SU(2)

The SU(2) algebra is familiar.¹ 3.1 https://www.w3.org/1998/Math/MathML">Jj, Jk=iϵjklJl https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429499210/072e8f02-fd02-46a7-9aee-b57ab6049c01/content/eqn0239.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> This is the simplest of the compact Lie algebras because ε_ijk for i, j, k = 1 to 3 is the simplest possible completely antisymmetric object with three indices. (3.1) is equivalent (in units in which ħ = 1) to the angular momentum algebra that you studied in quantum mechanics. In fact we will only do two things differently here. One is to label the generators by 1, 2 and 3 instead of x, y and z. This is obviously a great step forward. More important is the fact that we will not make any use of the operator J_a J_a. Initially, this will make the analysis slightly more complicated, but it will start us on a path that generalizes beautifully to all the other compact Lie algebras.