View source for Killing form

{{MSC|17B}}
{{TEX|done}}

The ''Killing form'' is 
a certain bilinear form on a finite-dimensional [[Lie algebra|Lie algebra]], 
introduced by W. Killing . Let $\def\f#1{\mathfrak{#1}}\f G$ be a finite-dimensional Lie algebra over a field $k$. By the Killing form on $\f G$ is meant the bilinear form

$$\def\tr{\textrm{tr}\;}\def\ad{\textrm{ad}\;}B(x,y) = \tr(\ad x \cdot \ad y),\quad x,y\in \f G $$
where $\tr$ denotes the trace of a linear operator, and $\ad x$ is the image of $x$ under the adjoint representation of $\f G$ (cf. also
[[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]), i.e. the linear operator on the vector space $\f G$ defined by the rule $z\mapsto [z,x]$, where $[\;,\;]$ is the commutation operator in the Lie algebra $\f G$. The Killing form is symmetric. The operators $\ad x$, $x\in \f G$, are skew-symmetric with respect to the Killing form, that is,

$$B([x,y],z) = B(x,[y,z])\quad \textrm{ for all } x,y,z\in \f G.$$
If $\f G_0$ is an ideal of $\f G$, then the restriction of the Killing form to $\f G_0$ is the same as the Killing form of $\f G_0$. Each commutative ideal $\f G_0$ is contained in the kernel of the Killing form. If the Killing form is non-degenerate, then the algebra $\f G$ is semi-simple (cf.
[[Lie algebra, semi-simple|Lie algebra, semi-simple]]).

Suppose that the characteristic of the field $k$ is 0. Then the radical of $\f G$ is the same as the orthocomplement with respect to the Killing form of the derived subalgebra $\f G' = [\f G,\f G]$. The algebra $\f G$ is solvable (cf.
[[Lie algebra, solvable|Lie algebra, solvable]]) if and only if $\f G\perp \f G'$, i.e. when $B([x,y],z) = 0$ for all $x,y,z\in \f G$ (Cartan's solvability criterion). If $\f G$ is nilpotent (cf.
[[Lie algebra, nilpotent|Lie algebra, nilpotent]]), then $B(x,y) = 0$ for all $x,y\in\f G$. The algebra $\f G$ is semi-simple if and only if the Killing form is non-degenerate (Cartan's semi-simplicity criterion).

Every complex semi-simple Lie algebra contains a real form $\Gamma$ (the compact Weyl form, see
[[Complexification of a Lie algebra|Complexification of a Lie algebra]]) on which the Killing form is negative definite.

====Comments====
The Killing form is a key tool in the Killing–Cartan classification of semi-simple Lie algebras over fields $k$ of characteristic 0. If $\textrm{char}\; k \ne 0$, the Killing form on a semi-simple Lie algebra may be degenerate.

The Killing form is also called the Cartan–Killing form.

Let $X_1,\dots,X_n$ be a basis for the Lie algebra $L_1$, and let the corresponding [[structure constant]]s be $\def\g{\gamma}\g_{ij}^k$, so that $[X_i,X_j] = \g_{ij}^k X_k$ ([[summation convention]]). Then in terms of these structure constants the Killing form is given by

$$B(X_a,X_b) = g_{ab} = \g_{ac}^d\g_{bd}^c$$
The metric (tensor) $g_{ab}$ is called the Cartan metric, especially in the theoretical physics literature. Using $g_{ab}$ one can lower indices (cf.
[[Tensor on a vector space|Tensor on a vector space]]) to obtain  "structure constants"  $\g_{abc} = g_{da} \g_{bc}^d$ which are completely anti-symmetric in their indices. (A direct consequence of the Jacobi identity and equivalent to the anti-symmetry of the operator $\ad y$ with respect to $B(x,z)$; cf. above.)

====References====
{|
|-
|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley  (1975)  (Translated from French)  {{MR|0682756}}  {{ZBL|0319.17002}} 
|-
|valign="top"|{{Ref|Ca}}||valign="top"|  E. Cartan,   "Sur la structure des groupes de transformations finis et continus", ''Oevres Complètes'', '''1''', CNRS  (1984)  pp. 137–288   {{ZBL|0007.10204}} JFM {{ZBL|59.0430.02}} JFM {{ZBL|25.0638.02}} 
|-
|valign="top"|{{Ref|Hu}}||valign="top"|  J.E. Humphreys,   "Introduction to Lie algebras and representation theory", Springer  (1972)  pp. §5.4  {{MR|0323842}}  {{ZBL|0254.17004}} 
|-
|valign="top"|{{Ref|Ka}}||valign="top"|  I. Kaplansky,   "Lie algebras and locally compact groups", Chicago Univ. Press  (1971)  {{MR|0276398}}  {{ZBL|0223.17001}} 
|-
|valign="top"|{{Ref|Ki}}||valign="top"|  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I"  ''Math. Ann.'', '''31'''  (1888)  pp. 252–290   JFM {{ZBL|20.0368.03}} 
|-
|valign="top"|{{Ref|Ki2}}||valign="top"|  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II"  ''Math. Ann.'', '''33'''  (1889)  pp. 1–48   JFM {{ZBL|20.0368.03}} 
|-
|valign="top"|{{Ref|Ki3}}||valign="top"|  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III"  ''Math. Ann.'', '''34'''  (1889)  pp. 57–122   JFM {{ZBL|21.0376.01}} 
|-
|valign="top"|{{Ref|Ki4}}||valign="top"|  W. Killing,   "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV"  ''Math. Ann.'', '''36'''  (1890)  pp. 161–189   
|-
|valign="top"|{{Ref|Na}}||valign="top"|  M.A. Naimark,   "Theory of group representations", Springer  (1982)  (Translated from Russian)  {{MR|0793377}}  {{ZBL|0484.22018}} 
|-
|valign="top"|{{Ref|Va}}||valign="top"|  V.S. Varadarajan,   "Lie groups, Lie algebras and their representations", Springer, reprint  (1984)  {{MR|0746308}}  {{ZBL|0955.22500}} 
|-
|}