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 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-degenerateif and only if the Lie algebra is a direct sum of simple Lie algebras.
Definition
Consider a Lie algebra over a field. Every element of defines the adjoint endomorphism of with the help of the Lie bracket, as Now, supposing 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.
Properties
The Killing form is bilinear and symmetric.
The Killing form is an invariant form, in the sense that it has the 'associativity' property
If a given Lie algebra is a direct sum of its ideals, then the Killing form of is the direct sum of the Killing forms of the individual summands.
Matrix elements
Given a basis of the Lie algebra, 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. In the above indexed definition, we are careful to distinguish upper and lower 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 such that the structure constants with all upper indices are completely antisymmetric. The Killing form for some Lie algebras are :
Connection with real forms
Suppose that 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. 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. This is a number between and the dimension of which is an important invariant of the real Lie algebra. In particular, a real Lie algebra is called compact if the Killing form is negative definite. 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 is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is, which are called its real forms. It turns out that every complex semisimple Lie algebra admits a unique 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. For example, the complex special linear algebra has two real forms, the real special linear algebra, denoted, and the special unitary algebra, denoted. 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 of real matrices with the unit determinant and the special unitary group, which is compact.
