Casimir element

Let $\mathfrak{g}$ have a nondegenerate symmetric invariant bilinear form on a Lie algebra $B$, and choose a basis $(x_i)$ with $B$-dual basis $(x^i)$. Its quadratic Casimir element isThe inverse-form tensor ${\sum }_i{x}_i\otimes x^i$ is independent of the basis and invariant under the Adjoint representation of a Lie algebra. Multiplying its factors in the universal enveloping algebra gives $[z,C_B]={\sum }_i([z,x_i]x^i+x_i[z,x^i])=0$, so $C_B$ is central. In a finite-dimensional irreducible complex representation, the Schur lemma makes its represented quadratic Casimir operator a scalar.^[ref]

With the Killing form on a complex semisimple Lie algebra, its Casimir eigenvalue on highest weight $\lambda$ is $(\lambda ,\lambda +2\rho )$, with the induced dual inner product and Weyl vector $\rho$. Scaling $B$ by $c\ne 0$ scales $C_B$ by $c^{-1}$. For the sl2 Lie algebra, the normalization $B(e,f)=1$, $B(h,h)=2$ gives $ef+fe+h^2/2$; the actual Killing form is four times this form, so its Casimir is one quarter of that expression.

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 $\mathrm{tr}(t_a{t}_b)=C(R){\delta }_{ab}$, taking the trace gives $C(R)\dim \mathfrak{g}=C_2(R)\dim R$.

For the defining representation of the Special orthogonal Lie algebra, Hermitian Lie algebra generators $t_{ij}=i(E_{ij}-E_{ji})/\sqrt{2}$ have trace normalization $\mathrm{tr}(t_{ij}{t}_{kl})={\delta }_{ik}{\delta }_{jl}$ for ordered pairs. Summing their squares gives ${\sum }_{i<j}{t}_{ij}^2=(N-1)I/2$. Consequently the Casimir eigenvalue is $(N-1)/2$ and the Dynkin index is one. Doubling the trace normalization doubles both numbers; a gauge coupling must be rescaled inversely with the Lie algebra generators. With long roots of squared length two, the first convention is $C_2(\lambda )=(\lambda ,\lambda +2\rho )/2$.

For $e_3=[e_1,e_2]$, $f_3=-[f_1,f_2]$, the operator $C=h_1^2+h_2^2+h_1{h}_2+3(h_1+h_2)+3{\sum }_{i=1}^3{f}_i{e}_i$ commutes with all generators. In its commutator with $e_1$, the Cartan polynomial contributes $3h_1{e}_1$, while the last sum contributes $-3h_1{e}_1$; the two mixed terms involving $f_2{e}_3$ cancel. On highest weight $(p,q)$ its eigenvalue is $p^2+q^2+pq+3p+3q$. This normalization is three times the conventional SU(3) quadratic Casimir eigenvalue.

For generators normalized by ${\mathrm{tr}}_3(t_a{t}_b)={\delta }_{ab}/2$, the quadratic Casimir operator on the irreducible SU(3) representation of Dynkin labels $(a,b)$ has the displayed scalar value. In SU(3) highest-weight coordinates $(p,q)$ it is $p^2+q^2+p+\sqrt{3}q$.

To derive this, write the operator as $H_1^2+H_2^2+{\sum }_{i=1}^3(E_{+}^i{E}_{-}^i+E_{-}^i{E}_{+}^i)$. On a highest-weight vector, the raising operators vanish, and the three root commutators sum to $H_1+\sqrt{3}{H}_2$. Centrality and the Schur lemma extend this value to the representation. The singlet, fundamental, adjoint and decuplet values are $0,4/3,3,6$, respectively. Tensor labels and multiplet dimensions are summarized in the Particle Data Group's SU(n) multiplet guide.

For traceless generators, the tensor product of Lie algebra representations has generators $t_a\otimes I+I\otimes s_a$. The trace of their summed squares is the sum of the two factor Casimir traces; the cross term vanishes because $\mathrm{tr}{t}_a=\mathrm{tr}{s}_a=0$. Decomposition into irreducible representations, with multiplicities counted, gives the displayed identity. Without tracelessness an extra term $2{\sum }_a\mathrm{tr}{t}_a\mathrm{tr}{s}_a$ is present. A one-dimensional abelian representation of charge one tensored with itself has Casimir four, not the sum two; semisimple Lie algebras avoid this problem because every generator is a sum of commutators.

The Casimir operator formed from the Killing form acts on an Irreducible Lie algebra representation with highest weight $\lambda$ by the scalar $(\lambda ,\lambda +2\rho )$, where $\rho$ is the half-sum of positive roots. It is zero on the trivial Lie algebra representation and nonzero on every nontrivial finite-dimensional irreducible complex representation of a semisimple Lie algebra.

For ${\mathfrak{sl}}_2$ with $[h,e]=2e$, $[h,f]=-2f$, and $[e,f]=h$, the Casimiracts on the irreducible highest-weight module $V(m)$ by the scalar $\frac{1}{2}m(m+2)$.