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Killing form

Context

\infty-Lie theory

∞-Lie theory

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Contents

Idea

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

Definition

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(ad(x)ad(y))

where ad(x)=[x,]:𝔤𝔤ad(x) = [x,-]:\mathfrak{g}\to \mathfrak{g} is the adad-operator giving the adjoint representation ad:𝔤Der(𝔤)ad: \mathfrak{g}\to Der(\mathfrak{g}).

In terms of a basis: if {t a}\{t_a\} is a basis for 𝔤\mathfrak{g} and {C a bc}\{C^a{}_{b c}\} 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} \,.

Properties

The Killing form is am invariant polynomial in that

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

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

For complex Lie algebras, nondegeneracy of the Killing form 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.

Generalizations

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(\rho(x)\rho(y)). 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(\mathfrak{g}).

Revised on September 6, 2010 11:48:18 by Urs Schreiber (134.100.32.213)