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.

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