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 a non-negative integer ${\displaystyle m}$ and have dimension ${\displaystyle m+1}$. In the physics literature, the representations are labeled by the quantity ${\displaystyle l=m/2}$, where ${\displaystyle l}$ is then either an integer or a half-integer, and the dimension is ${\displaystyle 2l+1}$.