Special linear Lie algebra

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the special linear Lie algebra of order n (denoted ) is the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 of matrices with trace zero and with the Lie bracket . This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras.

Representation Theory of

The simplest non-trivial Lie algebra is , consisting of two by two matrices with zero trace. There are three basis elements, ,, and , with



and


The commutators are

, , and

Let be a finite irreducible representation

Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...

 of , and let be an eigenvector of with the highest eigenvalue . Then,



or



Since is the eigenvector of highest eigenvalue, . Similarly, we can show that



and since h has a lowest eigenvalue, there is a such that . We will take the smallest such that this happens.

We can then recursively calculate



and we find



Taking , we get



Since we chose to be the smallest exponent such that , we conclude that
. From this, we see that

, , ...


are all nonzero, and it is easy to show that they are linearly independent.
Therefore, for each , there is a unique, up to isomorphism, irreducible representation of dimension spanned by elements , , ...
.

The beautiful special case of shows a general way to find irreducible representations of Lie Algebras. Namely, we divide the algebra to three subalgebras "h" (the Cartan Subalgebra), "e", and "f", which behave approximately like their namesakes in . Namely, in a irreducible representation, we have a "highest" eigenvector of "h", on which "e" acts by zero. The basis of the irreducible representation is generated by the action of "f" on the highest eigenvectors of "h".