Killing form

Killing form
Jgmoxness · CC BY-SA 3.0 · source
Name	Killing form
Field	Lie algebra
Introduced by	Élie Cartan
Related	Killing–Cartan classification, Cartan subalgebra, Cartan matrix, Weyl group, root system

Killing form is a canonical symmetric bilinear form defined on a finite-dimensional Lie algebra over a field of characteristic zero, introduced in the work of Wilhelm Killing and formalized by Élie Cartan. It plays a central role in the classification of semisimple Lie algebras, the structure theory of semisimple Lie algebras, and the representation theory of compact Lie groups and complex Lie algebras. The form connects with spectral properties, root system geometry, and invariants used across mathematical physics and differential geometry.

Definition and basic properties

For a finite-dimensional Lie algebra g over a field F, the Killing form B is given by B(X,Y)=Tr(ad_X ad_Y), where ad: g→End(g) is the adjoint representation and Tr denotes the trace on linear operators. Basic properties include symmetry B(X,Y)=B(Y,X), invariance B([Z,X],Y)=B(X,[Z,Y]) for all X,Y,Z∈g, and bilinearity over F. The Killing form is functorial under Lie algebra homomorphisms between simple Lie algebras and behaves naturally with respect to decompositions into direct sums of ideals, central elements, and derived algebras encountered in the work on Levi decomposition and Cartan decomposition.

Nondegeneracy and Cartan's criterion

Cartan's criterion states that a finite-dimensional Lie algebra g over a field of characteristic zero is solvable if and only if the Killing form vanishes on the derived algebra [g,g], linking nondegeneracy to semisimplicity. Equivalently, g is semisimple iff the Killing form is nondegenerate; this result appears in Cartan's classification and underlies the Killing–Cartan classification of simple Lie algebras such as the classical series A_n, B_n, C_n, D_n and exceptional types E6, E7, E8, F4, G2. Over real numbers one distinguishes compact versus noncompact forms via the signature of the Killing form, a criterion used in the classification of real semisimple Lie algebras and in the study of Cartan involutions.

Computation and examples

Explicit computation of the Killing form for matrix Lie algebras uses the adjoint action and matrix trace. For g=sl(n,C), B(X,Y)=2n Tr(XY) up to normalization, while for so(n,R) and sp(2n,C) analogous formulas involve the defining representation and classical invariants. For the exceptional Lie algebras E6, E7, E8, F4, G2, tables of Killing form values appear in the classification literature of Élie Cartan and in works by Hermann Weyl and Nathan Jacobson. In representation theory one computes B on Cartan subalgebras using the root system and the Killing form identifies a natural inner product on the dual of a Cartan subalgebra, connecting to Dynkin diagram combinatorics and the construction of Cartan matrix.

Relationship to structure theory

The Killing form detects ideals, centers, and decomposition: the radical of B equals the maximal solvable ideal (the radical), so semisimplicity equates to trivial radical. Orthogonality with respect to B provides decompositions into ideals and underpins the Levi decomposition into a semisimple subalgebra and the radical. The form also interacts with Cartan subalgebra theory: roots α in the root system are evaluated via the Killing-induced inner product, and reflections in the associated Weyl group are defined using B. In classification, the signature and positivity properties of B on real forms guide the distinction between compact and split real forms, as in the classification of real forms of complex semisimple Lie algebras by Élie Cartan and later authors.

Applications and invariants

The Killing form yields invariants such as the quadratic Casimir element in the universal enveloping algebra U(g), central to representation theory and quantum field theory applications like the construction of Laplacians on homogeneous spaces and the classification of unitary representations of compact Lie groups. In differential geometry the form produces bi-invariant metrics on Lie groups when nondegenerate, and its signature determines Riemannian versus pseudo-Riemannian structures on homogeneous spaces such as symmetric spaces classified by Élie Cartan. In algebraic geometry and number theory, the Killing form enters the study of adjoint algebraic groups, Galois cohomology, and the Hasse principle for forms of simple algebraic groups. The Casimir eigenvalues computed from the Killing form classify irreducible representations in the work of Hermann Weyl and Harish-Chandra.

Generalizations and variants

Generalizations include the relative Killing form for representations other than the adjoint, defined by Tr(ρ(X)ρ(Y)) for a representation ρ, and the notion of invariant bilinear forms on Lie superalgebras appearing in Kac–Moody algebra and superalgebra theory. The Killing form extends to infinite-dimensional settings with caution: for Kac–Moody algebras one uses an invariant bilinear form that may be degenerate, and for vertex operator algebras analogous bilinear forms pair with conformal structures studied by Richard Borcherds. Over fields of positive characteristic the Killing form can behave pathologically and must be replaced by forms derived from cohomological or modular invariants in the literature of modular Lie algebras and Chevalley group constructions.

Category:Lie algebras