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 $[,]:𝔤\otimes 𝔤\to 𝔤$ which satisfies

$$\begin{array}{c}[x,x]=0,\text{for all}x\in 𝔤,\\ [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0,\text{for all}x,y,z\in 𝔤\text{.}\end{array}$$

The first relation is the skew-symmetric relation and is equivalent to $[x,y]=-[y,x],$ for all $x,y\in 𝔤,$ provided that $\text{char}k\ne 2\text{.}$ 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

$$[x,a]\in 𝔞,\text{for all}x\in 𝔤,\text{and}a\in 𝔞\text{.}$$

A Lie algebra $𝔤$ is abelian if $[x,y]=0$ 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

$$D^1𝔞=𝔞,\text{and}{D}^n𝔞=[D^{n-1}𝔞,D^{n-1}𝔞],\text{for}n\ge 2\text{.}$$

An ideal $a$ of $𝔤$ is solvable if there exists a positive integer $n$ such that $D^n𝔞=0\text{.}$ 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 $𝔤\text{-action}$

$$\begin{array}{ccc}𝔤\otimes V& ⟶& V\\ x\otimes v& ⟼& x·v=xv\end{array}$$

such that

$$[x,y]·v=x\left(yv\right)-y\left(xv\right),\text{for all}x,y\in 𝔤,\text{and}v\in V\text{.}$$

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

$$\begin{array}{cc}\begin{array}{cccc}\rho :& 𝔤& ⟶& \text{End}\left(V\right)\\ & x& ⟼& \rho \left(x\right)\end{array}& \text{such that}\rho \left([x,y]\right)=\rho \left(x\right)\rho \left(y\right)-\rho \left(y\right)\rho \left(x\right),\end{array}$$

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

$$\rho \left(x\right)v=xv,\text{for all}x\in 𝔤,\text{and}v\in V\text{.}$$

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

$$\begin{array}{ccc}𝔤\otimes 𝔤& ⟶& 𝔤\\ x\otimes y& ⟼& [x,y]\text{.}\end{array}$$

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

$${\text{ad}}_x\left(y\right)=[x,y],\text{for all}y\in 𝔤\text{.}$$

The representation

$$\begin{array}{cccc}\text{ad}:& 𝔤& ⟶& \text{End}\left(𝔤\right)\\ & x& ⟼& {\text{ad}}_x\end{array}$$

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 $⟨,⟩:𝔤\times 𝔤\to k$ given by

$$⟨x,y⟩=\text{Tr}\left({\text{ad}}_x{\text{ad}}_y\right),\text{for all}x,y\in 𝔤\text{.}$$

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

$$⟨[x,y],z⟩+⟨y,[x,z]⟩=0,\text{for all}x,y,z\in 𝔤\text{.}$$

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 $𝔤=[𝔤,𝔤]\text{.}$
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 $(i,j)$ 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

$$X_1^{-},X_2^{-},\dots ,X_r^{-},H_1,H_2,\dots ,H_r,X_1^{+},X_2^{+},\dots ,X_r^{+},$$

and relations

$$\begin{array}{cc}[H_i,H_j]=0,& \text{for all}1\le i,j\le r,\\ \begin{array}{c}[H_i,X_j^{+}]={\alpha }_j\left(H_i\right)X_j^{+},\\ [H_i,X_j^{-}]=-{\alpha }_j\left(H_i\right)X_j^{-},\end{array}& \text{for all}1\le i,j\le r,\\ [X_i^{+},X_j^{-}]={\delta }_{ij}{H}_i,& \text{for}1\le i,j\le r,\\ \begin{array}{c}\underset{\underset{-{\alpha }_j\left(H_i\right)+1\text{brackets}}{⏟}}{[X_i^{+},[X_i^{+},\dots [X_i^{+},}{X}_j^{+}\left]\right]\dots ]=0,\\ \underset{\underset{-{\alpha }_j\left(H_i\right)+1\text{brackets}}{⏟}}{[X_i^{-},[X_i^{-},\dots [X_i^{-},}{X}_j^{-}\left]\right]\dots ]=0,\end{array}& \text{for}i\ne j\text{.}\end{array}$$

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 $${𝔤}_C\simeq {𝔤}_{C^{\prime }}\text{if and only if}C=C^{\prime }\text{.}$$

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

$$\begin{array}{ccc}{𝔫}^{-}& =& \text{Lie subalgebra of}𝔤\text{generated by}{X}_1^{-},X_2^{-},\dots ,X_r^{-}\text{.}\\ 𝔥& =& ℂ\text{-span}\{H_1,H_2,\dots ,H_r\},\\ {𝔫}^{+}& =& \text{Lie subalgebra of}𝔤\text{generated by}{X}_1^{+},X_2^{+},\dots ,X_r^{+},\end{array}$$

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

$$𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+},$$

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 $r=\text{dim}𝔥\text{.}$

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 ${𝔥}^{*}={\text{Hom}}_{ℂ}(𝔥,ℂ)$ and define the fundamental weights ${\omega }_1,\dots ,{\omega }_r\in {𝔥}^{*}$ by

$${\omega }_i\left(H_j\right)={\delta }_{ij},\text{for}1\le i,j\le r\text{.}$$

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

$$\begin{array}{ccc}{V}_{\mu }& =& \{v\in V\mid hv=\mu \left(h\right)v,\text{for}h\in 𝔥\}\\ & =& \{v\in V\mid H_{i}v={\mu }_{i}v,\text{for}1\le i\le r\}\end{array}$$

is the $\mu \text{-weight}$ space of $V\text{.}$ Vectors $v\in V_{\mu }$ are weight vectors of $V$ of weight $\mu ,$ $\text{wt}\left(v\right)=\mu \text{.}$ 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 $\text{dim}\left(V_{\mu }\right)\text{.}$ 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

$$P^{+}=\sum _{i=1}^r\mathbb{N}{\omega }_i\text{and}P=\sum _{i=1}^r\mathbb{Z}{\omega }_i,\text{respectively,}$$

where $\mathbb{N}={\mathbb{Z}}_{\ge 0}\text{.}$

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 $P,$

$$V=\underset{\mu \in P}{⨁}{V}_{\mu }\text{.}$$

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 $\text{wt}\left(v^{+}\right)\in P^{+}\text{.}$
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

$$\begin{array}{c}\text{dim}{𝔤}_{\alpha }=1\text{for all roots}\alpha ,\\ {𝔫}^{-}=\underset{\alpha <0}{⨁}{𝔤}_{\alpha },𝔥={𝔤}_0,{𝔫}^{+}=\underset{\alpha >0}{⨁}{𝔤}_{\alpha },\text{and}𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}\text{.}\end{array}$$

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

$${\alpha }_i=\sum _{j=1}^r{\alpha }_i\left(H_j\right){\omega }_j,\text{for}1\le i\le r\text{.}$$

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

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 $\text{gcd}(d_1,\dots ,d_r)=1$ 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

$$\begin{array}{cc}\begin{array}{c}{A}_r,D_r,\\ E_6,E_7,E_8:\end{array}& d_i=1\text{for all}1\le i\le r,\\ B_r:& d_i=1\text{for}1\le i\le r-1,\text{and}{d}_r=2,\\ C_r:& d_i=2,\text{for}1\le i\le r-1,\text{and}{d}_r=1,\\ F_4:& d_1=d_2=1,\text{and}{d}_3=d_4=2,\\ G_2:& d_1=3,\text{and}{d}_2=1\text{.}\end{array}$$

Let ${\alpha }_1,\dots ,{\alpha }_r$ be the simple roots for $𝔤\text{.}$ Define

$${𝔥}_{ℝ}^{*}=\sum _{i=1}^rℝ{\alpha }_i,$$

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

$$({\alpha }_i,{\alpha }_j)=d_i{\alpha }_i\left(H_j\right),\text{for}1\le i,j\le r,$$

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

$$s_{\alpha }\left(\lambda \right)=\lambda -(\lambda ,{\alpha }^{\vee })\alpha ,\text{where}{\alpha }^{\vee }=\frac{2\alpha }{(\alpha ,\alpha )}\text{.}$$

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

$$W=⟨s_{\alpha }\mid \alpha \in R⟩\text{.}$$

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 $$\begin{array}{cccccc}& s_i^2& =& 1,& & 1\le i\le r,\\ & \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_i{s}_j{s}_i{s}_j\dots }& =& \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_j{s}_i{s}_j{s}_i\dots }& & \text{for}i\ne j,\end{array}$$ where $$\begin{array}{ccc}{m}_{ij}& =& \{\begin{array}{cc}2,& \text{if}{\alpha }_i\left(H_j\right){\alpha }_j\left(H_i\right)=0,\\ 3,& \text{if}{\alpha }_i\left(H_j\right){\alpha }_j\left(H_i\right)=1,\\ 4,& \text{if}{\alpha }_i\left(H_j\right){\alpha }_j\left(H_i\right)=2,\\ 6,& \text{if}{\alpha }_i\left(H_j\right){\alpha }_j\left(H_i\right)=3\text{.}\end{array}\end{array}$$

Let $w\in W\text{.}$ A reduced decomposition for $w$ is an expression

$$w=s_{i_1}{s}_{i_2}\dots s_{i_{\ell \left(w\right)}}$$

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 $${\beta }_1={\alpha }_{i_1},{\beta }_2=s_{i_1}\left({\alpha }_{i_2}\right),\dots ,{\beta }_N=s_{i_1}{s}_{i_2}\dots s_{i_{N-1}}\left({\alpha }_{i_N}\right),$$ 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

$$x\otimes y-y\otimes x-[x,y],\text{where}x,y\in 𝔤\text{.}$$

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

$$𝔘𝔤=\frac{T\left(𝔤\right)}{J}\text{.}$$

There is a canonical map

$$\begin{array}{cccc}{\alpha }_0:& 𝔤& ⟶& 𝔘𝔤\\ & x& ⟼& x+J\text{.}\end{array}$$

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

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

$$[x,y]=xy-yx,\text{for all}x,y\in A\text{.}$$

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

$${\text{Hom}}_{\text{alg}}(𝔘𝔤,A)={\text{Hom}}_{\text{Lie}}(𝔤,L\left(A\right))$$

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, $\Delta :𝔘𝔤\to 𝔘𝔤\otimes 𝔘𝔤,$ by $$\Delta \left(x\right)=x\otimes 1+1\otimes x,\text{for all}x\in 𝔤,$$
2. a counit, $\epsilon :𝔘𝔤\to k,$ by $$\epsilon \left(x\right)=0,\text{for all}x\in 𝔤,$$
3. and an antipode, $S:𝔘𝔤\to 𝔘𝔤,$ by $$S\left(x\right)=-x,\text{for all}x\in 𝔤\text{.}$$

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

$$\Delta \left(x\right)=1\otimes x+x\otimes 1\text{.}$$

It can be shown that if $\text{char}k=0$ then the subspace $g$ of $𝔘𝔤$ is the set of primitive elements of $𝔘𝔤\text{.}$ Thus, if $\text{char}k=0,$ 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.

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

$$X_1^{-},X_2^{-},\dots ,X_r^{-},H_1,H_2,\dots ,H_r,X_1^{+},X_2^{+},\dots ,X_r^{+},$$

with relations

$$\begin{array}{cc}[H_i,H_j]=0,& \text{for all}1\le i,j\le r,\\ \begin{array}{c}[H_i,X_j^{+}]={\alpha }_j\left(H_i\right)X_j^{+},\\ [H_i,X_j^{-}]=-{\alpha }_j\left(H_i\right)X_j^{-},\end{array}& \text{for all}1\le i,j\le r,\\ [X_i^{+},X_j^{-}]={\delta }_{ij}{H}_i,& \text{for}1\le i,j\le r,\\ \sum _{s+t=1-{\alpha }_j\left(H_i\right)}{(-1)}^s\left(\genfrac{}{}{0ex}{}{1-{\alpha }_j\left(H_i\right)}{s}\right){\left(X_i^{\pm }\right)}^s{X}_j^{\pm }{\left(X_i^{\pm }\right)}^t=0,& \text{for}i\ne j,\end{array}$$

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

$$\underset{\underset{\ell \text{brackets}}{⏟}}{[a,[a,\dots [a,}b\left]\right]\dots ]=\sum _{s+t=\ell }{(-1)}^s\left(\genfrac{}{}{0ex}{}{\ell }{s}\right)a^{s}b{a}^t,$$

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

$$𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+},$$

where

$$\begin{array}{ccc}{𝔫}^{-}& =& \text{Lie subalgebra of}𝔤\text{generated by}{X}_1^{-},X_2^{-},\dots ,X_r^{-}\text{.}\\ 𝔥& =& ℂ\text{-span}\{H_1,H_2,\dots ,H_r\},\\ {𝔫}^{+}& =& \text{Lie subalgebra of}𝔤\text{generated by}{X}_1^{+},X_2^{+},\dots ,X_r^{+}\end{array}$$

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

$$𝔘𝔤\cong 𝔘{𝔫}^{-}\otimes 𝔘𝔥\otimes 𝔘{𝔫}^{+},\text{as vector spaces.}$$

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

$$Q^{+}=\sum _i\mathbb{N}{\alpha }_i,\text{where}\mathbb{N}={\mathbb{Z}}_{\ge 0}\text{.}$$

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

$$\begin{array}{ccc}{\left(𝔘{𝔫}^{+}\right)}_{\nu }& =& \text{span-}\{X_{i_1}^{+}\dots X_{i_p}^{+}\mid X_{i_1}^{+}\dots X_{i_p}^{+}\text{has}{\nu }_j\text{-factors of type}{X}_j^{+}\}\\ {\left(𝔘{𝔫}^{-}\right)}_{\nu }& =& \text{span-}\{X_{i_1}^{-}\dots X_{i_p}^{-}\mid X_{i_1}^{-}\dots X_{i_p}^{-}\text{has}{\nu }_j\text{-factors of type}{X}_j^{-}\}\text{.}\end{array}$$

Then

$$𝔘{𝔫}^{-}=\underset{\nu \in Q^{+}}{⨁}{\left(𝔘{𝔫}^{-}\right)}_{\nu },\text{and}𝔘{𝔫}^{+}=\underset{\nu \in Q^{+}}{⨁}{\left(𝔘{𝔫}^{+}\right)}_{\nu },$$

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

$${\beta }_1={\alpha }_{i_1},{\beta }_2=s_{i_1}\left({\alpha }_{i_2}\right),\dots ,{\beta }_N=s_{i_1}{s}_{i_2}\dots s_{i_{N-1}}\left({\alpha }_{i_N}\right)\text{.}$$

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 $𝔤\text{.}$

$$\text{For each root}\alpha ,\text{fix an element}{X}_{\alpha }\in {𝔤}_{\alpha }\text{.}$$

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

$${𝔫}^{-}=\underset{\alpha <0}{⨁}{𝔤}_{\alpha },{𝔫}^{+}=\underset{\alpha >0}{⨁}{𝔤}_{\alpha },𝔥=\text{span-}\{H_1,H_2,\dots ,H_r\}\text{and}𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+},$$

it follows that

$$\begin{array}{cc}\{X_{{\beta }_1},\dots ,X_{{\beta }_N}\}& \text{is a basis of}{𝔫}^{+},\\ \{X_{-{\beta }_1},\dots ,X_{-{\beta }_N}\}& \text{is a basis of}{𝔫}^{-},\text{and}\\ \{H_1,H_2,\dots ,H_r\}& \text{is a basis of}𝔥\text{.}\end{array}$$

Then, by the Poincaré-Birkhoff-Witt theorem,

$$\begin{array}{cc}\{X_{{\beta }_1}^{p_1}{X}_{{\beta }_2}^{p_2}\dots X_{{\beta }_N}^{p_N}\mid p_1,\dots ,p_N\in {\mathbb{Z}}_{\ge 0}\}& \text{is a basis of}𝔘{𝔫}^{+},\\ \{X_{-{\beta }_1}^{n_N}{X}_{-{\beta }_2}^{n_2}\dots X_{-{\beta }_1}^{n_1}\mid n_1,\dots ,n_N\in {\mathbb{Z}}_{\ge 0}\}& \text{is a basis of}𝔘{𝔫}^{-},\text{and}\\ \{H_1^{s_1}{H}_2^{s_2}\dots H_1^{s_r}\mid s_1,\dots ,s_N\in {\mathbb{Z}}_{\ge 0}\}& \text{is a basis of}𝔘𝔥\text{.}\end{array}$$

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

$$c=\sum _i{b}_i{b}^i\text{.}$$

Then

$$c\text{is in the center of}𝔘𝔤\text{.}$$

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

$$\rho =\frac{1}{2}\sum _{\alpha >0}\alpha ,$$

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

$$c\text{acts on}{V}^{\lambda }\text{by the constant}(\lambda +\rho ,\lambda +\rho )-(\rho ,\rho ),$$

where the inner product on ${𝔥}_{ℝ}^{*}$ is as given in (2.7).

$$\begin{array}{cc}{A}_{r-1}:& α1 α2 α3 … αr-1 αr \\ B_r:& α1 α2 α3 … αr-1 αr \\ C_r:& α1 α2 α3 … αr-1 αr \\ D_r:& α1 α2 α3 … αr-2 αr-1 αr \\ E_6:& α1 α3 α4 α5 α6 α2 \\ E_7:& α1 α3 α4 α5 α6 α7 α2 \\ E_8:& α1 α3 α4 α5 α6 α7 α8 α2 \\ F_4:& α1 α2 α3 α4 \\ G_2:& α1 α2 \\ \multicolumn{2}{c}{\text{Table 1.}\text{Dynkin diagrams corresponding to finite dimensional complex simple Lie algebras}}\end{array}$$ $$\begin{array}{cccc}{A}_{r-1}:& \left(\begin{array}{cccccc}2& -1& 0& \cdots & & 0\\ -1& 2& -1& \cdots & & 0\\ 0& -1& 2& \cdots & & 0\\ ⋮& & & \ddots & & ⋮\\ 0& \cdots & & -1& 2& -1\\ 0& \cdots & & 0& -1& 2\end{array}\right)& B_r:& \left(\begin{array}{cccccc}2& -1& 0& \cdots & & 0\\ -1& 2& -1& \cdots & & 0\\ 0& -1& 2& \cdots & & 0\\ ⋮& & & \ddots & & ⋮\\ 0& \cdots & & -1& 2& -2\\ 0& \cdots & & 0& -1& 2\end{array}\right)\\ C_r:& \left(\begin{array}{cccccc}2& -1& 0& \cdots & & 0\\ -1& 2& -1& \cdots & & 0\\ 0& -1& 2& \cdots & & 0\\ ⋮& & & \ddots & & ⋮\\ 0& \cdots & & -1& 2& -1\\ 0& \cdots & & 0& -2& 2\end{array}\right)& D_r:& \left(\begin{array}{cccccc}2& -1& 0& \cdots & & 0\\ -1& 2& -1& \cdots & & 0\\ ⋮& & \ddots & & & ⋮\\ 0& \cdots & -1& 2& -1& -1\\ 0& \cdots & 0& -1& 2& 0\\ 0& \cdots & 0& -1& 0& 2\end{array}\right)\\ E_6:& \left(\begin{array}{cccccc}2& 0& -1& 0& 0& 0\\ 0& 2& 0& -1& 0& 0\\ -1& 0& 2& -1& 0& 0\\ 0& -1& -1& 2& -1& 0\\ 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& -1& 2\end{array}\right)& E_7:& \left(\begin{array}{ccccccc}2& 0& -1& 0& 0& 0& 0\\ 0& 2& 0& -1& 0& 0& 0\\ -1& 0& 2& -1& 0& 0& 0\\ 0& -1& -1& 2& -1& 0& 0\\ 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& 0& -1& 2\end{array}\right)\\ \multicolumn{4}{c}{\begin{array}{cc}{E}_8:& \left(\begin{array}{cccccccc}2& 0& -1& 0& 0& 0& 0& 0\\ 0& 2& 0& -1& 0& 0& 0& 0\\ -1& 0& 2& -1& 0& 0& 0& 0\\ 0& -1& -1& 2& -1& 0& 0& 0\\ 0& 0& 0& -1& 2& -1& 0& 0\\ 0& 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& 0& 0& -1& 2\end{array}\right)\end{array}}\\ F_4:& \left(\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -2& 0\\ 0& -1& 2& -1\\ 0& 0& -1& 2\end{array}\right)& & G_2:\left(\begin{array}{cc}2& -1\\ -3& 2\end{array}\right)\\ \multicolumn{4}{c}{\text{Table 2.}\text{Cartan matrices corresponding to finite dimensional complex simple Lie algebras}}\end{array}$$

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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