= Quadratic Casimir operator
{title2=$\sum_a t_at_a=C_2(R)I$}
For a <Lie algebra representation> with generators referred to an orthonormal invariant form, the sum $\sum_at_at_a$ commutes with every generator. Indeed, the <commutator> is proportional to a contraction of antisymmetric <structure constants of a Lie algebra> with the symmetric product $t_at_b+t_bt_a$. The <Schur lemma> makes it scalar in an irreducible complex representation; that scalar is its quadratic <Casimir eigenvalue>. With <trace> normalization $\operatorname{tr}(t_at_b)=C(R)\delta_{ab}$, taking the <trace> gives $C(R)\dim\mathfrak g=C_2(R)\dim R$.
Back to article page
