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
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...
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,
The commutators are
be a finite irreducible representation
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...
, and let
be an eigenvector of
with the highest eigenvalue
is the eigenvector of highest eigenvalue,
. Similarly, we can show that
and since h has a lowest eigenvalue, there is a
. We will take the smallest
such that this happens.
We can then recursively calculate
and we find
, 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
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".