In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.
Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras.
[1] The Killing form was essentially introduced into Lie algebra theory by Élie Cartan (1894) in his thesis.
In a historical survey of Lie theory, Borel (2001) has described how the term "Killing form" first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached.
[2] Some other authors now employ the term "Cartan-Killing form".
[citation needed] At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows that the Killing form (i.e. the degree 2 coefficient) is invariant, but he did not make much use of the fact.
A basic result that Cartan made use of was Cartan's criterion, which states that the Killing form is non-degenerate if and only if the Lie algebra is a direct sum of simple Lie algebras.
defines the adjoint endomorphism ad(x) (also written as adx) of
is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form with values in K, the Killing form on
, the matrix elements of the Killing form are given by Here in Einstein summation notation, where the cijk are the structure coefficients of the Lie algebra.
Taking the trace amounts to putting k = n and summing, and so we can write The Killing form is the simplest 2-tensor that can be formed from the structure constants.
In the above indexed definition, we are careful to distinguish upper and lower indices (co- and contra-variant indices).
This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors.
When the Lie algebra is semisimple over a zero-characteristic field, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes.
such that the structure constants with all upper indices are completely antisymmetric.
viewed in their fundamental matrix representation):[citation needed] The table shows that the Dynkin index for the adjoint representation is equal to twice the dual Coxeter number.
is a semisimple Lie algebra over the field of real numbers
By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries ±1.
By Sylvester's law of inertia, the number of positive entries is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis, and is called the index of the Lie algebra
which is an important invariant of the real Lie algebra.
is called compact if the Killing form is negative definite (or negative semidefinite if the Lie algebra is not semisimple).
Note that this is one of two inequivalent definitions commonly used for compactness of a Lie algebra; the other states that a Lie algebra is compact if it corresponds to a compact Lie group.
The definition of compactness in terms of negative definiteness of the Killing form is more restrictive, since using this definition it can be shown that under the Lie correspondence, compact Lie algebras correspond to compact semisimple Lie groups.
is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is
It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form
The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.
, and the special unitary algebra, denoted
of 2 × 2 real matrices with the unit determinant and the special unitary group
be a finite-dimensional Lie algebra over the field
It is easy to show that this is symmetric, bilinear and invariant for any representation