Representation theory of SU(2)

In the study of the <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> of <a href="/facts/Lie_groups/a9jCQyjF">Lie groups</a>, the study of representations of <a href="/facts/SU(2)/Qgy2orcu">SU(2)</a> is fundamental to the study of representations of <a href="/facts/Semisimple_Lie_group/m8jA3Bcf">semisimple Lie groups</a>. It is the first case of a Lie group that is both a <a href="/facts/Compact_group/Itl2HJaR">compact group</a> and a <a href="/facts/Non-abelian_group/KQYHrwJl">non-abelian group</a>. The first condition implies the representation theory is discrete: representations are <a href="/facts/Direct_sum_of_representations/x3ljVsP6">direct sums</a> of a collection of basic <a href="/facts/Irreducible_representation/Hu5geBgk">irreducible representations</a> (governed by the <a href="/facts/Peter%E2%80%93Weyl_theorem/vQfn6cQr">Peter–Weyl theorem</a>). The second means that there will be irreducible representations in dimensions greater than 1.
SU(2) is the <a href="/facts/Universal_covering_group/IrRjtTL2">universal covering group</a> of <a href="/facts/SO(3)/yXbKPzhT">SO(3)</a>, and so its representation theory includes that of the latter, by dint of a <a href="/facts/Rotation_group_SO(3)/yXbKPzhT">surjective homomorphism</a> to it. This underlies the significance of SU(2) for the description of non-relativistic <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a> in <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>; 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 
 
 
 
 m
 
 
 {\displaystyle m}
 
 and have dimension 
 
 
 
 m
 +
 1
 
 
 {\displaystyle m+1}
 
. In the physics literature, the representations are labeled by the quantity 
 
 
 
 l
 =
 m
 
 /
 
 2
 
 
 {\displaystyle l=m/2}
 
, where 
 
 
 
 l
 
 
 {\displaystyle l}
 
 is then either an integer or a half-integer, and the dimension is 
 
 
 
 2
 l
 +
 1
 
 
 {\displaystyle 2l+1}
 
.

Representation theory of SU(2) open-in-new

Representation theory of SU(2)