2 Construction of the universal enveloping algebra

Math564: Representation theory of sl

Talk 4: Universal enveloping algebra I - the PBW theorem

Mariya Stamatova University of Zürich

April 6, 2019

Abstract

Today we are going to introduce the notion of the universal enveloping algebraUpgqas well as the PBW theorem, which allows us to construct a basis forUpgq.

1 Some preliminaries and notations

Before we start with the construction of the universal enveloping algebra, we need to recall some algebraic structures and set notations that we are going to use throughout the entire talk.

Recall that an associative unital algebra over a fieldKis a pairpA,q, consisting of a vector spaceA, together with a bilinear multiplication:AAÑA,a, bÞÑab, which is associa- tive, i.e. for anya, b, c P Awe havepabqc apbcq. Unital means that there is an element 1 P A, such thata1 1a afor anya P A. An algebra homomorphism is a linear map ϕ:AÑB,xyÞÑϕpxyq ϕpxqϕpyqforA, Bassociative algebras andx, yPA.

Each associative algebrapA,qcan be turned into a Lie algebra by replacing the multiplica- tion with the commutator, i.e. for any two elementsa, bPAwe havera, bs abba. We shall denote this Lie algebra asA^pq.

Consider the case ifVis a vector space, then the space of all endomorphisms ofVhas the natural structure of an associative unital algebra with multiplication being the composition of linear operators onV. Denote this associative algebraLpVqand its underlying Lie algebra as LpVq^pq.

To define ansl₂-module we need a Lie algebra homomorphism fromsl₂ toLpVq^pq, that is a linear mapϕ: sl₂ÑLpVq, which satisfies:

ϕprx, ysq rϕpxq, ϕpyqs ϕpxqϕpyq ϕpyqϕpxqfor allx, yPsl₂. Using the notation from the first talk, we getHϕphq,FϕpfqandEϕpeq.

This is a general construction, i.e. replacingsl₂with any Lie algebraggives the notion of a module over any Lie algebra. The homomorphismϕis usually called a representation of the Lie algebra. As we dicussed in the first talk, we use module and representation interchangeably, since they are equivalent definitions.

2 Construction of the universal enveloping algebra

There is an issue with the Lie algebra homomorphismϕ: sl₂ÑLpVq^pq, namely in the non- trivial cases the image ofϕis not closed with respect to composition but it is closed only with respect to taking the commutator of the linear operators. To fix this problem we have to look for external algebraic objects whose properties are related tog(sometimes not obviously).

We shall define an associative algebra Upgq, called theuniversal enveloping algebra of gand show that it has the following properties:

• The Lie algebragis a canonical subalgebra ofUpgq^pq;

• Anyg-action on any vector space canonically extends to aUpgq-action on the same vector space;

• The extension and the restriction from Upgq to g are mutually inverse isomorphisms between the categoriesg-modandUpgq-mod.

The last property is very important, since it says that there is a one-to-one correspondence between g-modules and Upgq-modules. Thus, any g-module corresponds to a morphism of associative algebrasψ: Upgq ÑLpVqand the image of this morphism is always closed with respect to composition of operators. This implies that one should study the internal structure ofUpgq. A disadvantage of the universal enveloping algebra is that in any non-trivial case it is infinite-dimensional, while the Lie algebragis finite dimensional.

Definition 2.1. LetRxe, f, hybe the free associative algebra with generatorse,fandhand quotient it by the idealI, generated by the relationseffeh,heeh2e,hffh 2f.

We call the quotientRxe, f, hy{Ithe universal enveloping algebra of a Lie algebragand denote it asUpgq.

Remark 2.2. We will identify the elements ofRxe, f, hywith their images inUpgq. Lemma 2.3. (a)There is a unique linear map: gÑUpgqsatisfying:

peq e, pfq f, phq h.

(b)The map is a linear homomorphism of Lie algebras: gÑUpgq^pq. Proof. (a)Clear, since generators are mapped to generators.

(b)Follows from the definition ofUpgqandpreserving the Lie bracket.

Remark 2.4. The mapis called canonical embedding ofgintoUpgq^pq. This map is injective, which will be proved later.

The main result of this section is the following universal property ofUpgq:

Theorem 2.5. LetAbe any associative algebra andϕ: gÑA^pqbe any homomorphism of Lie algebras. There exists a unique homomorphismϕs : Upgq ÑAof associative algebras, such that the diagram commutes:

g A

Upgq

ϕ

D!sϕ

i.e. we haveϕsϕ.

Proof. We need to prove the existence and uniqueness ofϕ. We shall begin with the existence.s Consider the discussed free associative algebraRxe, f, hy. For any associative algebraAwe have the unique homomorphismψ:RÑA, defined via

ψpeq ϕpeq, ψpfq ϕpfq, ψphq ϕphq (2.1) Consider the natural projectionπ : R Upgq Rxe, f, hy{I. Let K Kerpπq. Then we have:

ψpeffeq ψpeqψpfq ψpfqψpeq ϕpeqϕpfq ϕpfqϕpeq rϕpeq, ϕpfqs ϕpre,fs

ϕphq ψphq.

This means thatψpeffehq 0. Similarly,ψpheeh2eq 0andψphffh 2fq 0. This implies that the image of the kernel ψpKq is trivial. Therefore, ψ factors through Rxe, f, hy{KUpgq.

Denote byϕs the homomorphismϕs : Upgq, then the compositionϕ sϕfollows.

Now we prove the uniqueness ofϕ. Sinces ψis unique, as we already said, then it implies the uniqueness ofϕ, sinceϕ sϕgives the formulas 2.1.

Similarly as for many algebraic objects, the universal enveloping algebra is defined uniquely up to isomorphism.

Proposition 2.6. LetUpgq¹ be another associative algebra such that there exists a fixed homo- morphism¹ : gÑ pUpgq¹q^pqof Lie algebras having the universal property. Then we have that Upgqis canonically isomorphic toUpgq¹.

Proof. Set A=Upgq andϕ=. We obtainϕs id_Upgq and we know that the identity map is unique.

TakeUpgq¹=Aandϕ ¹. From the universal property in Theorem 2.5 we get a homomor- phisms¹ : Upgq ÑUpgq¹. Similarly, the universal property ofUpgq¹ gives another homomor- phisms: Upgq¹ÑUpgq.

As next, consider the compositionss¹ s id_Upgq1 ands s¹ id_Upgq, which gives the claim.

g Upgq¹

Upgq

¹ id_{Upgq 1}

s

id_Upgq

s¹

The universal property allows us to find some relations between theg-modules andUpgq- modules.

Remark 2.7. IfAandBare associative algebras andψ:AÑBa homomorphism of algebras, thenψ:A^pqÑB^pqis a homomorphism of Lie algebras.

Proposition 2.8. (a)LetVbe ag-module defined via the Lie algebra homomorphismϕ: g Ñ LpVq^pq. Then the homomorphismϕs : Upgq ÑLpVq, given by the universal property, endowsV with the canonical structure of aUpgq-module.

(b)LetV be aUpgq-module given byψ : Upgq Ñ LpVq. Then the compositionψis a Lie algebra homomorphism fromg to LpVq^pq, which endowsV with the canonical structure of a g-module.

(c) LetV and W be two g-modules with the induced structures ofUpgq-modules, given by (a).

Then Hom_gpV, Wq Hom_UpgqpV, Wq.

(d)LetVandWbe twoUpgq-modules with the induced structures ofg-modules given by (b). Then we have Hom_UpgqpV, Wq HomgpV, Wq.

(e)The operations in (a) and (b) are mutually inverse to each other.

Proof. (a)Follows immediately from the universal property.

(b)ψis a Lie algebra homomorphism. This is what Remark 2.7 implies.

(c)and(d)hold true, since for anyg-module and the associatedUpgq-moduleVthe image ofg inLpVq^pqis generated by the same elements as the image ofUpgqinLpVq.

(e)Follows from the definition ofand its uniqueness.

Letg-modbe the category of all leftg-modules andUpgq-modbe the category of all left Upgq-modules. The next corollary defines this very important relation.

Corollary 2.9. The operations defined in Proposition 2.8(a)and(b)are mutually inverse isomor- phisms between the categoriesg-modandUpgq-mod, i.e. there is a functorF: g-modÑUpgq- mod, another functorG: Upgq-modÑUpgq-modand two natural isomorphismsδ:FGñ Id_Upgqmodandη:GFñId_gmod.

Remark 2.10. The equivalence of the categories defined above allows us to use the notions of ag-module andUpgq-module interchangeably.

IfV is ag-module withvPV anduPUpgq, then we denote the action ofuonvbyupvq. In particular,epvq Epvq,fpvq Fpvq,hpvq Hpvq.

Remark 2.11. There is another way to constructUpgq.

First, define the tensor algebraTpgqof a Lie algebragas follows:

T⁰:C,T¹ :g,T²:gbg, more generallyTⁿ:gloooooooomoooooooonbgb bg

ntimes

.

Then, the tensor algebra is the associative unital algebraTpgq : T⁰ `T<sup>1</sup>`T²`. . . with multiplication given by concatenation of tensor words and the empty word for a unit element.

LetJdenote the two-sided ideal ofTpgq, generated by all elements of the formxbyybx rx, ys, wherex, y Pg. Then the universal algebraUpgqis constructed by taking the quotient of the tensor algebra by the idealJ, namelyTpgq{J.

3 The PBW theorem

The definition ofUpgqdoesn’t give us enough information about this algebra, for instance we don’t know if it is finite-dimensional or infinite- dimensional. It is not even clear yet that is injective.

This part of the talk will be focussed on the construction of an explicit basis ofUpgq. Theorem 3.1(Poincaré-Birkhoff-Witt). The settfⁱh^je^k:i, j, kPN0uis a basis ofUpgq. Remark 3.2. The theorem is usually called the PBW theorem. The monomials fⁱh^je^k are called standard monomials. They form a basis of the polynomial algebraCrf, h, es, which is commutative, unlikeUpgq.

Before we prove Theorem 3.1 we need two intermediate results.

Lemma 3.3. The standard monomials generateUpgq.

Proof. Recall the free algebraRxe, f, hy. Its basis is given by arbitrary monomialsx₁x₂. . . x_k wherekPN0andx_i P te, f, hufor alli1, . . . k.

We need to prove that each such monomial can be written as a linear combination of standard monomials.

We shall use induction onk:

• k1: nothing to prove;

• Fork¡1consider some monomialx₁x₂. . . x_kas above.

We call a pair of indices pi, jq, 1 ¤ i   j ¤ kan inversion, of one of the following situations holds true:

# x_i h x_jf,

# x_ie x_jf,

# x_ie x_jh.

Proceed by induction on the number of inversions inx₁x₂. . . x_k. If the monomials are already ordered, i.e. there are no inversions, we getx₁x₂. . . x_k, which is standard.

Otherwise, fix an inversionpi, i 1q:

x₁. . . x_i1x_ix_{i 1} x_{i 2}. . . x_kx₁. . . x_i1 x_{i 1}x_i . . . x_k x₁. . . x_i1rx_i, x_{i 1}sx_{i 2}. . . x_k. We notice that the bracket rx_i, x_{i 1}s can take values in the set th,2e,2fu, the second summand is of degree k1. The first summand has one inversion less than x₁x₂. . . x_k, hence the claim holds true for anyk.

Remark 3.4. Consider the vector spaceCra, b, cs. By using the induction on the degree of a monomial, we can describe the actions ofE,FandHonV:

Fpaⁱb^jc^kq a^{i 1}b^jc^k, (3.1) Hpaⁱb^jc^kq

#

b^{j 1}c^k, ifi0

FpHpaⁱ¹b^jc^kqq 2aⁱb^jc^k, otherwise (3.2)

Epaⁱb^jc^kq

$'

&

'%

c^{k 1}, ifi, j0

HpEpb^j1c^kqq 2Epb^j1c^kq, otherwise FpEpa^a1b^jc^kqq Hpaⁱ¹b^jc^kqifi0,

(3.3)

wherei, j, kPN0.

We can modify a little the last two equations above to get:

Hpaⁱb^jc^kq

#

b^{j 1}c^k, ifi0

FpHpaⁱ¹b^jc^kqq rH, Fsaⁱ¹b^jc^k, otherwise; (3.2*)

Epaⁱb^jc^kq

$'

&

'%

c^{k 1}, ifi, j0

HpEpb^j1c^kqq rE, Hspb^j1c^kq, ifi0, j0, FpEpa^a1b^jc^kqq rE, Fspaⁱ¹b^jc^kqifi0,

(3.3*)

wherei, j, kPN0.

Lemma 3.5. The equations (3.1)-(3.3) from the remark above define onV the structure of ag- module.

Proof. We have to check the three relations for theg-structure.

• We begin with the relationrH, Fs 2F.

Fori, j, kPN0we have:

HpFpaⁱb^jc^kqq^p3.1qHpa^{i 1}b^jc^kq^p3.2q

FpHpaⁱb^jc^kqq 2a^{i 1}b^jc^kp3.1qFpHpaⁱb^jc^kqq 2Fpaⁱb^jc^kq, which implies thatrH, Fs 2F.

• Now we have to prove the relationrE, Fs 2F. Fori, j, kPN0we have:

EpFpaⁱb^jc^kqq^p3.1qEpa^{i 1}b^jc^kq^p3.3qFpEpaⁱb^jc^kqq Hpaⁱb^jc^kq, and the relation is proved.

• As next, we prove the relationrH, Es 2E, which we rewrite asEHHE 2E.

For anyj, kPN0andi0we have:

EpHpb^jc^kqq Epb^{j 1}c^kq^p3.3qHpEpb^jc^kqq 2Epb^jc^kq and the relationrH, Es 2Eis proved on monomials of the formb^jc^k.

The part with the proof of this relation for monomialsaⁱb^jc^k, whereiPNandj, kPN0

is more complicated.

For this we use induction oni:

1. The casei0is done.

2. For the casei¥1, write the relationrH, Es 2EasHEEH2E0.

ApplyHEEH2Eto the monomialaⁱb^jc^k, use the equations (3.1)-(3.3), together with the modified equations (3.2*) and (3.3*) to obtain:

pHEEH2Eqpaⁱb^jc^kq

pHFE HrE, Fs EFHErH, Fs 2FE2rE, Fsqpaⁱ¹b^jc^kq. (3.4) By induction we have2FEFrE, Hs, usingrH, Fs 2F, we have:

HrE, Fs HEFHFE, ErH, Fs EHFEFH, 2rE, Fs rE,rH, Fss. Applying these relations to (3.4) gives us:

pHEEH2Eqpaⁱb^jc^kq prF,rE, Hss rE,rH, Fssqpaⁱ¹b^jc^kq. (3.5) We know thatrE, Fs H, so we can add the zero term0 rH, Hs rH, Hs rH,rE, Fss rH,rF, Essto the equation (3.5) and get:

pHEEH2Eqpaⁱb^jc^kq

prlooooooooooooooooooooooomooooooooooooooooooooooonF,rE, Hss rE,rH, Fss rH,rF, Essq

Jacobi identity forLpVq^pq

paⁱ¹b^jc^kq.

Hence, the last relation is satisfied.

Now we are ready to prove the PBW theorem.

Proof. To prove that the standard monomials form a basis inUpgqwe need to show:

• They generateUpgq;

• The are linearly independent.

The first part was proved in Lemma 3.3.

What is left to prove is the linear independency.

Consider now theUpgq-moduleVfrom Lemma 3.5.

Then for alli, j, kPN0for the constant polynomial1PVwe have:

FⁱH^jK^kp1q aⁱb^jc^k. The elementsaⁱb^jc^kPVare linearly independent.

Hence, the linear operatorsFⁱH^jE^kare linearly independent as well.

These linear operators are exactly the images of the standard monomials under the homomor- phism, defining theUpgq-structure onV, it follows that the standard monomials are linearly independent and this proves the statement of the theorem.

Corollary 3.6. The canonical embedding: gãÝÑUpgq^pqis injective.

Proof. We know that the elementse,fandhform a basis ofgand that the elementspeq e, pfq fandphq hare linearly independent inUpgqby PBW.

This means we can identifygwithpgq.