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.
Throughout, we work over , or even an algebraically closed field of characteristic zero.
Definition 1 A Lie algebra is a finite-dimensional vector space with a Lie bracket satisfying:
- The bracket is -bilinear in both variables.
- for any .
- . 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 are Lie algebras. Then is a Lie algebra as well, if the Lie bracket is defined as:
This is the direct sum
Lie algebras arise frequently in a more interesting way:
Definition 2 Let be a finite-dimensional associative -algebra. Then the Lie bracket of is defined as ; this makes 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, iff commute with each other.
Many of the important Lie algebras in fact arise in this way, from rings, especially matrix rings:
Definition 3 Fix . We define as the Lie algebra coming as above from the ring of -by- complex-valued matrices, with the usual Lie bracket.
It is often of interest to consider Lie subalgebras of . One of the most important is :
Definition 4 is the space of complex -by- matrices whose trace is zero.
One must actually check that is indeed a Lie subalgebra. But this follows from the identity
if are -by- matrices. So , for any .
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.
In other words it is a Lie algebra-homomorphism . This resembles the notion of group representations, where we had group-homomorphisms .
One can also phrase the above definition in the following way.
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 on means a map if has dimension . Then, we define for ,
this gives a map , and since 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 ; this, however, is better saved for a later post with more generality.
So, I’m planning to continue this series with a post on and its irreducible representations, and then possibly talk about Engel and Lie, semisimple Lie algebras, and further into representation theory.