 Quadratic Casimir operator

ID: quadratic-casimir-operator

For a Lie algebra representation with generators referred to an orthonormal invariant form, the sum

\sum_{a} t_{a} t_{a}

commutes with every generator. Indeed, the commutator is proportional to a contraction of antisymmetric structure constants of a Lie algebra with the symmetric product

t_{a} t_{b} + t_{b} t_{a}

. The Schur lemma makes it scalar in an irreducible complex representation; that scalar is its quadratic Casimir eigenvalue. With trace normalization

tr (t_{a} t_{b}) = C (R) δ_{ab}

, taking the trace gives

C (R) dim g = C_{2} (R) dim R

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