Inmathematics, theKilling form, named afterWilhelm Killing, is asymmetric bilinear form that plays a basic role in the theories ofLie groups andLie algebras.Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to thesemisimplicity of the Lie algebras.^[1]

The Killing form was essentially introduced into Lie algebra theory byÉlie 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 theSéminaire Bourbaki; it arose as amisnomer, 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 wasCartan's criterion, which states that the Killing form is non-degenerate if and only if the Lie algebra is adirect sum of simple Lie algebras.^[2]

Consider aLie algebra${\displaystyle \mathfrak{g}}$ over afieldK. Every elementx of${\displaystyle \mathfrak{g}}$ defines theadjoint endomorphismad(x) (also written asad_x) of${\displaystyle \mathfrak{g}}$ with the help of the Lie bracket, as

${\displaystyle \mathrm{ad}(x)(y)=[x,y].}$

Now, supposing${\displaystyle \mathfrak{g}}$ is of finite dimension, thetrace of the composition of two such endomorphisms defines asymmetric bilinear form

${\displaystyle B(x,y)=\mathrm{trace}(\mathrm{ad}(x)\circ \mathrm{ad}(y)),}$

with values inK, theKilling form on${\displaystyle \mathfrak{g}}$.

The following properties follow as theorems from the above definition.

• The Killing formB is bilinear and symmetric.
• The Killing form is an invariant form, as are all other forms obtained fromCasimir operators. Thederivation of Casimir operators vanishes; for the Killing form, this vanishing can be written as

${\displaystyle B([x,y],z)=B(x,[y,z])}$

where [ , ] is theLie bracket.

• If${\displaystyle \mathfrak{g}}$ is a complexsimple Lie algebra then any invariant symmetric bilinear form on${\displaystyle \mathfrak{g}}$ is a scalar multiple of the Killing form. This is no longer true if${\displaystyle \mathfrak{g}}$ is simple but not complex; key concept: absolutely simple Lie algebra.
• The Killing form is also invariant underautomorphismss of the algebra${\displaystyle \mathfrak{g}}$, that is,

${\displaystyle B(s(x),s(y))=B(x,y)}$

fors in${\displaystyle \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{g})}$.

• TheCartan criterion states that a Lie algebra issemisimple if and only if the Killing form isnon-degenerate.
• The Killing form of anilpotent Lie algebra is identically zero.
• IfI,J are twoideals in a Lie algebra${\displaystyle \mathfrak{g}}$ with zero intersection, thenI andJ areorthogonal subspaces with respect to the Killing form.
• The orthogonal complement with respect toB of an ideal is again an ideal.^[3]
• If a given Lie algebra${\displaystyle \mathfrak{g}}$ is a direct sum of its idealsI₁,...,I_n, then the Killing form of${\displaystyle \mathfrak{g}}$ is the direct sum of the Killing forms of the individual summands.

Given a basise_i of the Lie algebra${\displaystyle \mathfrak{g}}$, the matrix elements of the Killing form are given by

${\displaystyle B_{ij}=\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{a}\mathrm{d}(e_i)\circ \mathrm{a}\mathrm{d}(e_j)).}$

Here

${\displaystyle \left(\text{ad}(e_i)\circ \text{ad}(e_j)\right)(e_k)=[e_i,[e_j,e_k]]=[e_i,{c_{jk}}^m{e}_m]={c_{im}}^n{c_{jk}}^m{e}_n}$

inEinstein summation notation, where thec_ij^k are thestructure coefficients of the Lie algebra. The indexk functions as column index and the indexn as row index in the matrixad(e_i)ad(e_j). Taking the trace amounts to puttingk =n and summing, and so we can write

${\displaystyle B_{ij}={c_{im}}^n{c_{jn}}^m}$

The Killing form is the simplest 2-tensor that can be formed from the structure constants. The form itself is then${\displaystyle B=B_{ij}{e}^i\otimes e^j.}$

In the above indexed definition, we are careful to distinguish upper and lower indices (co- andcontra-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 issemisimple over a zero-characteristic field, its Killing form is nondegenerate, and hence can be used as ametric tensor to raise and lower indexes. In this case, it is always possible to choose a basis for${\displaystyle \mathfrak{g}}$ such that the structure constants with all upper indices arecompletely antisymmetric.

The Killing form for some Lie algebras${\displaystyle \mathfrak{g}}$ are (forX,Y in${\displaystyle \mathfrak{g}}$ viewed in their fundamental matrix representation):^{[citation needed]}

${\displaystyle \mathfrak{g}}$	${\displaystyle B(X,Y)}$	Classification	Dual coxeter number
${\displaystyle \mathfrak{g}\mathfrak{l}(n,\mathbb{R})}$	${\displaystyle 2n\text{tr}(XY)-2\text{tr}(X)\text{tr}(Y)}$	-	-
${\displaystyle \mathfrak{s}\mathfrak{l}(n,\mathbb{R}),n\ge 2}$	${\displaystyle 2n\text{tr}(XY)}$	${\displaystyle A_{n-1}}$	${\displaystyle n}$
${\displaystyle \mathfrak{s}\mathfrak{u}(n),n\ge 2}$	${\displaystyle 2n\text{tr}(XY)}$	${\displaystyle A_{n-1}}$	${\displaystyle n}$
${\displaystyle \mathfrak{s}\mathfrak{o}(n),n\ge 2}$	${\displaystyle (n-2)\text{tr}(XY)}$	${\displaystyle B_m,n=2m+1}$ for${\displaystyle n}$ odd.${\displaystyle D_m,n=2m}$ for${\displaystyle n}$ even.	${\displaystyle n-2}$
${\displaystyle \mathfrak{s}\mathfrak{o}(n,\mathbb{C}),n\ge 2}$	${\displaystyle (n-2)\text{tr}(XY)}$	${\displaystyle B_m,n=2m+1}$ for${\displaystyle n}$ odd.${\displaystyle D_m,n=2m}$ for${\displaystyle n}$ even.	${\displaystyle n-2}$
${\displaystyle \mathfrak{s}\mathfrak{p}(2n,\mathbb{R}),n\ge 1}$	${\displaystyle 2(n+1)\text{tr}(XY)}$	${\displaystyle C_n}$	${\displaystyle n+1}$
${\displaystyle \mathfrak{s}\mathfrak{p}(2n,\mathbb{C}),n\ge 1}$	${\displaystyle 2(n+1)\text{tr}(XY)}$	${\displaystyle C_n}$	${\displaystyle n+1}$

The table shows that theDynkin index for the adjoint representation is equal to twice thedual Coxeter number.

Suppose that${\displaystyle \mathfrak{g}}$ is asemisimple Lie algebra over the field of real numbers${\displaystyle \mathbb{R}}$. ByCartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries±1. BySylvester'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 theindex of the Lie algebra${\displaystyle \mathfrak{g}}$. This is a number between0 and the dimension of${\displaystyle \mathfrak{g}}$ which is an important invariant of the real Lie algebra. In particular, a real Lie algebra${\displaystyle \mathfrak{g}}$ is calledcompact if the Killing form isnegative definite (or negative semidefinite if the Lie algebra is not semisimple). Note that this is one of two inequivalent definitions commonly used forcompactness of a Lie algebra; the other states that a Lie algebra is compact if it corresponds to acompact 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 theLie correspondence, compact Lie algebras correspond to compact semisimple Lie groups.

If${\displaystyle {\mathfrak{g}}_{\mathbb{C}}}$ is a semisimple Lie algebra over thecomplex numbers, then there are several non-isomorphic real Lie algebras whosecomplexification is${\displaystyle {\mathfrak{g}}_{\mathbb{C}}}$, which are called itsreal forms. It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form${\displaystyle \mathfrak{g}}$. 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 complexspecial linear algebra${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ has two real forms, the real special linear algebra, denoted${\displaystyle \mathfrak{s}\mathfrak{l}(2,\mathbb{R})}$, and thespecial unitary algebra, denoted${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$. The first one is noncompact, the so-calledsplit real form, and its Killing form has signature(2, 1). The second one is the compact real form and its Killing form is negative definite, i.e. has signature(0, 3). The corresponding Lie groups are the noncompact group${\displaystyle \mathrm{S}\mathrm{L}(2,\mathbb{R})}$ of2 × 2 real matrices with the unit determinant and the special unitary group${\displaystyle \mathrm{S}\mathrm{U}(2)}$, which is compact.

Let${\displaystyle \mathfrak{g}}$ be a finite-dimensional Lie algebra over the field${\displaystyle K}$, and${\displaystyle \rho :\mathfrak{g}\to \text{End}(V)}$ be a Lie algebra representation. Let${\displaystyle {\text{Tr}}_V:\text{End}(V)\to K}$ be the trace functional on${\displaystyle V}$. Then we can define the trace form for the representation${\displaystyle \rho }$ as

${\displaystyle {\text{Tr}}_{\rho }:\mathfrak{g}\times \mathfrak{g}\to K,}$

${\displaystyle {\text{Tr}}_{\rho }(X,Y)={\text{Tr}}_V(\rho (X)\rho (Y)).}$

Then the Killing form is the special case that the representation is the adjoint representation,${\displaystyle {\text{Tr}}_{\text{ad}}=B}$.

It is easy to show that this is symmetric, bilinear and invariant for any representation${\displaystyle \rho }$.

If furthermore${\displaystyle \mathfrak{g}}$ is simple and${\displaystyle \rho }$ is irreducible, then it can be shown${\displaystyle {\text{Tr}}_{\rho }=I(\rho )B}$ where${\displaystyle I(\rho )}$ is theindex of the representation.

• Casimir invariant
• Killing vector field

• Lie groups
• Lie algebras

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