## Basic concepts of Lie algebras

References:

1. Karin Erdmann & Mark J. Wildon, Introduction to Lie Algebras, Springer Undergraduate Mathmatics Series;
2. Roger W. Carter, Lie Algebras and Root Systems, in the book Lectures on Lie groups and Lie algebras, by Roger William Carter, Graeme Segal, Ian Grant Macdonald;
3. Jean-Pierre Serre, Complex Semisimple Lie Algebras, translated from the French by G. A. Jones;
4. James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9 (GTM 9), Springer-Verlag;
5. William Fulton & Joe Harris, Representation theory, a first course, GTM 129, Springer-Verlag;
6. J. Dixmier, Algèbres de Lie, Les Cours de Sorbonne, rédaction de A. Pereira Gomes.

1 and 3 are highly recommended.

$\S 1$. Basic concepts of Algebras.

Let $F$ be a field and $A$ a vector space over $F$. We say that $A$ is an $F$algebra, if $A$ is simultaneously a ring, and the ring structure and the vector space structure on $A$ is compatible, in the sense that: for any $x, y \in A$ and $\lambda \in F^{\times}$, we have

$\displaystyle \lambda (xy)=(\lambda x)y=x(\lambda y)$.

The dimension of $A$ over $F$ as a vector space, is called the dimension of the algebra $A$

The algebra $A$ is called

• associative, iff (short for if and only if) for all $x, y, z \in A$, we have $(xy)z=x(yz)$;
• commutative, iff for all $x, y \in A$, one has $xy=yx$;
• a Lie algebra, iff the multiplication $xy$, usually written as $[xy]$, fulfilling that
• (i) $[xx]=0,\; \forall x \in A$; and
• (ii) for any $x, y, z \in A$, we have the Jacobi identity:

$\displaystyle [x[yz]]+[y[zx]]+[z[xy]]=0$.

Exercise. Check that the Lie bracket [,] on a Lie algebra $A$ satisfies that for all $x, y \in A$, we have $[xy]=-[yx]$ (anti-symmetry). And $[xy]=-[yx] \Leftrightarrow [xx]=0$, provided that the characteristic of $F$ is not 2.

$\S 2$. Examples of Lie algebras.

1. Any $F$-vector space $V$ has a Lie bracket defined by $[uw]=0$ for all $u, w \in V$. This is the abelian Lie algebra structure on $V$. In particular, the field $F$ may be regarded as a one dimensional abelian Lie algebra.

2. Let $A$ be an associative $F$-algebra. Define the bracket product $[,]: A \times A \to A$ by $[xy]=xy-yx$ for all $x, y \in A$. Then $(A, [,])$ becomes a Lie algebra.

By this construction, we noted that $\mathrm{M}(n, F)$, the collection of all the $n \times n$ matrix over $F$, with the bracket product defined by $[PQ]=PQ-PQ\; \forall P, Q \in \mathrm{M}(n, F)$, is a Lie algebra. This Lie algebra is usually denoted by $\mathfrak{gl}(n, F)$. We will illustrate this notation later.

Furthermore, given a vector space $V$ over $F$ of dimension $n$, we can identify End($V$), the collection of all the endomorphism of $V$, to $\mathrm{M}(n, F)$ (by choosing a certain $F$-basis of $V$). Hence End($V$) is also a Lie algebra. We denote it by $\mathfrak{gl}(V)$.

3. Let $A$ be an $F$-algebra. The derivated algebra of $A$, consisting of all the $F$-linear mapping $D: A \to A$, such that $D(ab)=aD(b)+D(a)b, \forall a, b \in A$, is a Lie algebra. We denote this algebra of derivations of $A$ by Der$A$.

4. Poisson algebra. This example originates in Physics. Let $A$ be a commutative ring, and the usual multiplication of $A$ is simply written by $xy\; \forall x,y \in A$. Define a bilinear mapping $\displaystyle \{, \}: A \times A \to A,\; (x, y) \mapsto {xy}$. This bilinear mapping is called a Poisson bracket, if it fulfills the following three conditions:

(i) {xy}=-{yx};

(ii) {x, yz}={xy}z+y{xz};

(iii) {x{yz}}={{xy}z}+{y{xz}}

for all the $x, y, z \in A$.

Exercise. Verify that every Poisson algebra is a Lie algebra!

We give a concrete example of Poisson algebra. Denote the set of all the smooth functions (a function is smooth if it is infinitely differentiable) over $\mathbb{R}^{2n}$ by $C^{\infty}(\mathbb{R}^{2n})$. For $f, g \in C^{\infty}(\mathbb{R}^{2n})$, defined the Poisson bracket of  as follows:

$\displaystyle \{f, g\}=\sum_{i=1}^n\,\big(\frac{\partial f}{q_i}\frac{\partial g}{p_i} - \frac{\partial f}{p_i}\frac{\partial g}{q_i}\big)$,

where the coordinates are denoted by $(q_1, \ldots, q_n, p_1, \ldots , p_n) \in \mathbb{R}^{2n}$.

One can check that such defined ${f, g}$ satisfies all the requirements of a Poisson algebra, thus it is also a Lie algebra.

$\S 3$. Subalgebras, ideals and centre of a Lie algebra.

Given a Lie algebra $L$, we can define a Lie subalgebra of $L$ to be a subspace $L_1$, such that $[ab] \in L_1, \forall a, b \in L_1$, or formally, $[L_1, L_1] \subset L_1$.

Examples of Lie subalgebras. Let $\mathfrak{sl}(n, F)$ be the subspace of $\mathfrak{gl}(n, F)$ consisting of all the matrices of trace 0. Let $\mathfrak{b}(n, F)$ (resp. $\mathfrak{n}(n, F)$) be the upper triangular (resp. strictly upper triangular) matrices in $\mathfrak{gl}(n, F)$. Then $\mathfrak{sl}(n, F)$, $\mathfrak{b}(n, F)$ and $\mathfrak{n}(n, F)$ are all Lie subalgebra of $\mathfrak{gl}(n, F)$.

We can also define an ideal of a Lie algebra $L$ to be a subspace $I$ of $L$ such that $[LI] \subset I$ or $[IL] \subset I$. The fact that $[xy]=-[yx]$ for a Lie algebra $L$ release us to distinguish the left and right ideal.

We point out that $\mathfrak{sl}(n, F)$ is an ideal of $\mathfrak{gl}(n, F)$, and $\mathfrak{n}(n, F)$ is an ideal of $\mathfrak{b}(n, F)$.

Note. An ideal is always an subalgebra. However, a subalgeba need not to be an ideal. Remember in mind that $\mathfrak{n}(n, F)$ is a subalgebra of $\mathfrak{gl}(n, F)$. But if $n>2$, then it is not an ideal. Verification is left as an exercise.

Immediately we note that $L$  itself and $\{0\}$ are always ideals of $L$. They are called the trivial ideals. If the only ideals of $L$ are the trivial ones, we say that $L$ is simple.

Another important example of  an ideal is the centre. For a Lie algebra $L$, the centre $Z(L)$ of $L$ is defined by $Z(L) = \{x \in L : [xy]=0\; \forall y \in L\}$. Thus $L$ is abelian iff $L=Z(L)$.

Exercise. Let $I, J$ be two ideals of a Lie algebra $L$. Show that

$\displaystyle I+J:=\{a+b : a \in I, \; b \in J\}$

and

$\displaystyle [IJ]:=\{\sum_i\,[a_ib_i] : a_i \in I, \; b_i \in J\}$

are both ideals of $L$.

$\S 4$. Homomorphism and quotient algebras.

Let $L_1$ and $L_2$ be two Lie algebras over $F$. An $F$-linear mapping $f: L_1 \to L_2$ is called a homomorphism of Lie algebras if $f([xy])=[f(x)f(y)], \; \forall x, y \in L_1$. A homomorphism is called an isomorphism if in addition, it is bijective. It is easy to check that the kernel of a homomorphism $f$, Ker $f$, is an ideal of $L_1$, and the image of $f$, Im$f$ is a subalgebra of $L_2$.

Example of isomorphism. Recall that we have $\mathfrak{gl}(V) \simeq \mathfrak{gl}(n, F)$, if $V$ is an $F$-vector space of dimension $n$.

An utmost important example of homomorphism is the famous adjoint homomorphism of a Lie algebra $L$. It is defined by

$\displaystyle \mathrm{ad}: L \to \mathfrak{gl}(L), \mathrm{ad}\, (y) = [xy],\; \forall y \in L$.

The kernel of ad $L$ is $Z(L)$, the centre of $L$.

For an ideal $I$ of a Lie algebra $L$, the quotient vector space $L/I$ has a Lie algebra structure; for any $x, y \in L$, define the Lie bracket in $L/I$ by $[x+I, y+I]=[xy]+I$. To check this Lie bracket is well-defined we use the bilinearlity of the Lie bracket in $L$ and the definition of an ideal. We left it as an exercise to check that $L/I$ is a Lie algebra with the bracket defined as above.

Remark. 关于同态（同构）的基本定理，以及在同态之下，理想的对应关系，略！

Problems.

1. 设$x_1, \ldots, x_n$$L$的一组基，因此$[x_ix_j]=\sum_{k=1}^n\, a_{ij}^k\,x_k$. 这些$a_{ij}^k, i, j, k\in \{1, \ldots, n\}$称为结构常数(structure constant). 证明：

(i) $a_{ii}^k=0=a_{ij}^k+a_{ji}^k$;

(ii) (Cartan-Maurer euqation) $\sum_k \, (a_{ij}^ka_{kl}^m+a{jl}^ka_{ki}^m+a_{li}^ka_{kj}^m)=0$.

(iii)每个 $a_{ij}^k$都满足(1,2)-张量的变换规则.

2. 设$A$是一个代数, $D, E$$A$的导出代数（简称导数）, 即$D, E \in \mathrm{Der}(A)$. 证明:

(i) $[D, E]=DE-ED$$A$的导数. 这里乘法是运算的复合.

(ii) 举例说明, $DE$可以不是一个导数.

3. 对任意$x\in L$, 伴随同态ad$x$$L$的一个导数. 从而ad$L \subset \mathrm{Der}(L)$.

4. 设$K$$L$的子代数（甚至仅仅是子空间即可）. $K$$L$中的正规化子(normaliser) $N_L(K)$定义为$N_L(K)=\{x \in L : [xK] \subset K\}$$K$$L$中的中心化子(centraliser) $C_L(K)$定义为$N_L(K)=\{x \in L : [xK]=0\}$. 证明: $N_L(K)$ 与 $C_L(K)$都是子代数.

5. Show that $\mathfrak{sl}(2, F)$ is simple provided that char $F \neq 2$.