Killing form


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The Killing form or Cartan-Killing form is a binary invariant polynomial that is present on any finite-dimensional Lie algebra.


Given a finite-dimensional kk-Lie algebra 𝔤\mathfrak{g} its Killing form B:𝔤𝔤kB:\mathfrak{g}\otimes \mathfrak{g}\to k is the symmetric bilinear form given by the formula

B(x,y)=tr(ad(x)ad(y)), B(x,y) \,=\, tr\big( ad(x) \circ ad(y) \big) \,,


ad(x)[x,]:𝔤𝔤 ad(x) \,\coloneqq\, [x,-] \,:\, \mathfrak{g} \longrightarrow \mathfrak{g}

is the linear map given by the adjoint action of xx, hence the value on xx of the adjoint representation ad:𝔤Der(𝔤)ad \colon \mathfrak{g} \to Der(\mathfrak{g}).

If {t a}\{t_a\} is a linear basis for 𝔤\mathfrak{g} and {C a bc}\{C^a{}_{b c}\} are the structure constants of the Lie algebra in this basis (defined by [t a,t b]= cC ab ct c[t_a, t_b] = \sum_c C^c_{a b} t_c), then

B(t a,t b)= c,dC c adC bc d. B(t_a, t_b) \,=\, \sum_{c,d} C^c{}_{a d} C^{d}_{b c} \,.


The Killing form is am invariant polynomial in that

B([x,y],z)=B(x,[y,z]) B\big([x,y],z\big) \,=\, B\big(x, [y,z] \big)

for all x,y,z𝕘x,y,z \in \mathbb{g}. This follows from the cyclic invariance of the trace],

For complex Lie algebras 𝔤\mathfrak{g}, nondegeneracy of the Killing form (i.e. being the metric making 𝔤\mathfrak{g} a metric Lie algebra) is equivalent to semisimplicity of 𝔤\mathfrak{g}.

For simple complex Lie algebras, any invariant nondegenerate symmetric bilinear form is proportional to the Killing form.


Sometimes one considers more generally a Killing form B ρB_\rho for a more general faithful finite-dimensional representation ρ\rho, B ρ(x,y)=tr(ρ(x)ρ(y))B_\rho(x,y) = tr\big(\rho(x)\rho(y)\big). If the Killing form is nondegenerate and x 1,,x nx_1,\ldots,x_n is a basis in LL with x 1 *,,x n *x_1^*,\ldots,x_n^* the dual basis of 𝔤 *\mathfrak{g}^*, with respect to the Killing form for ρ\rho, then the canonical element r= ix ix i *r = \sum_i x_i\otimes x_i^* defines the Casimir operator C(ρ)=(ρρ)(r)C(\rho) =(\rho\otimes\rho)(r) in the representation ρ\rho; regarding that the representation is faithful, if the ground field is \mathbb{C}, by Schur's lemma C(ρ)C(\rho) is a nonzero scalar operator. Instead of Casimir operators in particular faithful representations it is often useful to consider an analogous construction within the universal enveloping algebra, the Casimir element in U(𝔤)U(\mathfrak{g}).


See also: