The following topic came up in the context of my project, which has been expanding to include new areas of mathematics that I did not initially anticipate. Consequently, I have had to learn about several new areas of mathematics; this is, of course, a common experience at RSI. For me, the representation theory of Lie algebras has been one of those areas, and I will post here about it to help myself understand it. Right here, I’ll aim to cover the groundwork necessary to get to the actual representation theory in a future post.

Lie Algebras

Throughout, we work over ${{\mathbb C}}$, or even an algebraically closed field of characteristic zero.

Definition 1 A Lie algebra is a finite-dimensional vector space ${L}$ with a Lie bracket ${[\cdot, \cdot]: L \times L \rightarrow L}$ satisfying:

• The bracket ${[\cdot, \cdot]: L \times L \rightarrow L}$ is ${{\mathbb C}}$-bilinear in both variables.
• ${[A,B] = -[B,A]}$ for any ${A,B \in L}$.
• ${[A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0}$. This is the Jacobi identity.

To elucidate the meaning of the conditions, let’s look at a few examples.

Here are two easy ones:

Example 1 Any vector space is a Lie algebra with the bracket the zero map. Such Lie algebras are called commutative.

Example 2 Suppose ${L_1, L_2}$ are Lie algebras. Then ${L_1 \oplus L_2}$ is a Lie algebra as well, if the Lie bracket is defined as:

$\displaystyle [(x_1, x_2), (y_1, y_2)] = ( [x_1, x_2], [y_1, y_2] ).$

This is the direct sum

Lie algebras arise frequently in a more interesting way:

Definition 2 Let ${R}$ be a finite-dimensional associative ${{\mathbb C}}$-algebra. Then the Lie bracket of ${a,b \in R}$ is defined as ${[a,b] = ab - ba}$; this makes ${R}$ into a Lie algebra, as is easily checked (but associativity is important).

Note in particular the resemblance of the Lie bracket to commutators in group theory. For instance, ${[a,b]=0}$ iff ${a,b}$ commute with each other.

Many of the important Lie algebras in fact arise in this way, from rings, especially matrix rings:

Definition 3 Fix ${n \in \mathbb{N}}$. We define ${\mathfrak{gl}_n}$ as the Lie algebra coming as above from the ring of ${n}$-by-${n}$ complex-valued matrices, with the usual Lie bracket.

It is often of interest to consider Lie subalgebras of ${\mathfrak{gl}_n}$. One of the most important is ${\mathfrak{sl}_n}$:

Definition 4 ${\mathfrak{sl}_n}$ is the space of complex ${n}$-by-${n}$ matrices whose trace is zero.

One must actually check that ${\mathfrak{sl}_n}$ is indeed a Lie subalgebra. But this follows from the identity

$\displaystyle Tr(AB) = Tr(BA)$

if ${A,B}$ are ${n}$-by-${n}$ matrices. So ${Tr([A,B])=0}$, for any ${A,B}$.

Representations

In general, a representation of some algebraic object is an action of that object on a vector space in a manner compatible with its algebraic structure. In our case, this will mean respecting the bracket.

Definition 5 A representation of ${L}$ is a ${{\mathbb C}}$-linear map ${\rho: L \rightarrow M_n({\mathbb C})}$ (${M_n({\mathbb C})}$ is the space of ${n}$-by-${n}$ matrices, which is an associative algebra) such that

$\displaystyle \rho([X,Y]) = [\rho(X), \rho(Y)]. \ \ \ \ \ (1)$

In other words it is a Lie algebra-homomorphism ${L \rightarrow \mathfrak{gl}_n}$. This resembles the notion of group representations, where we had group-homomorphisms ${G \rightarrow GL_n}$.

One can also phrase the above definition in the following way.

Example 3 A (finite-dimensional) representation of ${L}$ is a finite-dimensional vector space ${V}$ with a ${{\mathbb C}}$-bilinear map ${L \times V \rightarrow V}$, say denoted by ${M, v \rightarrow Mv}$, such that

$\displaystyle [M_1, M_2] v = M_1 (M_2 v) - M_2( M_1 v).\ \ \ \ \ (2)$

The equivalence of this phrasing with the previous one follows from the correspondence between linear transformations and matrices. Looking at the (initial) definition, a representation of ${L}$ on ${V}$ means a map ${\rho: L \rightarrow \mathfrak{gl}_n}$ if ${V}$ has dimension ${n}$. Then, we define for ${X \in L, v \in V}$,

$\displaystyle Xv = \rho(X) v;$

this gives a map ${L \times V \rightarrow V}$, and since ${\rho}$ is a Lie homomorphism, (2) is satisfied. Similarly, one can work the other way.

There is a parallel here with the notion of modules and even group representations, which arise if one takes an “enveloping algebra” of ${L}$; this, however, is better saved for a later post with more generality.

So, I’m planning to continue this series with a post on ${\mathfrak{sl}_2}$ and its irreducible representations, and then possibly talk about Engel and Lie, semisimple Lie algebras, and further into representation theory.