5. The Poincar´ e-Birkhoff-Witt theorem 96
5.3. The Poincar´ e-Birkhoff-Witt theorem
There are several interrelated (but still different and non-equivalent) facts referred to as Poincar´e-Birkhoff-Witt theorems in literature. All of them are similar in that each of them has a condition (such as: k is a Q-algebra, or: k is a field, or: g is a free k-module), under which they claim that the universal enveloping algebra U(g) of a k-Lie algebra g is ”similar” in a certain way to the symmetric algebra Symg of the k-module g as a k-module, as a g-module, or as a k-algebra. What this ”similar”
means depends on which of the various Poincar´e-Birkhoff-Witt theorems we consider.
There is usually no isomorphism of k-algebras, but there is often an isomorphism of k-modules, sometimes one ofg-modules, and often one between the associated graded k-algebras. Here we are interested in the latter. The fact we are going to use is the following one:
Proposition 5.8. Let k be a commutative ring. Let g be ak-Lie algebra.
(a) Assume that the Lie algebra g satisfies the n-PBW condition for every n ∈ N. Then, the associated graded k-algebra gr (U(g)) is isomorphic to the symmetric algebra Symg = L
p∈N
Sympg as a k-algebra. (Here, the associated graded k-algebra gr (U(g)) is defined as the direct sum L
p∈N
grp(U(g)), with multiplication defined by
(up)p∈
ui·vp−i are to be understood as residue classes of certain ele-ments of U≤p(g) modulo U≤(p−1)(g).)
(b)Letn∈N. Assume that the Lie algebrag satisfies then-PBW condition. Then, Ker (grnψ) = gradg,n(Kn(g)). Here, ψ is defined as in Proposition 5.5, and Kn(g) is defined as in Definition 5.1 (applied to V =g).
Proof of Proposition 5.8. (a) We assumed that the Lie algebra g satisfies the n-PBW condition for every n∈N. Thus, for every n ∈N, the map PBWg,n : Symng→ grn(U(g)) is a k-module isomorphism. The direct sum L
p∈N
PBWg,p of these maps is thus a k-module isomorphism from L
p∈N
Sympg= Symg to L
p∈N
grp(U(g)) = gr (U(g)).
It only remains to show that this is a k-algebra isomorphism. This is straightforward and left to the reader (especially given that we will not be using this fact anyway).
(b) We assumed that the Lie algebra g satisfies the n-PBW condition. This means that the map PBWg,n : Symng → grn(U(g)) is a k-module isomorphism. Thus,
To get anything useful out of Proposition 5.8, we need to know some simple condi-tions under which the n-PBW condition is guaranteed to hold (the n-PBW condition itself is rather hard to check in most cases, particularly if we want to check it for all n∈N at once). These are the essence of the Poincar´e-Birkhoff-Witt theorems:21
21Note that Theorem 5.9 (at least part(a), but probably some of the other parts as well) depends
Theorem 5.9. Letk be a commutative ring. Letg be a k-Lie algebra. Let n∈N. (a) Ifg is a free k-module, theng satisfies the n-PBW condition.
(b) If k is a Q-algebra, then gsatisfies the n-PBW condition.
(c) Ifg is a projective k-module, theng satisfies the n-PBW condition.
(d) If the (additive) abelian group g is torsion-free, then g satisfies the n-PBW condition.
(e) If k is a Dedekind domain, then g satisfies the n-PBW condition.
(f )If the k-module gis the direct sum of cyclic modules (where a module is said to be cyclic if it is generated by one element), theng satisfies the n-PBW condition.
(g) Ifg is a flat k-module, theng satisfies the n-PBW condition.
Calling this theorem ”Poincar´e-Birkhoff-Witt theorem” is an anachronism, although a rather convenient one. The first to discover anything related to this result was apparently Poincar´e in 1900; it was a weak version of Theorem 5.9 which required k to be a field of characteristic 0 and claimed that if (v1, v2, ..., vm) is a basis of the k-vector space g, then (vi1 ⊗vi2 ⊗...⊗vin)n∈N; (i1,i2,...,in)∈{1,2,...,m}n;
i1≤i2≤...≤in
is a basis of the k-vector space U(g) (see Theorem 5.10 for why this is weaker than Theorem 5.9). This can be shown to be equivalent to the claim that g satisfies the n-PBW condition for everyn ∈N. Whether Poincar´e’s proof of this is correct is still a matter of controversy.
More historical details, along with a modernized version of Poincar´e’s original alleged proof, can be found in [20].
A comprehensive proof of Theorem 5.9 can be found in Higgins’s paper [10, Theorems 6 and 7 and Corollary 2] (where he proves all parts except for(d)and(g), but(d)can be derived from(b) by tensoring with Q, and(g) can be derived from his Theorem 8 combined with Lazard’s theorem that any flat module is a direct limit of free modules).
However, its parts(a)and(b)are proven more frequently in different sources: Theorem 5.9(a)is shown in most references which consider the Poincar´e-Birkhoff-Witt theorem:
for example, [1, Theorem 1.3.1], [13, Theorem 8.2.2], [5,§17.3], [4, §2.1], [11,§1.9], [21, Theorem 5.15], [8, §2.7, Th´eor`eme 1], [7, Theorem 3.3.1], [12, Part I, Chapter III, Theorem 4.3], [16, Theorem 2], [15, Theorem 6.5] give proofs (and while most of these sources superficially requirek to be a field, the only condition they actually use is that g be a free k-module). [24, Part 1, Chapter 1,§1.3.7] proves Theorem 5.9 (b) as well, and so do [22,§2.5] and [23, §5.5]. Theorem 5.9(e)and (f )were also shown by Pierre Cartier in [28].
In the case when g is a freek-module, the following is also known as ”the Poincar´ e-Birkhoff-Witt theorem”, as it is an equivalent version of Theorem 5.9 (a):
Theorem 5.10. Let k be a commutative ring. Let g be a k-Lie algebra. Assume that the k-module g has a basis (ei)i∈I, where I is a totally ordered set. Then, (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in
is a basis of the k-module U(g).
Convention 5.11. Here and in the following, we use the notationU≤n(g) for every n ∈Nas defined in Proposition 5.5. This means that for everyn ∈N, we denote by U≤n(g) the k-submodule ψ g⊗≤n
of U(g).
on the axiom of choice (or at least its proof does). So do some of its consequences which we are going to use.
Theorem 5.10 appears, for instance, in literature as [30, Theorem 5.1.1], [17, Theorem 3.1], [3, Section V.2, Theorem 3], [16, Theorem 5], [31, Chapter XIII, Theorem 3.1], [8,§2.7, Corollaire 3], [27, Chapter III, Theorem 3.8]22 or [7, Theorem 3.2.2]. It can be easily derived from Theorem 5.9(a), but the other direction is more standard: Almost all proofs of Theorem 5.9(a) proceed by deriving Theorem 5.10 first ([4, §2.1], [5] and [11, §1.9] are very explicit about doing so - for example, Theorem 5.10 is [4, Lemma 2.1.8] and [5,§17.3, Corollary C]). Also, one of the most translucent proofs for Theorem 5.10 is given in [6] and in [29, §7.1].
Remark 5.12. Some sources prove Theorem 5.9(a)only in the case when I is well-ordered (and not just totally well-ordered). However, once Theorem 5.9 (a) is proven in this case, we can easily see that Theorem 5.9(a) holds for any totally ordered setI.
Here is why: The only difficult part of Theorem 5.9 (a) is the linear independency of the family (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in
. To prove this independency, it is enough to show that the family (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈Jn;
i1≤i2≤...≤in
is linearly independent for every finite subset J of I. But for every finite subsetJ of I, we can extend the ordering on J to a well-ordering of I (without changing the order of the elements of J), and apply the proof from the well-ordered case.
This all requires the axiom of choice, but then again, in all applications of Poincar´ e-Birkhoff-Witt I have seen, the basis is countable and thus a well-ordering is easy to find.
We note that a part of Theorem 5.10 holds more generally:
Proposition 5.13. Let k be a commutative ring. Let g be a k-Lie algebra. Let (ei)i∈I be a generating set of the k-moduleg, whereI is a totally ordered set. Then, (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in
is a generating set of the k-moduleU(g).
This is actually the easier part of Theorem 5.10, and is proven by induction in almost every text on Lie algebras. The same argument shows the following strengthening of this proposition:
Proposition 5.14. Let k be a commutative ring. Let g be a k-Lie algebra. Let (ei)i∈I be a generating set of the k-module g, where I is a totally ordered set. Let m ∈N. Then, the family (ei1 ⊗ei2⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in; n≤m
is a generating set of the k-module U≤m(g).
This proposition appears, e. g., in [4, Lemma 2.1.6] and [12, Part I, Chapter III, Lemma 4.4] (although in an unnecessarily restrictive version).
We notice the following slight strengthening of Theorem 5.10:
Corollary 5.15. Let k be a commutative ring. Let g be a k-Lie algebra. Assume that the k-module ghas a basis (ei)i∈I, where I is a totally ordered set. Let m∈N. Then, the family (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in; n≤m
is a basis of the k-module U≤m(g).
22Note that [27, Chapter III, Theorem 3.8] requireskto beC, but this requirement is neither necessary for the theorem nor used in the proof. The proof works just as well for the general case.
Proof of Corollary 5.15. The family (ei1⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in;n≤m
is a gener-ating set of the k-module U≤m(g) (by Proposition 5.14) and linearly independent (by Theorem 5.10). Therefore it is a basis of thek-module U≤m(g). This proves Corollary 5.15.
Note that while most literature does not explicitly mention Corollary 5.15, it often tacitly uses it (for example, when deriving Theorem 5.9 from Theorem 5.10).
Our next results are concerned with the case when his a free k-module and satisfies g=h⊕N for some free k-moduleN. This requirement is harsh in comparison to what we have required in previous sections, but it still encompasses the situation whenk is a field, and besides is satisfied for many standard cases such as (g,h) = (glnk,slnk) even if k is just a commutative ring with 1.
We are going to prove the following consequence of Poincar´e-Birkhoff-Witt:
Proposition 5.16. Let k be a commutative ring. Let g be a k-Lie algebra. Let m ∈N.
Let h be a Lie subalgebra of g such that h is a free k-module and such that there exists a free k-submodule N of g such thatg=h⊕N.
Then, U≤m(g)∩(U(g)·h) = U≤(m−1)(g)·h. (Here, we are using the notation of Definition 5.11, and we are abbreviating the k-submodule U(g)·ψ(h) of U(g) by U(g)·h.)
Before we sketch a proof of Proposition 5.16, let us recall a basic from linear algebra:
Lemma 5.17. Let k be a commutative ring. Let V be a free k-module with basis (eκ)κ∈K, where K is a set. Let X and Y be two subsets of K. Then,
heκ | κ∈Xi ∩ heκ | κ∈Yi=heκ | κ∈X∩Yi. (Here we are using Convention 1.28.)
Proof of Proposition 5.16. Let (ei)i∈P be a basis of the k-module h, and let (ei)i∈Q be a basis of the k-module N. Assume WLOG that the setsP and Qare disjoint. Let I =P ∪Q. Choose a well-ordering on I such that every element of Q is smaller than any element ofP. Then, (ei)i∈I is a basis of thek-moduleh⊕N =g. Proposition 5.10 thus yields that (ei1 ⊗ei2⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in
is a basis of thek-module U(g).
Let I∗ be the disjoint union of the sets In for all n ∈ N. In other words, let I∗ be the set of all finite sequences of elements of I. In particular, the empty sequence (i. e., the only element ofI0) is an element ofI∗.
For every two elementsκ∈I∗ andκ0 ∈I∗, we define an elementκ·κ0 ∈I∗ as follows:
Writeκ in the formκ= (i1, i2, ..., in) and writeκ0 in the formκ0 = (j1, j2, ..., jm); then setκ·κ0 = (i1, i2, ..., in, j1, j2, ...., jm).
Let K be the set
{(i1, i2, ..., in) | (i1, i2, ..., in)∈I∗; i1 ≤i2 ≤...≤in}
= [
n∈N
{(i1, i2, ..., in) | (i1, i2, ..., in)∈In; i1 ≤i2 ≤...≤in}. In other words, K is the set of all increasing finite sequences of elements of I.
Let us note that everyq ∈K∩Q∗ and p∈K∩P∗ satisfyq·p∈K. (This is because we have chosen a well-ordering on I such that every element of Q is smaller than any element of P.)
For every κ∈I∗, define an element eκ of U(g) by
eκ =ei1 ⊗ei2 ⊗...⊗ein, where (i1, i2, ..., in) is such that κ= (i1, i2, ..., in). Then, (eκ)κ∈K is a basis of thek-moduleU(g) (since this is just another way to state our knowledge that (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in
is a basis of the k-module U(g)). Thus, U(g) =heκ | κ∈Ki.
Let X be the subset {(i1, i2, ..., in)∈K | in∈P} of K.
Let Y be the subset {(i1, i2, ..., in)∈K | n ≤m} of K.
Clearly, X∩Y is the subset{(i1, i2, ..., in)∈K | in∈P; n ≤m} of K.
We notice that (eκ)κ∈Y is a basis of the k-moduleU≤m(g) (since this is just another way to state our knowledge that the family (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈In;
i1≤i2≤...≤in; n≤m
is a basis of thek-module U≤m(g)).
The definition of eκ readily yields
eκ·eκ0 =eκ·κ0 for every κ∈I∗ and κ0 ∈I∗. (69) Also note that
e(i)=ei for every i∈I, (70)
where ei means ei = ψ(ei) ∈ U(g) on the right hand side (this is a slight abuse of notation, but legitimate in view of the Poincar´e-Birkhoff-Witt theorem).
Now let us show that U(g)·h =heκ | κ∈Xi. In fact, U(g)
| {z }
=heκ | κ∈Ki
· h
|{z}
=hei |i∈Pi (since (ei)i∈P is a basis ofh)
=heκ | κ∈Ki · hei | i∈Pi
=heκei | (κ, i)∈K×Pi. (71) Hence, in order to prove that U(g)·h = heκ | κ∈Xi, it is enough to show that heκei | (κ, i)∈K ×Pi=heκ | κ∈Xi. In order to do so, we must prove that
heκ | κ∈Xi ⊆ heκei | (κ, i)∈K×Pi (72) and
heκei | (κ, i)∈K ×Pi ⊆ heκ | κ∈Xi. (73) Proof of (72). Everyκ∈X can be written in the form (i1, i2, ..., in) for some n∈N and (i1, i2, ..., in)∈K satisfying in∈P. Thus,
eκ =e(i1,i2,...,in)=e(i1,i2,...,in−1)·(in)=e(i1,i2,...,in−1)
| {z }
∈U(g)
e(in)
|{z}
=ein (by (70))
(by (69))
∈U(g) ein
|{z}
∈h(sincein∈P)
⊆U(g)·h.
We have thus shown that every κ∈X satisfieseκ ∈U(g)·h. Therefore,
heκ | κ∈Xi ⊆U(g)·h. (74) Combined with (71), this proves (72).
Proof of (73). Let (κ, i) ∈K ×P be arbitrary. We only have to prove that eκei ∈ heκ0 | κ0 ∈Xi (because once this is shown for every (κ, i) ∈ K ×P, it will become clear that heκei | (κ, i)∈K×Pi ⊆ heκ0 | κ0 ∈Xi = heκ | κ∈Xi, and this will prove (73)).
Sinceκ∈K, we can writeκin the form (i1, i2, ..., in) for somen ∈Nand (i1, i2, ..., in)∈ In satisfying i1 ≤ i2 ≤ ... ≤ in. Let in+1 = i. Let ν be the smallest integer in {1,2, ..., n+ 1} such that iν ∈ P (such a ν exists since in+1 = i ∈ P). Then, i1, i2, ..., iν−1 lie in Q whereas iν, iν+1, ..., in lie in P (this is because i1 ≤ i2 ≤ ... ≤ in and because every element ofQ is smaller than any element of P). Since in+1 =i also lies in P, we conclude that all of the elements iν, iν+1, ..., in+1 lie in P. This yields (iν, iν+1, ..., in+1)∈P∗.
Sincei1,i2,...,iν−1lie inQ, we have (i1, i2, ..., iν−1)∈Q∗. Combined with (i1, i2, ..., iν−1)∈ K (since i1 ≤i2 ≤...≤iν−1), this yields (i1, i2, ..., iν−1)∈K ∩Q∗.
Now we have
eκei =e(i1,i2,...,in)ei (since κ= (i1, i2, ..., in))
=e(i1,i2,...,in) ein+1
| {z }
=e(in+1) (by (70))
(sincei=in+1)
=e(i1,i2,...,in)e(in+1) =e(i1,i2,...,in)·(in+1) (by (69))
=e(i1,i2,...,in+1) =e(i1,i2,...,iν−1)·(iν,iν+1,...,in+1) =e(i1,i2,...,iν−1)e(iν,iν+1,...,in+1) (by (69)). But we can identify the universal enveloping algebraU(h) with a k-submodule ofU(g) - namely, with the k-submodule heκ | κ∈P∗iof U(g). The elemente(iν,iν+1,...,in+1) of U(g) lies in this submodule U(h) (because (iν, iν+1, ..., in+1)∈P∗).
Applying Proposition 5.13 to h and (ei)i∈P instead of g and (ei)i∈I, we see that (ei1 ⊗ei2 ⊗...⊗ein)n∈N; (i1,i2,...,in)∈Pn;
i1≤i2≤...≤in
is a generating set of the k-module U(h). In other words,
U(h) =
*
ei1 ⊗ei2 ⊗...⊗ein
| {z }
=e(i1,i2,...,in)
| n ∈N; (i1, i2, ..., in)∈Pn; i1 ≤i2 ≤...≤in
| {z }
this is equivalent to (i1,i2,...,in)∈K
+
=
e(i1,i2,...,in) | n ∈N; (i1, i2, ..., in)∈Pn; (i1, i2, ..., in)∈K
=
e(i1,i2,...,in) | n ∈N; (i1, i2, ..., in)∈K∩Pn
=heκ | κ∈K∩P∗i. Therefore,e(iν,iν+1,...,in+1) = P
λ∈K∩P∗
ρλeλfor some scalarsρλ ∈k(becausee(iν,iν+1,...,in+1) ∈ U(h)). Consider these scalars ρλ. It is easily seen that ρ() = 0, where () denotes the empty sequence (in fact, if the map εU(g) : U(g) → k is defined as in Propo-sition 1.78, then PropoPropo-sition 1.78 (b) yields that εU(g) e(iν,iν+1,...,in+1)
= 0 (because (iν, iν+1, ..., in+1) is not the empty sequence) on one hand butεU(g)
P
λ∈K∩P∗
ρλeλ
=ρ()
on the other, so that we obtain 0 = ρ()). Therefore, e(iν,iν+1,...,in+1) = P but this would be a slight overkill. Instead we only need the weaker result that heκ | κ∈X∩Yi ⊆ U≤(m−1)(g)·h, which can be seen exactly the same way as we
Next we prove a result that generalizes Lemma 4.3 of [2]:
23This is becauseλ6= () and λ∈P∗.