S U (2)

are

E_{n} = Sym^{n} (C^{2})

, one for each

n \geq 0

n + 1

. Equivalently they are spin-

j

representations with

j = n /2

. The defining representation has

n = 1

, and

n = 0

J_{x}, J_{y}, J_{z}

with

[J_{x}, J_{y}] = i J_{z}

cyclically. The actual SU(2) Lie algebra generators are

L_{a} = - i J_{a}

, so

[L_{x}, L_{y}] = L_{z}

. On an orthonormal weight basis

v_{m}

m = - j, - j + 1, \dots, j

, the complexified action is

J_{z} v_{m} = m v_{m}, J_{\pm} v_{m} = (j \mp m) (j \pm m + 1) v_{m \pm 1}, J_{\pm} = J_{x} \pm i J_{y} .

The Casimir operator in this normalization is

C = - \sum_{a} L_{a}^{2} = \sum_{a} J_{a}^{2}

C ∣_{E_{n}} = j (j + 1) I = \frac{n ( n + 2 )}{4} I .

Other Lie-algebra metric normalizations rescale this scalar. On the diagonal subgroup

diag (e^{i t}, e^{- i t})

, the weights of

E_{n}

are

n, n - 2, \dots, - n

, all with multiplicity one.

 Ancestors (10)

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