II. Lie algebras and enveloping algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 16 October 2012

All of the statements in §1 are proved in [Ser1987] Chapts. I-III. The statements in §3, except possibly (3.6), are proved in [Dix1994] Chapt. 2. The proof that the Lie algebra can be recovered from its enveloping algebra (3.6) can be found in [Bou1972] II §1.4. The classification theorem for semisimple Lie algebras, Theorem (2.2), is proved in [Ser1987] VI §4 Theorem 7. The results in (2.4) and (2.5) on the classification of finite dimensional modules for simple Lie algebras are proved in [Ser1987] VII §1-4. Theorem (2.8) is proved in [Bou1972] Chapt 6 §1.3 and Proposition (2.8) is proved in [Bou1972] Chapt 6 § 1.6 Cor. 2 and Cor. 3.

Semisimple Lie algebras

Definition of a Lie algebra

Let k be a field. A Lie algebra over k is a vector space 𝔤 over k with a bracket [,]:𝔤𝔤𝔤 which satisfies

[x,x]=0,for all x𝔤, [x,[y,z]]+ [z,[x,y]]+ [y,[z,x]]=0, for allx,y,z𝔤.

The first relation is the skew-symmetric relation and is equivalent to [x,y]=-[y,x], for all x,y𝔤, provided that chark2. The second relation is the Jacobi identity. A Lie algebra 𝔤 over k is finite dimensional if it is finite dimensional as a vector space over k, and it is complex if k=.

Definition of a simple Lie algebra

An ideal of 𝔤 is a subspace 𝔞𝔤 such that

[x,a]𝔞, for allx𝔤,and a𝔞.

A Lie algebra 𝔤 is abelian if [x,y]=0 for all x,y𝔤. A finite dimensional Lie algebra 𝔤 over a field k of characteristic 0 is simple if

  1. 𝔤 is not the one dimensional abelian Lie algebra,
  2. The only ideals of 𝔤 are 0 and 𝔤.

Definition of the radical of a Lie algebra

Let 𝔤 be a finite dimensional Lie algebra over a field k of characteristic 0. If 𝔞𝔤 is an ideal of 𝔤 define

D1𝔞=𝔞,and Dn𝔞= [ Dn-1𝔞, Dn-1𝔞 ] ,forn2.

An ideal a of 𝔤 is solvable if there exists a positive integer n such that Dn𝔞=0. The radical of 𝔤 is the largest solvable ideal of 𝔤. A finite dimensional Lie algebra is semisimple if its radical is 0.

Definition of simple modules for a Lie algebra

Let 𝔤 be a Lie algebra over a field k. A 𝔤-module is a vector space V over k with a 𝔤-action

𝔤V V xv x·v=xv

such that

[x,y]·v=x(yv) -y(xv),for all x,y𝔤,andv V.

A representation of 𝔤 on a vector space G is a map

ρ: 𝔤 End(V) x ρ(x) such thatρ ([x,y])=ρ (x)ρ(y)-ρ (y)ρ(x),

for all x,y𝔤. Every 𝔤-module V determines a representation of 𝔤 on V (and vice versa) by the formula

ρ(x)v=xv, for allx𝔤,and vV.

A submodule of a 𝔤-module V is a subspace WV such that xwW for all x𝔤 and wW. A simple or irreducible 𝔤-module is a 𝔤-module V such that the only submodules of V are 0 and V. A 𝔤-module V is completely decomposable if V is a direct sum of simple submodules.

Definition of the adjoint representation of a Lie algebra

Let 𝔤 be a finite dimensional Lie algebra over a field k. The vector space 𝔤 is a 𝔤-module where the action of 𝔤 on 𝔤 is given by

𝔤𝔤 𝔤 xy [x,y].

The linear transformation of 𝔤 determined by the action of an element x𝔤 is denoted adx. Thus,

adx(y)= [x,y],for all y𝔤.

The representation

ad: 𝔤 End(𝔤) x adx

is the adjoint representation of 𝔤.

Definition of the Killing form

Let 𝔤 be a finite dimensional Lie algebra over a field k. The Killing form on g is the symmetric bilinear form ,: 𝔤×𝔤k given by

x,y= Tr(adxady) ,for allx,y𝔤.

The Killing form , is invariant, i.e.

[x,y],z+ y,[x,z] =0,for allx,y,z𝔤.

Characterizations of semisimple Lie algebras

A finite dimensional Lie algebra 𝔤 over a field k of characteristic 0 is semisimple if any of the following equivalent conditions holds:

  1. 𝔤 is a direct sum of simple Lie subalgebras.
  2. The radical of 𝔤 is 0.
  3. Every finite dimensional 𝔤 module is completely decomposable and 𝔤=[𝔤,𝔤].
  4. The killing form on 𝔤 is non-degenerate.

Finite dimensional complex simple Lie algebras

Dynkin diagrams and Cartan matrices

A Dynkin diagram is one of the graphs in Table 1. A Cartan matrix is one of the matrices in Table 2. The (i,j) entry of a Cartan matrix is denoted αj(Hi). Notice that every Cartan matrix satisfies the conditions,

  1. αi(Hi)=2, for all 1ir,
  2. αj(Hi) is a non positive integer, for all ij,
  3. αi(Hj)=0 if and only if αj(Hi)=0.

If C is a Cartan matrix the vertices of the corresponding Dynkin diagram are labeled by αi, 1ir, such that αi(Hj) αj(Hi) is the number of lines connecting vertex αi to vertex αj. If αj(Hi)> αi(Hj) then there is a > sign on the edge connecting vertex αj to vertex αi, with the point towards αi. With these conventions it is clear that the Cartan matrix contains exactly the same information as the Dynkin diagram; each can be constructed from the other.

Classification of finite dimensional complex simple Lie algebras

Fix a Cartan matrix C= (αj(Hi)) 1i,jr . Let 𝔤C be the Lie algebra over given by generators

X1-, X2-, , Xr-, H1,H2,,Hr, X1+, X2+, , Xr+,

and relations

[Hi,Hj]=0, for all1i,jr, [Hi,Xj+]= αj(Hi) Xj+, [Hi,Xj-]= -αj(Hi) Xj-, for all1i,jr, [Xi+,Xj-] =δijHi, for1i,jr, [Xi+, [Xi+, [Xi+, -αj(Hi) +1brackets Xj+]]]=0, [Xi-, [Xi-, [Xi-, -αj(Hi) +1brackets Xj-]]]=0, forij.

Let C be a Cartan matrix and let 𝔤C be the Lie algebra defined above.

  1. The Lie algebra 𝔤C is a finite dimensional complex simple Lie algebra
  2. EVery finite dimensional complex simple Lie algebra is isomorphic to 𝔤C for some Cartan matrix C.
  3. If C, C are Cartan matices then 𝔤C𝔤C if and only ifC=C.

Triangular decomposition

Fix a Cartan matrix C= (αi(Hj)) 1i,jr and let 𝔤=𝔤C. Define

𝔫- = Lie subalgebra of𝔤 generated byX1-, X2-,, Xr-. 𝔥 = -span {H1,H2,,Hr} , 𝔫+ = Lie subalgebra of𝔤 generated byX1+, X2+,, Xr+,

The elements X1-, X2-, , Xr-, H1,,Hr, X1+, X2+, , Xr+ are linearly independent in 𝔤 and

𝔤=𝔫-𝔥 𝔫+,

The Lie subalgebra 𝔥𝔤 is a Cartan subalgebra of 𝔤 and the Lie subalgebra 𝔟=𝔥𝔫+ is a Borel subalgebra of 𝔤. The rank of 𝔤 is r=dim𝔥.

Weights and weight spaces

Fix a Cartan matrix C= (αj(Hi)) 1i,jr and let 𝔤=𝔤C. Let 𝔥*=Hom (𝔥,) and define the fundamental weights ω1,,ωr 𝔥* by

ωi(Hj)= δij,for 1i,jr.

Let V be a 𝔤--module and let μ=i=1r μiωi𝔥*. The subspace

Vμ = { vV hv=μ(h)v, forh𝔥 } = { vV Hiv=μiv, for1ir }

is the μ-weight space of V. Vectors vVμ are weight vectors of V of weight μ, wt(v)=μ. The weights of the 𝔤-module V are the elements μ𝔥* such that Vμ0. If μ is a weight of V, the multiplicity of μ in V is dim(Vμ). A highest weight vector in a 𝔤-module V is a weight vector vV such that 𝔫+v=0 or, equivalently, a weight vector vV such that Xi+v=0, for 1ir.

The set of dominant integral weights P+ and the weight lattice P are the subsets of 𝔥* given by

P+=i=1r ωiandP= i=1rωi ,respectively,

where =0.

Classification of simple 𝔤-module

Let 𝔤 be a finite dimensional complex simple Lie algebra. Every finite dimensional 𝔤--module V is a direct sum of its weight spaces and all weights of V are elements of P,

V=μPVμ.

Let 𝔤 be a finite dimensional complex simple Lie algebra.

  1. Every finite dimensional irreducible 𝔤--module V contains a unique, up to constant multiples, highest weight vector v+V and wt(v+) P+.
  2. Conversely, if λP+, then there is a unique (up to isomorphism) finite dimensional irreducible 𝔤--module, Vλ, with highest weight vector of weight λ.

Roots and the root lattice

Fix a Cartan matrix C= (αj(Hi)) 1i,jr and let 𝔤=𝔤C. The adjoint action of 𝔤 on 𝔤 (see (1.5)) makes 𝔤 into a finite dimensional 𝔤-module. An element αP, α0 is a root if the weight space 𝔤α0. A root is positive, α>0, if 𝔤α𝔫+ and negative, α<0, if 𝔤α𝔫-. We have

dim𝔤α=1 for all rootsα, 𝔫-=α<0 𝔤α,𝔥=𝔤0 ,𝔫+= α>0𝔤α, and𝔤=𝔫- 𝔥𝔫+.

The roots αi, 1ir, given by 𝔤αi=Xi+ are the simple roots. The Cartan matrix is the transition matrix between the simple roots and the fundamental weights,

αi=j=1r αi(Hj)ωj ,for1ir.

The root lattice is the lattice QP𝔥* given by Q=i=1r αi.

The inner product 𝔥*

Let 𝔤 be a finite dimensional complex simple Lie algebra and let C= (αj(Hi)) 1i,jr be the corresponding Cartan matrix. There exist unique positive integers d1,d2,,dr such that gcd ( d1,,dr ) =1 and the matrix (diαj(Hi)) 1i,jr is symmetric. The integers d1,d2,,dr are given explicitly by

Ar,Dr, E6,E7, E8: di=1for all 1ir, Br: di=1for 1ir-1,and dr=2, Cr: di=2,for 1ir-1, anddr=1, F4: d1=d2=1, andd3=d4=2, G2: d1=3,and d2=1.

Let α1,,αr be the simple roots for 𝔤. Define

𝔥*= i=1r αi,

so that 𝔥* is a real vector space of dimension r. Define an symmetric inner product on 𝔥* by

(αi,αj)= diαi(Hj), for1i,jr,

where the values αj(Hi) are the entries of the Cartan matrix corresponding to 𝔤.

The Weyl group corresponding to 𝔤

Let 𝔤 be a finite dimensional complex simple Lie algebra and let R be the set of roots of 𝔤 and let α1,,αr be the simple roots. For each root αR define a linear transformation of 𝔥* by

sα(λ)=λ- (λ,α)α, whereα= 2α(α,α).

The Weyl group corresponding to 𝔤 is the group of linear transformations of 𝔥* generated by the reflections sα, αR,

W= sα αR .

The simple reflections in W are the elements si=sαi, 1ir.

Let 𝔤 be a finite dimensional complex simple Lie algebra and let W be the Weyl group corresponding to 𝔤.

  1. The Weyl group W is a finite group.
  2. The Weyl group W can be presented by generators s1,,sr and relations si2 = 1, 1ir, sisj sisj mij factors = sjsi sjsi mij factors forij, where mij = { 2, ifαi (Hj)αj (Hi)= 0, 3, ifαi (Hj)αj (Hi)= 1, 4, ifαi (Hj)αj (Hi)= 2, 6, ifαi (Hj)αj (Hi)= 3.

Let wW. A reduced decomposition for w is an expression

w=si1 si2 si(w)

of w as a product of generators which is as short as possible. The length (w) of this expression is the length of w.

Let 𝔤 be a finite dimensional simple complex Lie algebra and let W be the Weyl group corresponding to 𝔤.

  1. There is a unique longest element w0 in W.
  2. Let w0=si1 siN be a reduced decomposition for the longest element of W. Then the elements β1=αi1, β2=si1 (αi2), ,βN=si1 si2siN-1 (αiN), are the positive roots of 𝔤.

Enveloping algebras

Motivation for the enveloping algebra

A Lie algebra 𝔤 is not an algebra, at least as defined in I (1.1), because the bracket is not associative. We would like to find an algebra, or even better a Hopf algebra, 𝔘𝔤, for which the category of modules for 𝔘𝔤 is the same as the category of modules for 𝔤. In other words we want 𝔘𝔤 to carry all the information that 𝔤 does and to be a Hopf algebra.

Definition of the enveloping algebra

Let 𝔤 be a Lie algebra over k. Let T(𝔤)= k0 𝔤k be the tensor algebra of 𝔤 and let J be the ideal of T(𝔤) generated by the tensors

xy-yx- [x,y],where x,y𝔤.

The enveloping algebra of 𝔤, 𝔘𝔤, is the associative algebra

𝔘𝔤= T(𝔤) J .

There is a canonical map

α0: 𝔤 𝔘𝔤 x x+J.

The algebra 𝔘𝔤 can be given by the following universal property:

Let α:𝔤A be a mapping of 𝔤 into an associative algebra A over k such that

α([x,y])= α(x)α(y)- α(y)α(x),

for all x,y𝔤, and let 1 and 1A denote the identities in 𝔘𝔤 and A respectively. Then there exists a unique algebra homomorphism τ:𝔘𝔤A such that τ(1)=1A and α=τα0, i.e. the following diagram commutes.

𝔤 α0 𝔘𝔤 α τ A

A functional way of realising the enveloping algebra

If A is an algebra over k, as defined in I (1.1), then define a bracket on A by

[x,y]=xy-yx, for allx,yA.

This defines a Lie algebra structure on A and we denote the resulting Lie algebra by L(A) to distinguish it from A. L is a functor from the category of algebras to the category of Lie algebras. 𝔘 is a functor from the category of Lie algebras to the category of algebras. In fact 𝔘 is the left adjoint of the functor L since

Homalg (𝔘𝔤,A)= HomLie (𝔤,L(A))

for all Lie algebras 𝔤 and all algebras A.

The enveloping algebra is a Hopf algebra

The enveloping algebra 𝔘𝔤 of 𝔤 is a Hopf algebra if we define

  1. a comultiplication, Δ: 𝔘𝔤𝔘𝔤 𝔘𝔤, by Δ(x)=x 1+1x,for all x𝔤,
  2. a counit, ε:𝔘𝔤 k, by ε(x)=0, for allx𝔤,
  3. and an antipode, S:𝔘𝔤 𝔘𝔤, by S(x)=-x, for allx𝔤.

Modules for the enveloping algebra and the Lie algebra are the same!

Every 𝔤-module M is a 𝔘𝔤-module and vice versa, since there is a unique extension of the action of 𝔤 on M to a 𝔘𝔤-action on M.

The Lie algebra can be recovered from its enveloping algebra!

An element x of a Hopf algebra A is primitive if

Δ(x)=1x +x1.

It can be shown that if chark=0 then the subspace g of 𝔘𝔤 is the set of primitive elements of 𝔘𝔤. Thus, if chark=0, we can "determine" the Lie algebra 𝔤 from the algebra 𝔘𝔤 and the Hopf algebra structure on it.

A basis for the enveloping algebra

The following statement is the Poincaré-Birkhoff-Witt theorem.

Suppose that 𝔤 has a totally ordered basis (xi) iA . Then the elements

xi1 xi2 xin

in the enveloping algebra 𝔘𝔤, where i1i2 in is an arbitrary increasing finite sequence of elements of Λ, form a basis a 𝔘𝔤.

The enveloping algebra of a complex simple Lie algebra

A presentation by generators and relations

Let 𝔤 be a finite dimensional complex simple Lie algebra and let C= (αj(Hi)) 1i,jr be the corresponding Cartan matrix. Then the enveloping algebra 𝔘𝔤 of 𝔤 can be presented as the algebra over generated by

X1-, X2-, , Xr-, H1,H2,, Hr, X1+, X2+, , Xr+,

with relations

[Hi,Hj]=0, for all1i,jr, [Hi,Xj+]= αj(Hi) Xj+, [Hi,Xj-]= -αj(Hi) Xj-, for all1i,jr, [Xi+,Xj-] =δijHi, for1i,jr, s+t=1-αj(Hi) (-1)s ( 1-αj(Hi) s ) (Xi±)s Xj± (Xi±)t =0, forij,

where, if a,b𝔘𝔤, we use the notation [a,b]=ab-ba. Note that since

[a, [a, [a, brackets b]]]= s+t= (-1)s (s) asbat,

for any two elements a,b𝔘𝔤 and any positive integer , the relations for 𝔘𝔤 are exactly the same as the relations for 𝔤 given in (2.2).

Triangular decomposition

Let 𝔤 be a finite dimensional complex simple Lie algebra as presented in (2.2). Recall from (2.3) that 𝔤 has a decomposition

𝔤=𝔫- 𝔥𝔫+,

where

𝔫- = Lie subalgebra of𝔤 generated byX1-, X2-,, Xr-. 𝔥 = -span {H1,H2,,Hr} , 𝔫+ = Lie subalgebra of𝔤 generated byX1+, X2+,, Xr+

It follows from this and the Poincaré-Birkhoff-Witt theorem that

𝔘𝔤𝔘𝔫- 𝔘𝔥𝔘𝔫+, as vector spaces.

Grading on 𝔘𝔫+ and 𝔘𝔫-

Let 𝔤 be a finite dimensional complex simple Lie algebra as presented in (2.2). Let α1,,αr be the simple roots for 𝔤 and let

Q+=iαi, where=0 .

For each element ν=i=1r νiαiQ+ define

(𝔘𝔫+)ν = span- { Xi1+ Xip+ Xi1+ Xip+has νj-factors of type Xj+ } (𝔘𝔫-)ν = span- { Xi1- Xip- Xi1- Xip-has νj-factors of type Xj- } .

Then

𝔘𝔫-= νQ+ (𝔘𝔫-)ν, and𝔘𝔫+= νQ+ (𝔘𝔫+)ν,

as vector spaces.

Pointcaré-Birkhoff-Witt bases of 𝔘𝔫-, 𝔘𝔥, and 𝔘𝔫+

Let 𝔤 be a finite dimensional complex simple Lie algebra as presented in (2.2), let 𝔫+, 𝔫- and 𝔥 be as in (2.3) and recall the root spaces 𝔤α from (2.6). Let W be the Weyl group corresponding to 𝔤. Fix a reduced decomposition of the longest element w0W, w0=si1 siN, and define

β1=αi1, β2=si1 (αi2),, βN=si1 si2siN-1 (αiN).

The elements β1,,βN are the positive roots 𝔤 and the elements -β1,,-βN are the negative roots of 𝔤.

For each rootα, fix an elementXα𝔤α .

Since 𝔤α is 1-dimensional Xα is uniquely defined, up to multiplication by a constant. Since

𝔫-=α<0 𝔤α, 𝔫+=α>0 𝔤α, 𝔥=span- {H1,H2,,Hr} and𝔤=𝔫- 𝔥𝔫+,

it follows that

{ Xβ1,, XβN } is a basis of𝔫+, { X-β1 ,, X-βN } is a basis of𝔫-,and { H1,H2 ,, Hr } is a basis of𝔥.

Then, by the Poincaré-Birkhoff-Witt theorem,

{ Xβ1p1 Xβ2p2 XβNpN p1,,pN 0 } is a basis of𝔘𝔫+, { X-β1nN X-β2n2 X-β1n1 n1,,nN 0 } is a basis of𝔘𝔫- ,and { H1s1 H2s2 H1sr s1,,sN 0 } is a basis of𝔘𝔥.

The Casimir element in 𝔘𝔤

Let 𝔤 be a finite dimensional simple complex Lie algebra and let , be the Killing form on 𝔤 (see (1.6)). Let {bi} be a basis of 𝔤 and let {bi} be the dual basis of 𝔤 with respect to the Killing form. Let c be the element of the enveloping algebra 𝔘𝔤 of 𝔤 given by

c=ibibi.

Then

cis in the center of𝔘𝔤.

Any central element of 𝔘𝔤 must act on each finite dimensional simple module by a constant. For each dominant integral weight λ let Vλ be the finite dimensional simple 𝔘𝔤-module indexed by λ (see (2.5)). Let ρ be the element of 𝔥* given by

ρ=12α>0 α,

where the sum is over all positive roots for 𝔤. Then the element

cacts onVλ by the constant (λ+ρ,λ+ρ)- (ρ,ρ),

where the inner product on 𝔥* is as given in (2.7).

Ar-1: α1 α2 α3 αr-1 αr Br: α1 α2 α3 αr-1 αr Cr: α1 α2 α3 αr-1 αr Dr: α1 α2 α3 αr-2 αr-1 αr E6: α1 α3 α4 α5 α6 α2 E7: α1 α3 α4 α5 α6 α7 α2 E8: α1 α3 α4 α5 α6 α7 α8 α2 F4: α1 α2 α3 α4 G2: α1 α2 Table 1.Dynkin diagrams corresponding to finite dimensional complex simple Lie algebras Ar-1: ( 2 -1 0 0 -1 2 -1 0 0 -1 2 0 0 -1 2 -1 0 0 -1 2 ) Br: ( 2 -1 0 0 -1 2 -1 0 0 -1 2 0 0 -1 2 -2 0 0 -1 2 ) Cr: ( 2 -1 0 0 -1 2 -1 0 0 -1 2 0 0 -1 2 -1 0 0 -2 2 ) Dr: ( 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 -1 0 0 -1 2 0 0 0 -1 0 2 ) E6: ( 2 0 -1 0 0 0 0 2 0 -1 0 0 -1 0 2 -1 0 0 0 -1 -1 2 -1 0 0 0 0 -1 2 -1 0 0 0 0 -1 2 ) E7: ( 2 0 -1 0 0 0 0 0 2 0 -1 0 0 0 -1 0 2 -1 0 0 0 0 -1 -1 2 -1 0 0 0 0 0 -1 2 -1 0 0 0 0 0 -1 2 -1 0 0 0 0 0 -1 2 ) E8: ( 2 0 -1 0 0 0 0 0 0 2 0 -1 0 0 0 0 -1 0 2 -1 0 0 0 0 0 -1 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 -1 0 0 0 0 0 0 -1 2 ) F4: ( 2 -1 0 0 -1 2 -2 0 0 -1 2 -1 0 0 -1 2 ) G2: ( 2-1 -32 ) Table 2. Cartan matrices corresponding to finite dimensional complex simple Lie algebras

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

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