Construction of Finite-Dimensional Unitary Representations of Lie Groups

Explains how to construct finite-dimensional unitary representations of Lie groups.

Reference: (Peskin & Schroeder, 1995)

In gauge theory, local gauge transformations are unitary transformations of the fields. These are generated by finite-dimensional Hermitian representations of Lie algebras, for which it is sufficient that the Lie algebra is compact.

Here we explain some basic facts about Lie algebras.

An algebra that does not have a commutative Lie algebra $\mathfrak{u}(1)$ as an invariant subalgebra (ideal) is called semisimple, and an algebra with no non-trivial ideals is called simple. By definition, a simple Lie algebra is semisimple. Furthermore, any semisimple Lie algebra can be expressed as a direct sum of simple Lie algebras. Therefore, to study Lie algebras, it is sufficient to study simple Lie algebras.

In constructing a gauge theory, what we need to consider is finding compact simple Lie algebras. Actually, this condition is quite restrictive; there exist only 3 infinite families of algebras (called classical algebras) and 5 exceptional types.

The corresponding classical groups are as follows:

The exceptional types are denoted as:

Incidentally, $E_6$ and $E_8$ are considered for applications in grand unified theories.

Construction of Representations

I hope to write about this someday.

References