Killing form

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.
History and name.
The Killing form was essentially introduced into Lie algebra theory by in his thesis. The name "Killing form" first appeared in a paper of Armand Borel in 1951, but he stated in 2001 that he doesn't remember why he chose it. Borel admits that the name seems to be a misnomer, and that it would be more correct to call it the "Cartan form". Wilhelm Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra is 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 this fact. A basic result 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.
Definition.
Consider a Lie algebra formula_1 over a field . Every element of formula_1 defines the adjoint endomorphism (also written as ) of formula_1 with the help of the Lie bracket, as
Now, supposing formula_1 is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form
with values in , the Killing form on formula_1.
Properties.
The following properties follow as theorems from the above definition.
Matrix elements.
Given a basis of the Lie algebra formula_1, the matrix elements of the Killing form are given by
Here
in Einstein summation notation, where the are the structure coefficients of the Lie algebra. The index functions as column index and the index as row index in the matrix . Taking the trace amounts to putting and summing, and so we can write
The Killing form is the simplest 2-tensor that can be formed from the structure constants. The form itself is then formula_21
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. In this case, it is always possible to choose a basis for formula_1 such that the structure constants with all upper indices are completely antisymmetric.
The Killing form for some Lie algebras formula_1 are (for in formula_1 viewed in their fundamental n by n (2n by 2n) representations):
Connection with real forms.
Suppose that formula_1 is a semisimple Lie algebra over the field of real numbers formula_26. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries . 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 formula_1. This is a number between and the dimension of formula_1 which is an important invariant of the real Lie algebra. In particular, a real Lie algebra formula_1 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 Lie groups.
If formula_30 is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is formula_30, which are called its real forms. It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form formula_1. The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.
For example, the complex special linear algebra formula_33 has two real forms, the real special linear algebra, denoted formula_34, and the special unitary algebra, denoted formula_35. The first one is noncompact, the so-called split real form, and its Killing form has signature . The second one is the compact real form and its Killing form is negative definite, i.e. has signature . The corresponding Lie groups are the noncompact group formula_36 of real matrices with the unit determinant and the special unitary group formula_37, which is compact.