j∈N is a family of formal distributions in U

w,w⁻¹

with the property that all but finitely manyj ∈_Nsatisfyc_j(w) =0, then ∑

j∈_N

c_j(w)_∂⁽_w^j) is a well-defined F-linear map F

w,w⁻¹

→ U

w,w⁻¹ (wherecj(w)∂⁽w^j) meanscj(w)◦∂⁽w^j), and wherecj(w) abbreviatesL_cj(w), accord-ing to Definition 2.21).³³ These maps will be called differential operators of finite order into U. In other words:

Definition 2.22. LetUbe anF-module. Adifferential operator of finite order into U means a mapF

w,w⁻¹

→U

w,w⁻¹

which has the form ∑

j∈_N

c_j(w)_∂⁽_w^j) for a family c_j(w)_j∈N of formal distributions inU

w,w⁻¹

with the prop-erty that all but finitely many j ∈_Nsatisfy c_j(w) =0.

Remark 2.23. Maps of the form ∑

j∈_Ncj(w)∂⁽w^j) are well-defined even if we do not require that all but finitely many j ∈ _N satisfy cj(w) = 0. We shall not make use of them in this generality, however.

It is clear that (for given U) the differential operators of finite order into U form an F-module. This F-module contains the operator c(w) for each w ∈ U

w,w⁻¹

(recall that “the operator c(w)” really means the map L_c(w) :

33This follows from the fact thatU

w,w⁻¹

is anF

w,w⁻¹

-module.

F

w,w⁻¹

→_Uw,w⁻¹), and when U =F, it also contains the operator ∂⁽w^j)

for everyj ∈_N.

We first notice that the “coefficients”c_j(w)of a differential operator ∑

j∈Nc_j(w)∂⁽w^j)

are uniquely determined by the operator. More precisely:

Proposition 2.24. Let U be an F-module. Let D be a differential operator of finite order into U. Then, there exists a unique family c_j(w)_j∈N of formal distributions in U

w,w⁻¹

with the properties that

• all but finitely many j∈ _Nsatisfyc_j(w) =0;

• we haveD = _∑

j∈_N

c_j(w)_∂⁽_w^j).

Proof of Proposition 2.24. The existence of such a family follows from the fact that D is a differential operator of finite order into U. It thus remains to prove its uniqueness. In other words, it remains to prove that if

c^h_j¹ⁱ(w)

j∈N and

c^h_j²ⁱ(w)

j∈N are two families c_j(w)_j∈N with the two properties stated in Proposition 2.24, then

c^h_j¹ⁱ(w)

j∈_N =c^h_j²ⁱ(w)

j∈_N. Thus, let

c^h_j¹ⁱ(w)

j∈_Nand

c^h_j²ⁱ(w)

j∈_Nbe two families c_j(w)_j∈N with the two properties stated in Proposition 2.24.

For everyj ∈N, define a formal distributions_j(w) ∈U

w,w⁻¹ by s_j(w) =c^h_j¹ⁱ(w)−c^h_j²ⁱ(w).

It is easy to show that

0 =

∑

j∈_N

s_j(w)_∂⁽_w^j). (29)

34

34Proof.The family

c^h1i_j (w)

j∈Nsatisfies the second of the two properties stated in Proposition 2.24. In other words, we have

D=

∑

j∈N

c^h1i_j (w)∂^(j)_w (30) The same argument, applied to the family

c^h2i_j (w)

j∈N instead of

c^h1i_j (w)

j∈N, yields D=

∑

j∈N

c^h2i_j (w)∂^(j)w .

We will now show that everyn ∈_Nsatisfies

sn(w) =0. (31)

Proof of (31): We shall prove (31) by strong induction overn.

So let N ∈ N. Assume that (31) holds for each n ∈ _Nsatisfying n < N. We need to show that (31) holds forn= N as well.

We know that (31) holds for eachn∈ _Nsatisfying n< N. In other words, sn(w) =0 for each n∈ _Nsatisfying n< N. (32) Applying both sides of the equality (29) to the Laurent polynomial w^N ∈ F

Subtracting this equality from (30), we obtain 0=

∑

Thus, s_N(w) = 0. In other words, (31) holds for n = N. This completes the induction step.

Now, we have proven (31) by induction. Thus, every n∈ _{Nsatisfies sn}(w) = 0. In other words, every n ∈ _N satisfies c^h_n¹ⁱ(w)−c^h_n²ⁱ(w) = 0 (since sn(w) is defined to bec^hn¹ⁱ(w)−c^hn²ⁱ(w)). In other words, everyn∈ _Nsatisfiesc^hn¹ⁱ(w) = c^h_n²ⁱ(w). Hence,

c^h_j¹ⁱ(w)

j∈_N = c^h_j²ⁱ(w)

j∈_N. This completes the proof of Proposition 2.24.

Next, we shall discuss composition of differential operators. Two differential operators of finite order into U do not necessarily have a well-defined com-position, even if U = F (because they send Laurent polynomials to formal distributions, but they cannot be applied to formal distributions). However, a composition can be defined in a particular setting:

Definition 2.25. Let U be anF-module.

(a)Let Dbe a differential operator of finite order into U. We can then write D in the form ∑

j∈Nc_j(w)∂⁽w^j) for a family c_j(w)_j∈N of formal distributions in U

w,w⁻¹

with the property that all but finitely many j ∈ _N satisfy cj(w) = 0. Consider this family. (Proposition 2.24 says that this family is unique.) We say that D is Laurent if all c_j(w) are Laurent polynomials (i.e., belong toF

w,w⁻¹ ).

It is clear that ifD is Laurent, thenD has the following two properties:

• The image of Dis contained inU

w,w⁻¹ .

• The operator D can be uniquely extended to a continuous F-linear map F

w,w⁻¹

→ U

w,w⁻¹

(which is again defined as ∑

j∈Nc_j(w)∂⁽_w^j)), since U

w,w⁻¹

is an F

w,w⁻¹

-module. We denote this extension by D again.

(b)Let Dbe any differential operator of finite order intoU, and letEbe any differential operator of finite order into F. We cannot define a map D◦_{E in} general.

However, ifE is Laurent, then D◦E is a well-defined map fromF

w,w⁻¹ toU

w,w⁻¹

(since the image of Eis contained inF

w,w⁻¹ ).

On the other hand, if D is Laurent, then D◦E is a well-defined map from F

w,w⁻¹

to U

w,w⁻¹

(but here, the D in D◦E does not mean the orig-inal map D : F

w,w⁻¹

→ _Uw,w⁻¹, but rather the extension of D to a continuousF-linear map F

w,w⁻¹

→U

w,w⁻¹ ).

Altogether, we thus know that there is a well-defined operatorD◦E when-ever at least one of Dand E is Laurent.

We shall now prove that the composition of two differential operators is again a differential operator, provided that it is well-defined:

Proposition 2.26. Let U be an F-module. Let D be any differential operator of finite order intoU. Let Ebe any differential operator of finite order intoF.

(a)Assume that at least one of D and E is Laurent. Then, the map D◦E is a differential operator of finite order intoU.

(b)If both D and Eare Laurent, then D◦Eis also Laurent.

Proof of Proposition 2.26. For every e(w) ∈ _Fw,w⁻¹

and every n ∈ _{N, we} have

∂⁽wⁿ⁾ ◦e(w) =

∑

n i=0

∂⁽wⁿ⁻ⁱ⁾(e(w))◦∂⁽wⁱ⁾ (33) (as maps fromF

w,w⁻¹ toF

w,w⁻¹ ). ³⁵

Now,Dis a differential operator of finite order intoF. Hence, we can writeD in the formD= _∑

j∈Nd_j(w)∂⁽w^j) for a family of d_j(w)_j∈N of formal distributions inF

w,w⁻¹

with the property that all but finitely manyj ∈_Nsatisfyd_j(w) = 0. Consider this family.

Also,E is a differential operator of finite order intoU. Hence, we can writeE in the form E= _∑

j∈Ne_j(w)∂⁽_w^j) for a family of e_j(w)_j∈N of formal distributions inU

w,w⁻¹

with the property that all but finitely manyj ∈_Nsatisfye_j(w) = 0. Consider this family.

For all i ∈ _{N and j} ∈ N, the product di(w)·_ej(w) in U

w,w⁻¹

is well-defined³⁶.

35Proof of (33): Lete(w) ∈_Fw,w⁻¹

and n∈ _{N. Let} g ∈_Fw,w⁻¹

be arbitrary. Applying Proposition 2.7(b)toFinstead ofU(and renaming the indeterminatezasw), we conclude that

∂⁽ⁿ⁾_w (ab) =

∑

n i=0

∂⁽ⁱ⁾_w (a)∂⁽ⁿ⁻ⁱ⁾_w (b) for anya∈F

w,w⁻¹

andb∈F

w,w⁻¹

. Applying this toa=gandb=e(w), we obtain

∂⁽ⁿ⁾_w (e(w)·g) =

∑

n i=0

∂⁽ⁱ⁾_w (g)∂⁽ⁿ⁻ⁱ⁾_w (e(w)) =

∑

n i=0

∂⁽ⁿ⁻ⁱ⁾_w (e(w))◦∂⁽ⁱ⁾_w

! (g). Hence,

∂⁽ⁿ⁾w ◦e(w)(g) =∂⁽ⁿ⁾w (e(w)·g) =

∑

n i=0

∂⁽ⁿ⁻ⁱ⁾w (e(w))◦∂⁽ⁱ⁾w

!

(g). (34) Let us now forget that we fixedg. We thus have proven (34) for everyg ∈F

w,w⁻¹ . In other words, we have∂⁽ⁿ⁾_w ◦e(w) = _∑ⁿ

i=0

∂⁽ⁿ⁻ⁱ⁾_w (e(w))◦∂⁽ⁱ⁾_w . This proves (33).

36Proof. We assumed that at least one of D and E is Laurent. We WLOG assume that D is

The operator D◦E is well-defined (since at least one of D and E is Laurent).

Laurent (since the proof in the case whenEis Laurent is similar). Then, alld_j(w)are Laurent polynomials. That is, d_i(w) ∈ U

w,w⁻¹

for every i ∈ _{N. Hence,} d_i(w)·e_j(w) is well-defined for all i ∈ _N and j ∈ _N(since the product of an element of U

w,w⁻¹

with an element ofF

w,w⁻¹

is always well-defined). Qed.

We haveD = _∑

(because the inner sum has only finitely many nonzero addends (since all but finitely manyj∈_Nsatisfyd_j(w)=0))

◦∂⁽w^m). (35)

Since all but finitely manym∈ _{Nsatisfy ∑}^m

i=0

m i





 ∑

n∈N;

n≥i

dn(w)·_∂⁽_wⁿ⁻ⁱ⁾(e_m−i(w))





= 0 ³⁷, this yields thatD◦Eis a differential operator of finite order intoU.

Propo-sition 2.26(a)is thus proven.

(b) Assume that D and E Laurent. Then, all d_j(w) are Laurent polynomials, and alle_j(w)are Laurent polynomials. Now, it is easy to see that

∑

m i=0

m i







∑

n∈_N;

n≥i

dn(w)·∂⁽wⁿ⁻ⁱ⁾(em−i(w))





 is a Laurent polynomial for each m ∈ _N (since the inner sum ∑

n∈_N;

n≥i

in this ex-pression has only finitely many nonzero terms). Thus, (35) shows that D◦E is Laurent. This proves Proposition 2.26(b).

We now define the notion of an adjoint differential operator:

Definition 2.27. Let U be an F-module. Let D be a differential operator of finite order intoU. We can then write Din the form ∑

j∈_N

c_j(w)∂⁽w^j) for a family cj(w)_j∈N of formal distributions inU

w,w⁻¹

with the property that all but finitely many j∈ _Nsatisfy c_j(w) =0. Consider this family. (Proposition 2.24 says that this family is unique.)

We denote by D^∗ the map

j

∑

∈N

(−1)^j∂⁽w^j)◦cj(w): Fh

w,w⁻¹i

→Uhh

w,w⁻¹ii ,

which is again a differential operator of finite order intoU(due to Proposition 2.26). This operator D^∗ is called the adjoint differential operator to D. If D is Laurent, thenD^∗ is Laurent (this follows easily from Proposition 2.26(b)).

We end this section with some results whose proofs we leave to the reader:

Proposition 2.28. Let U be an F-module. Let D be a Laurent differential operator of finite order intoU. Then, any f ∈ _Fw,w⁻¹

and g ∈ _Fw,w⁻¹ satisfy

Res(D^∗f)gdw=Res f (Dg)dw.

37This is easily derived from the facts that all but finitely manyj∈_Nsatisfyd_j(w) =0, and all but finitely manyj∈_Nsatisfye_j(w) =0.

In view of the fact that we regard Resf gdw as a sort of bilinear pairing be-tween f and g(see Section 3.1 below), this Proposition 2.28 likely explains where the name “adjoint differential operator” forD^∗ comes from.

Definition 2.29. Let U be an F-module. Let D be any differential operator of finite order into F. We can then write D in the form ∑

j∈_N

c_j(w)∂⁽w^j) for a family cj(w)_j∈N of formal distributions in F

w,w⁻¹

with the property that all but finitely many j ∈ _{N satisfy} c_j(w) = 0. Consider this family.

(Proposition 2.24 says that this family is unique.) Then, we define a map D(U) : U

w,w⁻¹

→ U

w,w⁻¹

by D(U) = _∑

j∈_Nc_j(w)∂⁽w^j) (this is the same expression as for D, but it is now acting onU

w,w⁻¹ ).

Proposition 2.30. Let U be an F-module. Let D be any differential opera-tor of finite order into U. Let E be any differential operator of finite order into F. Assume that at least one of D and E is Laurent. We can canoni-cally lift the differential operator D : F

w,w⁻¹

→ _Fw,w⁻¹

to a map De : U

w,w⁻¹

→ U

w,w⁻¹

by writing D in the form ∑

j∈_N

c_j(w)∂⁽w^j) (with cj(w)_j∈N being as usual) and defining De to be the map ∑

j∈_Ncj(w)∂⁽w^j) (but now acting onU

w,w⁻¹

). (This is well-defined because of Proposition 2.24.) Notice that if Dis Laurent, then so is D.e

Now, we have

(D◦E)^∗ = (E^∗)_U◦D^∗.

The proof of Proposition 2.30 can be obtained easily using Proposition 2.28, once one reduces to the case ofD and E both being Laurent (by F-linearity and continuity) and ofU =_F(by U-linearity).

Im Dokument Fundamentals of formal distributions (Lecture 3 of Vertex algebras by Victor Kac) (Seite 41-49)