Representation theory of SU(2)
In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter–Weyl theorem). The second means that there will be irreducible representations in dimensions greater than 1.
SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic spin in theoretical physics; see below for other physical and historical context.
As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by an integer or half-integer λ ≥ 0, and have dimension 2λ + 1.
The representations of the group are found by considering representations of , the Lie algebra of SU(2). In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of infinitesimal transformations, and their Lie groups consist of 'integrated' transformations. Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation. The group representations can be realized on spaces of polynomials in two complex variables. In what follows, we shall consider the complex Lie algebra (i.e. the complexification of the Lie algebra), which doesn't affect the representation theory. A reference for this material is Section 4.6 of (Hall 2015).
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