Representation Theory of SU(2) - Lie Algebra Representations

Lie Algebra Representations

Consider first representations of the Lie algebra

.

In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of infinitesimal transformations, and their Lie groups to 'integrated' transformations.

Then pass to the complex Lie algebra (i.e. complexify the Lie algebra). This doesn't affect the representation theory. The Lie algebra is spanned by three elements e, f and h with the Lie brackets

Since is semisimple, the representation ρ(h) is always diagonalizable (for complex number scalars). Its eigenvalues are called the weights.

Suppose x is an eigenvector of the weight α. Then,

In other words, e raises the weight by one and f reduces the weight by one. A consequence is that

is a Casimir invariant. By Schur's lemma, its action is proportional to the identity map, for irreducible representations. The constant of proportionality is conveniently written

λ(λ+1).