The decomposition theorem

The next fact is known as thedecomposition theorem:

Theorem 3.13. Let U be anF-module. Let a(z,w) ∈ U

z,z⁻¹,w,w⁻¹ be a local U-valued formal distribution. Then, a(z,w) uniquely decomposes as a

sum

∑

j∈N

c^j(w)∂⁽w^j)δ(z−w)

with c^j(w)_j∈N being a family of formal distributions c^j(w) ∈ U

w,w⁻¹ having the property that all but finitely many j ∈_Nsatisfy c^j(w) =0.

The formal distributions c^j(w)in this decomposition are given by

c^j(w) =Res(z−w)^ja(z,w)dz for all j∈ _N. (52) Before we prove this theorem, let us show a lemma:

Lemma 3.14. Let U be an F-module. Let b(z,w) ∈ U

z,z⁻¹,w,w⁻¹ . As-sume that

Res(z−w)ⁿb(z,w)dz=0 for all n ∈_N. (53) (a)Then,b(z,w) ∈ U

w,w⁻¹ [[z]].

(b)Assume thatb(z,w) is local. Then,b(z,w) =0.

Proof of Lemma 3.14. We haveb(z,w) ∈U

z,z⁻¹,w,w⁻¹

= U

w,w⁻¹ z,z⁻¹ . Hence, we can writeb(z,w) in the form b(z,w) = _∑

j∈Zz^jb_j(w) for someb_j(w) ∈ U

w,w⁻¹

. Consider theseb_j(w). Every n∈ _Nsatisfies

Res zⁿ

|{z}

=((z−w)+w)ⁿ

=_∑ⁿ

k=0

n k

(z−w)^kw^n−k

b(z,w)dz =

∑

n k=0

n k

Res(z−w)^kb(z,w)dz

| {z }

=0

(by (53), applied tokinstead ofn)

w^n−k =0.

Compared with Reszⁿ b(z,w)

| {z }

=_∑

j∈Zz^jb_j(w)

dz=Reszⁿ

∑

j∈_Z

z^jb_j(w)dz =Res

∑

j∈_Z

z^n+jb_j(w)dz=b−n−1(w),

this yields that b−n−1(w) = 0 for every n ∈ N. In other words, b_j(w) = 0 for every negative j∈ _{Z. Hence,}

b(z,w) =

∑

j∈Z

z^jb_j(w) =

∑

j∈N

z^jb_j(w) +

∑

j∈Z;

jis negative

z^jb_j(w)

| {z }

=₀

=

∑

j∈N

z^jb_j(w)

∈ Uhh

w,w⁻¹ii [[z]].

This proves Lemma 3.14(a).

(b)Lemma 3.14(a)yieldsb(z,w) ∈ U

w,w⁻¹

[[z]]. But U

w,w⁻¹ [[z]]

is an F[z,w]-module, and the element z−_{w of F}[z,w] acts as a non-zero-divisor on this module (i.e., every element α ∈ U

w,w⁻¹

[[z]] satisfies (z−w)α = 0 must itself satisfy α = 0) ⁴⁴. But b(z,w) is local. In other words, (z−w)^Nb(z,w) = 0 for some N ∈ N. Consider this N. We can can-cel the (z−w)^N from the equality (z−w)^Nb(z,w) = 0 (since z−w acts as a non-zero-divisor on the module U

w,w⁻¹

[[z]], andb(z,w) belongs to said module). As a result, we obtainb(z,w) =0. This proves Lemma 3.14(b).

Proof of Theorem 3.13. Uniqueness: Let us first show the uniqueness of the fam-ily c^j(w)_j∈N satisfying a(z,w) = _∑

j∈_Nc^j(w)∂⁽w^j)δ(z−w). For this, we let c^j(w)_j∈N be any family of formal distributions c^j(w) ∈ U

w,w⁻¹ hav-ing the property that all but finitely manyj ∈_Nsatisfyc^j(w) = 0. Assume that a(z,w) = _∑

j∈Nc^j(w)∂⁽w^j)δ(z−w). We need to prove that (52) holds.

For everyn∈ _{N, we have}

j

∑

∈_N

c^j(w)Res (z−w)ⁿ∂⁽_w^j)δ(z−w)

| {z }

=







∂⁽w^j−n)δ(z−w), if n≤ j;

0, ifn> j

(by Proposition 3.12(b), applied tonandj instead ofmandn)

dz

=

∑

j∈_N

c^j(w)Res (

∂⁽w^j−n)δ(z−w), ifn≤ j;

0, ifn >j dz

=

∑

j∈_N;

j≥n

c^j(w) Res∂⁽w^j−n)δ(z−w)dz

| {z }

=δj−n,0

(by Proposition 3.12(a), applied toj−ninstead ofn)

=

∑

j∈_N;

j≥n

c^j(w)δj−n,0 =cⁿ(w). (54)

Hence, for every j∈ _{N, we have} Res(z−w)ⁿ a(z,w)

| {z }

= _∑

j∈Nc^j(w)∂^(j)wδ(z−w)

dz=

∑

j∈N

c^j(w)Res(z−w)ⁿ∂⁽w^j)δ(z−w)dz

=cⁿ(w).

44Proof.Actually, U

w,w⁻¹

[[z]]is not just anF[z,w]-module, but also an F

w,w⁻¹ [[z]] -module (sinceU

w,w⁻¹

is anF

w,w⁻¹

-module). The elementz−wof F

w,w⁻¹ [[z]]

is invertible (since z−w = w zw⁻¹−1

), and thus acts as a non-zero-divisor on the F

w,w⁻¹

[[z]]-module U

w,w⁻¹

[[z]], qed.

Renaming n as j and interchanging the two sides of this equality, we obtain precisely (52). Thus, (52) holds, and so the uniqueness of the family c^j(w)_j∈N is proven.

Existence: Now, we shall show that the family c^j(w)_j∈N defined by (52) actually satisfies a(z,w) = _∑

j∈_Nc^j(w)∂⁽w^j)δ(z−_w) and has the property that all but finitely many j∈ _Nsatisfyc^j(w) =0.

Indeed, the latter property obviously holds (since a(z,w) is local). So it re-mains to prove that the family c^j(w)_j∈N defined by (52) satisfies

a(z,w) =

∑

j∈_N

c^j(w)∂⁽_w^j)δ(z−w). (55) Consider this family c^j(w)_j∈N. Since a(z,w) is local, we know that every sufficiently high j ∈_Nsatisfies (z−w)^ja(z,w) =0, and therefore

every sufficiently high j∈ _Nsatisfiesc^j(w) =0. (56) Thus, the sum ∑

j∈_Nc^j(w)∂⁽w^j)δ(z−w) is well-defined. We set b(z,w) = a(z,w)−

∑

j∈N

c^j(w)∂⁽w^j)δ(z−w).

Then, our goal is to showb(z,w) =0 (because this will immediately yield (55)).

We know that a(z,w) is local. Also, for every j ∈ N, the formal distribution

∂⁽_w^j)δ(z−w) is local (since Proposition 3.12 (b) yields(z−w)^j+1∂⁽_w^j)δ(z−w) = 0), and therefore the formal distributionc^j(w)∂⁽w^j)δ(z−w)is local as well. Hence, the sum ∑

j∈_Nc^j(w)∂⁽w^j)δ(z−_w) is local as well (since it has only finitely many nonzero addends – due to (56) – and these addends are local). Thus, b(z,w) is local (since we have defined b(z,w) as a difference between the local formal distributionsa(z,w) and ∑

j∈_Nc^j(w)∂⁽w^j)δ(z−w)).

For everyn∈ _{N, we have}

Res(z−w)ⁿ b(z,w)

| {z }

=a(z,w)− _∑

j∈Nc^j(w)∂^(j)_wδ(z−w)

dz

=_Res(z−w)ⁿa(z,w)dz

| {z }

=cⁿ(w) (by (52), applied toj=n)

−

∑

j∈_N

c^j(w)Res(z−w)ⁿ∂⁽w^j)δ(z−w)dz

| {z }

=cⁿ(w) (by (54))

=cⁿ(w)−cⁿ(w) =0.

Hence, Lemma 3.14(b)yieldsb(z,w) = 0 (sinceb(z,w)is local). This completes the proof of (55). Thus, the existence part of Theorem 3.13 is proven. The proof of Theorem 3.13 is thus complete.

For the next corollary, we shall use the notations of Section 2.7.

Corollary 3.15. Let U be an F-module. To every a = a(z,w) ∈ U

z,z⁻¹,w,w⁻¹

, let us associate theF-linear operator Da :Fh

w,w⁻¹i

→Uhh

w,w⁻¹ii , ϕ(w) 7→Resϕ(z)a(z,w)dz.

(a)For everyc(w) ∈_Uw,w⁻¹ _{and j} ∈_{N, we have}

Dc(w)∂^(j)_wδ(z−w) =c(w)∂⁽w^j) (57) as maps F

w,w⁻¹

→ U

w,w⁻¹

. (Here, c(w)_∂⁽_w^j) means the operator

∂⁽w^j), followed by multiplication with c(w).) In particular, if U = _{F, then} D_δ(z−w) =1 (the identity map, or, rather, the canonical inclusionF

w,w⁻¹

→ F

w,w⁻¹ ).

(b) Let a = a(z,w) ∈ U

z,z⁻¹,w,w⁻¹

. Prove that a(z,w) is local if and only if Da is a differential operator of finite order. (See Definition 2.22 for the definition of a differential operator of finite order.)

(c) Let a = a(z,w) ∈ U

z,z⁻¹,w,w⁻¹

be local. Then, show that a(w,z) is local and satisfies

D_a(w,z) =D_a(z,w)∗

. (58)

(See Definition 2.27 for the meaning of

D_a(z,w)∗

.) (d)Let a =a(z,w) ∈U

z,z⁻¹,w,w⁻¹

. Show that

D_ψ(w)a(z,w) =_ψ(w)◦D_a(z,w) for all ψ(w) ∈_F^hw,w⁻¹i

; (59) D_∂wa(z,w) =_∂w◦D_a(z,w); (60) D_ψ(z)a(z,w) = D_a(z,w)◦ψ(w) for all ψ(z) ∈_F^hz,z⁻¹i

; (61)

D_∂za(z,w) =−_Da(z,w)◦_{∂w. (62)} Proof of Corollary 3.15. (a) Let c(w) ∈ _Uw,w⁻¹ _{and j} ∈ _{N. Let ϕ}(w) ∈ F

w,w⁻¹

. We write ϕ(w) in the form ϕ(w) = _∑

m∈_Zϕmw^m, where all ϕm be-long to F and all but finitely many of these ϕm are zero. Then, ∂⁽_w^j)ϕ(w) =

m∑∈_Z

m j

ϕmw^m−j (by the definition of ∂⁽w^j)).

Now,

Dc(w)∂^(j)wδ(z−w)(ϕ(w)) =Resϕ(z)c(w)∂⁽_w^j)δ(z−w)dz

by the definition of D_c

(w)∂^(j)wδ(z−w)

Before we start proving Proposition 3.15(b), let us show two auxiliary results:

• Everya =a(z,w) ∈ U

z,z⁻¹,w,w⁻¹

satisfies

D_{(z−w)a(z,w)} = [Da,w]. (63) Here,wdenotes the continuousF-linear map

U

z,z⁻¹,w,w⁻¹

→U

z,z⁻¹,w,w⁻¹

which sends every

b ∈ U

z,z⁻¹,w,w⁻¹

to w·b. (In analogy to Proposition 2.3, we should have called it Lw instead ofw, but the notationw is shorter.)]

Proof of (63): Let a = a(z,w) ∈ U

z,z⁻¹,w,w⁻¹

. Then, every ϕ(w) ∈ F

w,w⁻¹

satisfies D_{(z−w)a(z,w)}(_ϕ(w))

=Resϕ(z) (z−w)a(z,w)dz

by the definition of D(z−w)a(z,w)

=Resϕ(z)za(z,w)dz−_wResϕ(z)a(z,w)dz

=Reszϕ(z)a(z,w)dz−wResϕ(z)a(z,w)dz. (64) Hence, every ϕ(w)∈ _Fw,w⁻¹ _satisfies

[Da,w]

| {z }

=Da◦w−w◦Da

(ϕ(w))

= (Da◦w−w◦Da) (ϕ(w)) = Da(wϕ(w))

| {z }

=Reszϕ(z)a(z,w)dz (by the definition ofDa)

−w Da(ϕ(w))

| {z }

=Resϕ(z)a(z,w)dz (by the definition ofDa)

=Reszϕ(z)a(z,w)dz−wResϕ(z)a(z,w)dz

=D(z−w)a(z,w)(ϕ(w)) (by (64)).

In other words,[Da,w] = D_{(z−w)a(z,w)}. This proves (63).

• If ana∈ U

z,z⁻¹,w,w⁻¹

satisfies Da =0, then

a =0. (65)

Proof of (65): Leta ∈U

z,z⁻¹,w,w⁻¹

be such thatDa =0. Writeain the forma = _∑

(n,m)∈Z²an,mzⁿw^m with an,m ∈ U. Then, everyi∈ _Zsatisfies Da

wⁱ

=Reszⁱ a(z,w)

| {z }

=a= _∑

(n,m)∈Z2

an,mzⁿw^m

dz (by the definition of Da)

=Reszⁱ

∑

(n,m)∈_Z²

an,mzⁿw^mdz =

∑

m∈Z

a−i−1,mw^m.

Hence, every i ∈ _Z satisfies ∑

m∈_Za−i−1,mw^m = Da

|{z}

=0

wⁱ

= 0. Comparing coefficients in this equality, we conclude that everyi∈ _Zandm∈ _Zsatisfy a−i−1,m = 0. In other words, every (n,m) ∈ _Z² satisfy an,m = 0. Hence, a=0. This completes our proof of (65).

(b) =⇒: Assume that a(z,w) is local. Theorem 3.13 thus yields that a(z,w) can be written in the form

j

∑

∈_N

c^j(w)∂⁽w^j)δ(z−_w)

with c^j(w)_j∈N being a family of formal distributions c^j(w) ∈ U

w,w⁻¹ having the property that all but finitely many j∈ _Nsatisfy c^j(w) =0. Consider this c^j(w)_j∈N. Sincea(z,w) = _∑

j∈Nc^j(w)∂⁽w^j)δ(z−w), we have Da =

∑

j∈N

Dc^j(w)∂^(j)wδ(z−w)

| {z }

=c^j(w)∂^(j)_w

(by (57), applied toc(w)=c^j(w))

(since Da dependsF-linearly ona)

=

∑

j∈_N

c^j(w)∂⁽_w^j). (66)

Thus, Da is a differential operator of finite order (since all but finitely many j ∈ _Nsatisfy c^j(w) =0). This proves the =⇒ direction of Proposition 3.15(b).

⇐=: Assume that Da is a differential operator of finite order. In other words, Da = _∑

j∈_Ncj(w)∂⁽w^j)for a family cj(w)_j∈Nof formal distributions inU

w,w⁻¹ with the property that all but finitely many j ∈ _{N satisfy cj}(w) = 0. Consider this family c_j(w)_j∈N.

Let us define theF-linear map w : U

z,z⁻¹,w,w⁻¹

→ U

z,z⁻¹,w,w⁻¹ as in (63). We have

h w,∂⁽wⁿ⁾

i

=−∂⁽wⁿ⁻¹⁾ for everyn∈ _N. (67)

45

All but finitely manyj ∈ _Nsatisfy cj(w) =0. Thus, there exists a J ∈ _Nsuch that every j∈ _Nsatisfying j ≥ J satisfies c_j(w) =0. Consider this J. Then,

Da =

∑

j∈_N

cj(w)∂⁽w^j) =

J−1 j

∑

=0

cj(w)∂⁽w^j) (68)

45Indeed, this is similar to Proposition 2.3; there are only two differences:

• We now have two variables z and w instead of a single variable z. But this does not change much because only the variable w is “active”. (If we identify U

z,z⁻¹,w,w⁻¹

with U

z,z⁻¹ w,w⁻¹

, then we are back in the single-variable case.)

• We are now denoting bywthe map that would be denoted by Lwin the terminology of Proposition 2.3.

These two differences are insubstantial, and the proof of Proposition 2.3 can be easily adapted to prove (67).

(by the definition of J). We are going to prove that(z−w)^Ja(z,w) = 0.

Indeed, let us show that D_(z−w)ka(z,w) =

J−1 j

∑

=k

c_j(w)∂⁽w^j−k) for everyk ∈ _N (69) (where any sum whose lower limit is larger than its upper limit is understood to be empty).

Proof of (69): We shall prove (69) by induction overk.

Induction base: For k = 0, the equality (69) follows immediately from (68).

Thus, the induction base is complete.

Induction step: Let K∈ N. Assume that (69) holds fork =K. We need to show

In other words, (69) holds for k = K+1. This completes the induction step.

Thus, (69) is proven.

Thus, (65) (applied to(z−w)^Ja(z,w)instead ofa) yields that(z−w)^J a(z,w) = 0. Hence,a(z,w)is local. This proves the ⇐=direction of Proposition 3.15(b).

(c)Clearly, a(w,z) is local.⁴⁶ It remains to prove that D_a(w,z) =D_a(z,w)∗

. In other words, it remains to prove that D_a(w,z) = (Da)^∗ (sincea(z,w) = a).

Theorem 3.13 yields thata(z,w)can be written in the form

j

∑

∈_N

c^j(w)∂⁽w^j)δ(z−w)

with c^j(w)_j∈N being a family of formal distributions c^j(w) ∈ U

w,w⁻¹ having the property that all but finitely many j∈ _Nsatisfy c^j(w) =0. Consider this c^j(w)_j∈N. Theorem 3.13 furthermore shows that thesec^j(w)are given by (52).

We havea(z,w) = _∑

j∈_Nc^j(w)∂⁽w^j)δ(z−w). Renaming the indeterminateszand wasw andz in this equality, we obtain

a(w,z) =

∑

j∈_N

c^j(z)∂⁽z^j) δ(w−z)

| {z }

=δ(z−w) (by Proposition 3.12(c))

=

∑

j∈_N

c^j(z) ∂⁽z^j)δ(z−w)

| {z }

=(−1)^j∂^(j)w δ(z−w) (by Proposition 3.12(e))

=

∑

j∈_N

c^j(z) (−1)^j∂⁽_w^j)δ(z−w) =

∑

j∈_N

(−1)^j c^j(z)∂⁽_w^j)

| {z }

=∂^(j)_w c^j(z)

(sincec^j(z)is a polynomial inz and thus commutes with∂^(j)w)

δ(z−w)

=

∑

j∈_N

(−1)^j∂⁽w^j)

c^j(z)δ(z−w).

From (66), we have Da = _∑

j∈_N

c^j(w)∂⁽_w^j), so that (Da)^∗ = _∑

j∈_N

(−1)^j∂⁽_w^j)c^j(w).

46Proof.We know thata(z,w)is local. Thus, there exists anN∈_Nsuch that(z−w)^Na(z,w) = 0. Consider thisN. Switching the indeterminateszandwin(z−w)^Na(z,w) =0, we obtain (w−z)^Na(w,z) = 0. Hence, (z−w)^N

| {z }

=(−1)^N(w−z)^N

a(w,z) = (−1)^N(w−z)^Na(w,z)

| {z }

=0

= _{0. Thus,} a(w,z)is local, qed.

Now, for every ϕ(w) ∈_Fw,w⁻¹

, we have D_a(w,z)(ϕ(w))

=Resϕ(z) a(w,z)

| {z }

=_∑

j∈N(−1)^j∂^(j)_w(^{cj(z)δ(z−w)})

dz

by the definition of D_a(w,z)

=Resϕ(z)

∑

j∈_N

(−1)^j∂⁽w^j)

c^j(z)δ(z−w)

! dz

=

∑

j∈_N

(−1)^jResϕ(z)∂⁽_w^j)

c^j(z)δ(z−w)dz

| {z }

=∂^(j)w Resϕ(z)c^j(z)δ(z−w)dz (since both multiplication withϕ(z)and

the mapq7→Resqdzcommute with∂^(j)w)

=

∑

j∈N

(−1)^j∂⁽w^j)Resϕ(z)c^j(z)δ(z−w)dz

| {z }

=Resδ(z−w)ϕ(z)c^j(z)dz

=ϕ(w)c^j(w)

(by Proposition 3.6(c), applied toϕ(z)c^j(z)instead ofa(z))

=

∑

j∈_N

(−1)^j∂⁽w^j)

ϕ(w)c^j(w)=

∑

j∈_N

(−1)^j∂⁽w^j)c^j(w)

!

| {z }

=(Da)^∗

(ϕ(w)) = (Da)^∗(ϕ(w)).

Hence,D_a(w,z) = (Da)^∗. This completes the proof of Proposition 3.15(c).

(d) Proof of (59): For every ψ(w) ∈ _Fw,w⁻¹

and ϕ(w) ∈ _Fw,w⁻¹ , we have

D_ψ(w)a(z,w)(ϕ(w)) =Resϕ(z)ψ(w)a(z,w)dz

by the definition of D_ψ(w)a(z,w)

=_ψ(w) Resϕ(z)a(z,w)dz

| {z }

=D_a(z,w)(ϕ(w)) (by the definition ofD_a(z,w))

=_ψ(w)D_a(z,w)(_ϕ(w))

=ψ(w)◦D_a(z,w)

(ϕ(w)). This yields (59).

Proof of (60): For every ϕ(w) ∈_Fw,w⁻¹

, we have D_∂wa(z,w)(ϕ(w)) =Resϕ(z)∂wa(z,w)dz

by the definition of D_∂wa(z,w)

=_∂w Resϕ(z)a(z,w)dz

| {z }

=D_a(z,w)(ϕ(w)) (by the definition ofD_a(z,w))

(since the mapq 7→Resqdzcommutes with ∂w)

=∂wD_a(z,w)(ϕ(w)) =∂w◦D_a(z,w)

(ϕ(w)). This yields (60).

Proof of (61): For every ϕ(w) ∈_Fw,w⁻¹

, we have D_ψ(z)a(z,w)(ϕ(w)) = Resϕ(z)ψ(z)a(z,w)dz

by the definition ofD_ψ(w)a(z,w)

=D_a(z,w)(ϕ(w)ψ(w)) by the definition of D_a(z,w)

=D_a(z,w)(_ψ(w)_ϕ(w)) = D_a(z,w)◦_ψ(w)(_ϕ(w)). This yields (61).

Proof of (62): For every ϕ(w) ∈_Fw,w⁻¹

, we have D_∂za(z,w)(_ϕ(w)) =Resϕ(z)_∂za(z,w)dz

by the definition of D_∂za(z,w)

=−Res∂z(ϕ(z))a(z,w)dz

| {z }

=D_a(z,w)(∂w(ϕ(w))) (by the definition ofD_a(z,w))

by (28), applied toU

w,w⁻¹

, ϕ(z) and a(z,w) instead ofU, g and f

=−D_a(z,w)(∂w(ϕ(w))) =−D_a(z,w)◦∂w

(ϕ(w)). This yields (62).

Im Dokument Fundamentals of formal distributions (Lecture 3 of Vertex algebras by Victor Kac) (Seite 66-76)