## II. Lie algebras and enveloping algebras

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 $\left[,\right]:\phantom{\rule{0.2em}{0ex}}𝔤\otimes 𝔤\to 𝔤$ which satisfies

$[x,x]=0,for all x∈𝔤, [x,[y,z]]+ [z,[x,y]]+ [y,[z,x]]=0, for allx,y,z∈𝔤.$

The first relation is the skew-symmetric relation and is equivalent to $\left[x,y\right]=-\left[y,x\right],$ for all $x,y\in 𝔤,$ provided that $\text{char}\phantom{\rule{0.2em}{0ex}}k\ne 2\text{.}$ 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 $k=ℂ\text{.}$

### Definition of a simple Lie algebra

An ideal of $𝔤$ is a subspace $𝔞\subseteq 𝔤$ such that

$[x,a]∈𝔞, for allx∈𝔤,and a∈𝔞.$

A Lie algebra $𝔤$ is abelian if $\left[x,y\right]=0$ for all $x,y\in 𝔤\text{.}$ 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 $𝔤\text{.}$

### Definition of the radical of a Lie algebra

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

$D1𝔞=𝔞,and Dn𝔞= [ Dn-1𝔞, Dn-1𝔞 ] ,forn≥2.$

An ideal $a$ of $𝔤$ is solvable if there exists a positive integer $n$ such that ${D}^{n}𝔞=0\text{.}$ The radical of $𝔤$ is the largest solvable ideal of $𝔤\text{.}$ 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 $k\text{.}$ A $𝔤\text{-module}$ is a vector space $V$ over $k$ with a $𝔤\text{-action}$

$𝔤⊗V ⟶ V x⊗v ⟼ x·v=xv$

such that

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

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 $x,y\in 𝔤\text{.}$ Every $𝔤\text{-module}$ $V$ determines a representation of $𝔤$ on $V$ (and vice versa) by the formula

$ρ(x)v=xv, for allx∈𝔤,and v∈V.$

A submodule of a $𝔤\text{-module}$ $V$ is a subspace $W\subseteq V$ such that $xw\in W$ for all $x\in 𝔤$ and $w\in W\text{.}$ A simple or irreducible $𝔤\text{-module}$ is a $𝔤\text{-module}$ $V$ such that the only submodules of $V$ are 0 and $V\text{.}$ A $𝔤\text{-module}$ $V$ 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 $k\text{.}$ The vector space $𝔤$ is a $𝔤\text{-module}$ where the action of $𝔤$ on $𝔤$ is given by

$𝔤⊗𝔤 ⟶ 𝔤 x⊗y ⟼ [x,y].$

The linear transformation of $𝔤$ determined by the action of an element $x\in 𝔤$ is denoted ${\text{ad}}_{x}\text{.}$ Thus,

$adx(y)= [x,y],for all y∈𝔤.$

The representation

$ad: 𝔤 ⟶ End(𝔤) x ⟼ adx$

is the adjoint representation of $𝔤\text{.}$

### Definition of the Killing form

Let $𝔤$ be a finite dimensional Lie algebra over a field $k\text{.}$ The Killing form on $g$ is the symmetric bilinear form $⟨,⟩:\phantom{\rule{0.2em}{0ex}}𝔤×𝔤\to k$ given by

$⟨x,y⟩= Tr(adxady) ,for allx,y∈𝔤.$

The Killing form $⟨,⟩$ is invariant, i.e.

$⟨[x,y],z⟩+ ⟨y,[x,z]⟩ =0,for allx,y,z∈𝔤.$

### 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 $𝔤=\left[𝔤,𝔤\right]\text{.}$
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 $\left(i,j\right)$ entry of a Cartan matrix is denoted ${\alpha }_{j}\left({H}_{i}\right)\text{.}$ Notice that every Cartan matrix satisfies the conditions,

1. ${\alpha }_{i}\left({H}_{i}\right)=2,$ for all $1\le i\le r,$
2. ${\alpha }_{j}\left({H}_{i}\right)$ is a non positive integer, for all $i\ne j,$
3. ${\alpha }_{i}\left({H}_{j}\right)=0$ if and only if ${\alpha }_{j}\left({H}_{i}\right)=0\text{.}$

If $C$ is a Cartan matrix the vertices of the corresponding Dynkin diagram are labeled by ${\alpha }_{i},$ $1\le i\le r,$ such that ${\alpha }_{i}\left({H}_{j}\right){\alpha }_{j}\left({H}_{i}\right)$ is the number of lines connecting vertex ${\alpha }_{i}$ to vertex ${\alpha }_{j}\text{.}$ If ${\alpha }_{j}\left({H}_{i}\right)>{\alpha }_{i}\left({H}_{j}\right)$ then there is a $>$ sign on the edge connecting vertex ${\alpha }_{j}$ to vertex ${\alpha }_{i},$ with the point towards ${\alpha }_{i}\text{.}$ 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 $C={\left({\alpha }_{j}\left({H}_{i}\right)\right)}_{1\le i,j\le r}\text{.}$ Let ${𝔤}_{C}$ be the Lie algebra over $ℂ$ given by generators

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

and relations

$[Hi,Hj]=0, for all1≤i,j≤r, [Hi,Xj+]= αj(Hi) Xj+, [Hi,Xj-]= -αj(Hi) Xj-, for all1≤i,j≤r, [Xi+,Xj-] =δijHi, for1≤i,j≤r, [Xi+, [Xi+, … [Xi+, ⏟ -αj(Hi) +1brackets Xj+]]…]=0, [Xi-, [Xi-, … [Xi-, ⏟ -αj(Hi) +1brackets Xj-]]…]=0, fori≠j.$

Let $C$ be a Cartan matrix and let ${𝔤}_{C}$ be the Lie algebra defined above.

1. The Lie algebra ${𝔤}_{C}$ is a finite dimensional complex simple Lie algebra
2. EVery finite dimensional complex simple Lie algebra is isomorphic to ${𝔤}_{C}$ for some Cartan matrix $C\text{.}$
3. If $C,$ ${C}^{\prime }$ are Cartan matices then $𝔤C≃𝔤C′ if and only ifC=C′.$

### Triangular decomposition

Fix a Cartan matrix $C={\left({\alpha }_{i}\left({H}_{j}\right)\right)}_{1\le i,j\le r}$ and let $𝔤={𝔤}_{C}\text{.}$ Define

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

The elements ${X}_{1}^{-},{X}_{2}^{-},\dots ,{X}_{r}^{-},{H}_{1},\dots ,{H}_{r},{X}_{1}^{+},{X}_{2}^{+},\dots ,{X}_{r}^{+}$ are linearly independent in $𝔤$ and

$𝔤=𝔫-⊕𝔥 ⊕𝔫+,$

The Lie subalgebra $𝔥\subseteq 𝔤$ is a Cartan subalgebra of $𝔤$ and the Lie subalgebra $𝔟=𝔥\oplus {𝔫}^{+}$ is a Borel subalgebra of $𝔤\text{.}$ The rank of $𝔤$ is $r=\text{dim}\phantom{\rule{0.2em}{0ex}}𝔥\text{.}$

### Weights and weight spaces

Fix a Cartan matrix $C={\left({\alpha }_{j}\left({H}_{i}\right)\right)}_{1\le i,j\le r}$ and let $𝔤={𝔤}_{C}\text{.}$ Let ${𝔥}^{*}={\text{Hom}}_{ℂ}\phantom{\rule{0.2em}{0ex}}\left(𝔥,ℂ\right)$ and define the fundamental weights ${\omega }_{1},\dots ,{\omega }_{r}\in {𝔥}^{*}$ by

$ωi(Hj)= δij,for 1≤i,j≤r.$

Let $V$ be a $𝔤-\text{-module}$ and let $\mu =\sum _{i=1}^{r}{\mu }_{i}{\omega }_{i}\in {𝔥}^{*}\text{.}$ The subspace

$Vμ = { v∈V∣ hv=μ(h)v, forh∈𝔥 } = { v∈V∣ Hiv=μiv, for1≤i≤r }$

is the $\mu \text{-weight}$ space of $V\text{.}$ Vectors $v\in {V}_{\mu }$ are weight vectors of $V$ of weight $\mu ,$ $\text{wt}\phantom{\rule{0.2em}{0ex}}\left(v\right)=\mu \text{.}$ The weights of the $𝔤\text{-module}$ $V$ are the elements $\mu \in {𝔥}^{*}$ such that ${V}_{\mu }\ne 0\text{.}$ If $\mu$ is a weight of $V,$ the multiplicity of $\mu$ in $V$ is $\text{dim}\phantom{\rule{0.2em}{0ex}}\left({V}_{\mu }\right)\text{.}$ A highest weight vector in a $𝔤\text{-module}$ $V$ is a weight vector $v\in V$ such that ${𝔫}^{+}v=0$ or, equivalently, a weight vector $v\in V$ such that ${X}_{i}^{+}v=0,$ for $1\le i\le r\text{.}$

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

$P+=∑i=1r ℕωiandP= ∑i=1rℤωi ,respectively,$

where $ℕ={ℤ}_{\ge 0}\text{.}$

### Classification of simple $𝔤\text{-module}$

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

$V=⨁μ∈PVμ.$

Let $𝔤$ be a finite dimensional complex simple Lie algebra.

1. Every finite dimensional irreducible $𝔤-\text{-module}$ $V$ contains a unique, up to constant multiples, highest weight vector ${v}^{+}\in V$ and $\text{wt}\phantom{\rule{0.2em}{0ex}}\left({v}^{+}\right)\in {P}^{+}\text{.}$
2. Conversely, if $\lambda \in {P}^{+},$ then there is a unique (up to isomorphism) finite dimensional irreducible $𝔤-\text{-module,}$ ${V}^{\lambda },$ with highest weight vector of weight $\lambda \text{.}$

### Roots and the root lattice

Fix a Cartan matrix $C={\left({\alpha }_{j}\left({H}_{i}\right)\right)}_{1\le i,j\le r}$ and let $𝔤={𝔤}_{C}\text{.}$ The adjoint action of $𝔤$ on $𝔤$ (see (1.5)) makes $𝔤$ into a finite dimensional $𝔤\text{-module.}$ An element $\alpha \in P,$ $\alpha \ne 0$ is a root if the weight space ${𝔤}_{\alpha }\ne 0\text{.}$ A root is positive, $\alpha >0,$ if ${𝔤}_{\alpha }\subseteq {𝔫}^{+}$ and negative, $\alpha <0,$ if ${𝔤}_{\alpha }\subseteq {𝔫}^{-}\text{.}$ We have

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

The roots ${\alpha }_{i},$ $1\le i\le r,$ given by ${𝔤}_{{\alpha }_{i}}=ℂ{X}_{i}^{+}$ are the simple roots. The Cartan matrix is the transition matrix between the simple roots and the fundamental weights,

$αi=∑j=1r αi(Hj)ωj ,for1≤i≤r.$

The root lattice is the lattice $Q\subseteq P\subseteq {𝔥}^{*}$ given by $Q=\sum _{i=1}^{r}ℤ{\alpha }_{i}\text{.}$

### The inner product ${𝔥}_{ℝ}^{*}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $C={\left({\alpha }_{j}\left({H}_{i}\right)\right)}_{1\le i,j\le r}$ be the corresponding Cartan matrix. There exist unique positive integers ${d}_{1},{d}_{2},\dots ,{d}_{r}$ such that $\text{gcd}\phantom{\rule{0.2em}{0ex}}\left({d}_{1},\dots ,{d}_{r}\right)=1$ and the matrix ${\left({d}_{i}{\alpha }_{j}\left({H}_{i}\right)\right)}_{1\le i,j\le r}$ is symmetric. The integers ${d}_{1},{d}_{2},\dots ,{d}_{r}$ are given explicitly by

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

Let ${\alpha }_{1},\dots ,{\alpha }_{r}$ be the simple roots for $𝔤\text{.}$ Define

$𝔥ℝ*= ∑i=1r ℝαi,$

so that ${𝔥}_{ℝ}^{*}$ is a real vector space of dimension $r\text{.}$ Define an symmetric inner product on ${𝔥}_{ℝ}^{*}$ by

$(αi,αj)= diαi(Hj), for1≤i,j≤r,$

where the values ${\alpha }_{j}\left({H}_{i}\right)$ are the entries of the Cartan matrix corresponding to $𝔤\text{.}$

### 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 ${\alpha }_{1},\dots ,{\alpha }_{r}$ be the simple roots. For each root $\alpha \in R$ define a linear transformation of ${𝔥}_{ℝ}^{*}$ by

$sα(λ)=λ- (λ,α∨)α, whereα∨= 2α(α,α).$

The Weyl group corresponding to $𝔤$ is the group of linear transformations of ${𝔥}_{ℝ}^{*}$ generated by the reflections ${s}_{\alpha },$ $\alpha \in R,$

$W= ⟨ sα∣ α∈R ⟩ .$

The simple reflections in $W$ are the elements ${s}_{i}={s}_{{\alpha }_{i}},$ $1\le i\le r\text{.}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $W$ be the Weyl group corresponding to $𝔤\text{.}$

1. The Weyl group $W$ is a finite group.
2. The Weyl group $W$ can be presented by generators ${s}_{1},\dots ,{s}_{r}$ and relations $si2 = 1, 1≤i≤r, sisj sisj… ⏟ mij factors = sjsi sjsi… ⏟ mij factors fori≠j,$ 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 $w\in W\text{.}$ A reduced decomposition for $w$ is an expression

$w=si1 si2… siℓ(w)$

of $w$ as a product of generators which is as short as possible. The length $\ell \left(w\right)$ of this expression is the length of $w\text{.}$

Let $𝔤$ be a finite dimensional simple complex Lie algebra and let $W$ be the Weyl group corresponding to $𝔤\text{.}$

1. There is a unique longest element ${w}_{0}$ in $W\text{.}$
2. Let ${w}_{0}={s}_{{i}_{1}}\dots {s}_{{i}_{N}}$ be a reduced decomposition for the longest element of $W\text{.}$ Then the elements $β1=αi1, β2=si1 (αi2),… ,βN=si1 si2…siN-1 (αiN),$ are the positive roots of $𝔤\text{.}$

## 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 $𝔤\text{.}$ 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 $k\text{.}$ Let $T\left(𝔤\right)=\underset{k\ge 0}{⨁}{𝔤}^{\otimes k}$ be the tensor algebra of $𝔤$ and let $J$ be the ideal of $T\left(𝔤\right)$ generated by the tensors

$x⊗y-y⊗x- [x,y],where x,y∈𝔤.$

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 $\alpha :\phantom{\rule{0.2em}{0ex}}𝔤\to A$ 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 $x,y\in 𝔤,$ and let 1 and ${1}_{A}$ denote the identities in 𝔘𝔤 and $A$ respectively. Then there exists a unique algebra homomorphism $\tau :\phantom{\rule{0.2em}{0ex}}𝔘𝔤\to A$ such that $\tau \left(1\right)={1}_{A}$ and $\alpha =\tau \circ {\alpha }_{0},$ 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

$[x,y]=xy-yx, for allx,y∈A.$

This defines a Lie algebra structure on $A$ and we denote the resulting Lie algebra by $L\left(A\right)$ to distinguish it from $A\text{.}$ $L$ 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 $A\text{.}$

### The enveloping algebra is a Hopf algebra

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

1. a comultiplication, $\phantom{\rule{1em}{0ex}}\Delta :\phantom{\rule{0.2em}{0ex}}𝔘𝔤\to 𝔘𝔤\otimes 𝔘𝔤,$ by $Δ(x)=x⊗ 1+1⊗x,for all x∈𝔤,$
2. a counit, $\phantom{\rule{1em}{0ex}}\epsilon :\phantom{\rule{0.2em}{0ex}}𝔘𝔤\to k,$ by $ε(x)=0, for allx∈𝔤,$
3. and an antipode, $\phantom{\rule{1em}{0ex}}S:\phantom{\rule{0.2em}{0ex}}𝔘𝔤\to 𝔘𝔤,$ by $S(x)=-x, for allx∈𝔤.$

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

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

### The Lie algebra can be recovered from its enveloping algebra!

An element $x$ of a Hopf algebra $A$ is primitive if

$Δ(x)=1⊗x +x⊗1.$

It can be shown that if $\text{char}\phantom{\rule{0.2em}{0ex}}k=0$ then the subspace $g$ of $𝔘𝔤$ is the set of primitive elements of $𝔘𝔤\text{.}$ Thus, if $\text{char}\phantom{\rule{0.2em}{0ex}}k=0,$ 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 ${\left({x}_{i}\right)}_{i\in A}\text{.}$ Then the elements

$xi1 xi2 … xin$

in the enveloping algebra $𝔘𝔤,$ where ${i}_{1}\le {i}_{2}\le \dots \le {i}_{n}$ is an arbitrary increasing finite sequence of elements of $\Lambda ,$ form a basis a $𝔘𝔤\text{.}$

## 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 $C={\left({\alpha }_{j}\left({H}_{i}\right)\right)}_{1\le i,j\le r}$ be the corresponding Cartan matrix. Then the enveloping algebra 𝔘𝔤 of $𝔤$ can be presented as the algebra over $ℂ$ generated by

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

with relations

$[Hi,Hj]=0, for all1≤i,j≤r, [Hi,Xj+]= αj(Hi) Xj+, [Hi,Xj-]= -αj(Hi) Xj-, for all1≤i,j≤r, [Xi+,Xj-] =δijHi, for1≤i,j≤r, ∑s+t=1-αj(Hi) (-1)s ( 1-αj(Hi) s ) (Xi±)s Xj± (Xi±)t =0, fori≠j,$

where, if $a,b\in 𝔘𝔤,$ we use the notation $\left[a,b\right]=ab-ba\text{.}$ Note that since

$[a, [a, … [a, ⏟ ℓbrackets b]]…]= ∑s+t=ℓ (-1)s (ℓs) asbat,$

for any two elements $a,b\in 𝔘𝔤$ and any positive integer $\ell ,$ 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

$𝔤=𝔫-⊕ 𝔥⊕𝔫+,$

where

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

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

$𝔘𝔤≅𝔘𝔫-⊗ 𝔘𝔥⊗𝔘𝔫+, as vector spaces.$

### Grading on $𝔘{𝔫}^{+}$ and $𝔘{𝔫}^{-}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra as presented in (2.2). Let ${\alpha }_{1},\dots ,{\alpha }_{r}$ be the simple roots for $𝔤$ and let

$Q+=∑iℕαi, whereℕ=ℤ≥0 .$

For each element $\nu =\sum _{i=1}^{r}{\nu }_{i}{\alpha }_{i}\in {Q}^{+}$ 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

$𝔘𝔫-= ⨁ν∈Q+ (𝔘𝔫-)ν, and𝔘𝔫+= ⨁ν∈Q+ (𝔘𝔫+)ν,$

as vector spaces.

### Pointcaré-Birkhoff-Witt bases of $𝔘{𝔫}^{-},$$𝔘𝔥,$ and $𝔘{𝔫}^{+}$

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 ${𝔤}_{\alpha }$ from (2.6). Let $W$ be the Weyl group corresponding to $𝔤\text{.}$ Fix a reduced decomposition of the longest element ${w}_{0}\in W,$ ${w}_{0}={s}_{{i}_{1}}\dots {s}_{{i}_{N}},$ and define

$β1=αi1, β2=si1 (αi2),…, βN=si1 si2…siN-1 (αiN).$

The elements ${\beta }_{1},\dots ,{\beta }_{N}$ are the positive roots $𝔤$ and the elements $-{\beta }_{1},\dots ,-{\beta }_{N}$ are the negative roots of $𝔤\text{.}$

$For each rootα, fix an elementXα∈𝔤α .$

Since ${𝔤}_{\alpha }$ is 1-dimensional ${X}_{\alpha }$ is uniquely defined, up to multiplication by a constant. Since

$𝔫-=⨁α<0 𝔤α, 𝔫+=⨁α>0 𝔤α, 𝔥=span- {H1,H2,…,Hr} and𝔤=𝔫- ⊕𝔥⊕𝔫+,$

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,

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

### 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 $\left\{{b}_{i}\right\}$ be a basis of $𝔤$ and let $\left\{{b}^{i}\right\}$ 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 $\lambda$ let ${V}^{\lambda }$ be the finite dimensional simple $𝔘𝔤\text{-module}$ indexed by $\lambda$ (see (2.5)). Let $\rho$ be the element of ${𝔥}_{ℝ}^{*}$ given by

$ρ=12∑α>0 α,$

where the sum is over all positive roots for $𝔤\text{.}$ Then the element

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

where the inner product on ${𝔥}_{ℝ}^{*}$ 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.