SU(2) Lie algebra ( $su (2)$ )

The SU(2) Lie algebra is the compact real form of

sl_{2} (C)

. In the Hermitian physics convention its generators satisfy

[J_{i}, J_{j}] = i ϵ_{ijk} J_{k}

Killing form of the SU(2) Lie algebra ( $κ (T^{a}, T^{b}) = - 2 δ^{ab}$ )

In the anti-Hermitian basis

T^{a} = - i σ^{a} /2

[T^{a}, T^{b}] = ε^{ab c} T^{c}

κ (T^{a}, T^{b}) = - 2 δ^{ab}

. This is negative definite; the positive internal form for compact Yang-Mills theory is its negative.

With Hermitian angular-momentum generators

[J_{a}, J_{b}] = i ϵ_{ab c} J_{c}

2 δ_{ab}

and its inverse contraction is

C_{κ} = \frac{1}{2} \sum_{a} J_{a}^{2}

. Equivalently the anti-Hermitian basis

T_{a} = - i J_{a}

- 2 δ_{ab}

and gives the same operator. In the complex highest-weight basis,

C_{κ} = H^{2} /8 + (E^{+} E^{-} + E^{-} E^{+}) /4

. It acts on spin

j

j (j + 1) /2

, so it differs in normalization from the usual angular-momentum quadratic Casimir operator.

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