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A certain bilinear form on a finite-dimensional [[Lie algebra|Lie algebra]], introduced by W. Killing . Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554001.png" /> be a finite-dimensional Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554002.png" />. By the Killing form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554003.png" /> is meant the bilinear form
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{{MSC|17B}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554004.png" /></td> </tr></table>
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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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554005.png" /> denotes the trace of a linear operator, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554006.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554007.png" /> under the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554008.png" /> (cf. also [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]), i.e. the linear operator on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k0554009.png" /> defined by the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540011.png" /> is the commutation operator in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540012.png" />. The Killing form is symmetric. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540014.png" />, are skew-symmetric with respect to the Killing form, that is,
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$$\def\tr{\textrm{tr}\;}\def\ad{\textrm{ad}\;}B(x,y) = \tr(\ad x, \ad y),\quad x,y\in \f G $$
 +
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540015.png" /></td> </tr></table>
+
$$B([x,y],z) = B(x,[y,z])\quad \textrm{ for all } x,y,z\in \f G.$$
 
+
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.
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540016.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540017.png" />, then the restriction of the Killing form to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540018.png" /> is the same as the Killing form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540019.png" />. Each commutative ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540020.png" /> is contained in the kernel of the Killing form. If the Killing form is non-degenerate, then the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540021.png" /> is semi-simple (cf. [[Lie algebra, semi-simple|Lie algebra, semi-simple]]).
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[[Lie algebra, semi-simple|Lie algebra, semi-simple]]).
 
 
Suppose that the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540022.png" /> is 0. Then the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540023.png" /> is the same as the orthocomplement with respect to the Killing form of the derived subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540024.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540025.png" /> is solvable (cf. [[Lie algebra, solvable|Lie algebra, solvable]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540026.png" />, i.e. when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540028.png" /> (Cartan's solvability criterion). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540029.png" /> is nilpotent (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540031.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540032.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540033.png" /> (the compact Weyl form, see [[Complexification of a Lie algebra|Complexification of a Lie algebra]]) on which the Killing form is negative definite.
 
 
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  W. Killing,  "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I"  ''Math. Ann.'' , '''31'''  (1888)  pp. 252–290</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  W. Killing,  "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II"  ''Math. Ann.'' , '''33'''  (1889)  pp. 1–48</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  W. Killing,  "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III"  ''Math. Ann.'' , '''34'''  (1889)  pp. 57–122</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  W. Killing,  "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV"  ''Math. Ann.'' , '''36'''  (1890)  pp. 161–189</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Cartan,  "Sur la structure des groupes de transformations finis et continus" , ''Oevres Complètes'' , '''1''' , CNRS  (1984)  pp. 137–288</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I. Kaplansky,  "Lie algebras and locally compact groups" , Chicago Univ. Press  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Naimark,  "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR></table>
 
  
 +
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====
 
====Comments====
The Killing form is a key tool in the Killing–Cartan classification of semi-simple Lie algebras over fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540034.png" /> of characteristic 0. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540035.png" />, the Killing form on a semi-simple Lie algebra may be degenerate.
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The Killing form is a key tool in the Killing–Cartan classification of semi-simple Lie algebras over fields $k$ of characteristic 0. If $\textrm{char}\; k \ne$, the Killing form on a semi-simple Lie algebra may be degenerate.
  
 
The Killing form is also called the Cartan–Killing form.
 
The Killing form is also called the Cartan–Killing form.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540036.png" /> be a basis for the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540037.png" />, and let the corresponding structure constants be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540038.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540039.png" /> (summation convention). Then in terms of these structure constants the Killing form is given by
+
Let $X_1,\dots,X_n$ be a basis for the Lie algebra $L_1$, and let the corresponding structure constants 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.)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540040.png" /></td> </tr></table>
 
  
The metric (tensor) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540041.png" /> is called the Cartan metric, especially in the theoretical physics literature. Using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540042.png" /> one can lower indices (cf. [[Tensor on a vector space|Tensor on a vector space]]) to obtain  "structure constants"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540043.png" /> which are completely anti-symmetric in their indices. (A direct consequence of the Jacobi identity and equivalent to the anti-symmetry of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540044.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055400/k05540045.png" />; cf. above.)
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. O'Raifeartaigh,  "Group structure of gauge theories" , Cambridge Univ. Press (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Varadarajan,  "Lie groups, Lie algebras and their representations" , Springer, reprint (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.E. Humphreys,  "Introduction to Lie algebras and representation theory" , Springer  (1972)  pp. §5.4</TD></TR></table>
+
{|
 +
|-
 +
|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}}
 +
|-
 +
|}

Revision as of 12:27, 22 May 2012

2020 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]

The Killing form is a certain bilinear form on a finite-dimensional 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

$$\def\tr{\textrm{tr}\;}\def\ad{\textrm{ad}\;}B(x,y) = \tr(\ad x, \ad y),\quad x,y\in \f G $$ 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), 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).

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) 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), 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) 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 $\textrm{char}\; k \ne$, 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 constants 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) 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

[Bo] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[Ca] 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
[Hu] J.E. Humphreys, "Introduction to Lie algebras and representation theory", Springer (1972) pp. §5.4 MR0323842 Zbl 0254.17004
[Ka] I. Kaplansky, "Lie algebras and locally compact groups", Chicago Univ. Press (1971) MR0276398 Zbl 0223.17001
[Ki] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I" Math. Ann., 31 (1888) pp. 252–290 JFM Zbl 20.0368.03
[Ki2] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II" Math. Ann., 33 (1889) pp. 1–48 JFM Zbl 20.0368.03
[Ki3] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III" Math. Ann., 34 (1889) pp. 57–122 JFM Zbl 21.0376.01
[Ki4] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV" Math. Ann., 36 (1890) pp. 161–189
[Na] M.A. Naimark, "Theory of group representations", Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[Va] V.S. Varadarajan, "Lie groups, Lie algebras and their representations", Springer, reprint (1984) MR0746308 Zbl 0955.22500
How to Cite This Entry:
Killing form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Killing_form&oldid=12372
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article