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''or algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108801.png" />''
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The linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108802.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108803.png" /> in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108804.png" /> (or in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108805.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108806.png" />) mapping each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108807.png" /> to the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108808.png" /> of the inner automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a0108809.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088010.png" /> is a linear group in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088011.png" />, then
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''or algebraic group $G$''
  
<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/a/a010/a010880/a01088012.png" /></td> </tr></table>
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The ''adjoint representation'' of a Lie group $G$ is
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the linear representation $\def\Ad{\textrm{Ad}\;} \Ad $ of $G$ in the tangent space $T_e(G)$ (or in the Lie algebra $\def\g{\mathfrak{g}}\g$ of $G$), given by the mapping which sends each $a\in G$ to the differential $\def\Int{\textrm{Int}\;}\Ad a = d(\Int a)_e$ of the inner automorphism $\Int a: x\mapsto axa^{-1}$. If $G\subseteq \def\GL{\textrm{GL}} \GL(V)$ is a linear group of isomorphisms of a vector space $V$, then
  
The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088013.png" /> contains the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088014.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088015.png" /> is connected and if the ground field has characteristic zero, coincides with this centre. The differential of the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088016.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088017.png" /> coincides with the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088019.png" />.
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$$(\Ad a)X = aXa^{-1}, \quad X\in T_e(G) = \g\subset \textrm{End}(V).$$
  
The adjoint representation of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088020.png" /> is the linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088021.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088022.png" /> into the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088023.png" /> acting by the formula
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The kernel $\ker \Ad$ contains the centre of $G$, and if $G$ is connected and if the ground field has characteristic zero, it coincides with this centre. The differential of the adjoint representation of $\def\ad{\textrm{ad}\;} G$ at $e$ coincides with the adjoint representation $\ad$ of $\g$, defined as follows:
  
<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/a/a010/a010880/a01088024.png" /></td> </tr></table>
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The adjoint representation of a Lie algebra $\g$ is the linear representation $\ad$ of the algebra $\g$ into the module $\g$ acting by the formula
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088025.png" /> is the bracket operation in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088026.png" />. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088027.png" /> is the centre of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088028.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088029.png" /> are derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088030.png" /> and are called inner derivations. The image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088031.png" /> is called the adjoint linear Lie algebra and is an ideal in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088032.png" /> of all derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088033.png" />, moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088034.png" /> is the one-dimensional cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088036.png" />, defined by the adjoint representation. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010880/a01088038.png" /> is a semi-simple Lie algebra over a field of characteristic zero.
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$$(\ad x)y = [x,y],\quad x,y\in \g,$$
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where $[\;,\;]$ is the bracket operation in the algebra $\g$. The kernel $\ker \ad$ is the centre of the Lie algebra $\g$. The operators $\ad x$ are derivations of $\g$ and are called inner derivations. The image $\ad \g$ is called the adjoint linear Lie algebra and is an ideal in the Lie algebra $\def\Der{\textrm{Der}\;}\Der \g$ of all derivations of $\g$, moreover $\Der \g/\ad\g$ is the one-dimensional cohomology space $H^1(\g,\g)$ of $\g$, defined by the adjoint representation. In particular, $\ad \g = \Der\g$ if $\g$ is a semi-simple Lie algebra over a field of characteristic zero.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson,  "Lie algebras" , Interscience (1962)  ((also: Dover, reprint, 1979))  {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press (1958)  (Translated from Russian)  {{MR|0201557}} {{ZBL|0882.01025}} {{ZBL|0534.22001}} {{ZBL|0079.03903}} {{ZBL|0058.26003}} {{ZBL|0022.17104}} {{ZBL|0016.20305}} {{ZBL|0016.20304}} {{ZBL|0015.24901}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin (1965)  (Translated from French)  {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.E. Humphreys,  "Linear algebraic groups" , Springer (1975)  {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>
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{|
 
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|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French)  {{MR|0682756}} {{ZBL|0319.17002}}
 
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====Comments====
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|valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys,  "Linear algebraic groups", Springer (1975)  {{MR|0396773}} {{ZBL|0325.20039}}
 
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|valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson,  "Lie algebras", Interscience (1962)  ((also: Dover, reprint, 1979))  {{MR|0143793}} {{ZBL|0121.27504}}  
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French)  {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR></table>
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|valign="top"|{{Ref|Po}}||valign="top"| L.S. Pontryagin,  "Topological groups", Princeton Univ. Press (1958(Translated from Russian)    {{MR|0201557}} {{ZBL|0022.17104}}
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|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre,  "Lie algebras and Lie groups", Benjamin (1965)  (Translated from French)  {{MR|0218496}} {{ZBL|0132.27803}}
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Latest revision as of 16:57, 27 March 2012

2020 Mathematics Subject Classification: Primary: 17-XX Secondary: 22-XX [MSN][ZBL]

or algebraic group $G$

The adjoint representation of a Lie group $G$ is the linear representation $\def\Ad{\textrm{Ad}\;} \Ad $ of $G$ in the tangent space $T_e(G)$ (or in the Lie algebra $\def\g{\mathfrak{g}}\g$ of $G$), given by the mapping which sends each $a\in G$ to the differential $\def\Int{\textrm{Int}\;}\Ad a = d(\Int a)_e$ of the inner automorphism $\Int a: x\mapsto axa^{-1}$. If $G\subseteq \def\GL{\textrm{GL}} \GL(V)$ is a linear group of isomorphisms of a vector space $V$, then

$$(\Ad a)X = aXa^{-1}, \quad X\in T_e(G) = \g\subset \textrm{End}(V).$$

The kernel $\ker \Ad$ contains the centre of $G$, and if $G$ is connected and if the ground field has characteristic zero, it coincides with this centre. The differential of the adjoint representation of $\def\ad{\textrm{ad}\;} G$ at $e$ coincides with the adjoint representation $\ad$ of $\g$, defined as follows:

The adjoint representation of a Lie algebra $\g$ is the linear representation $\ad$ of the algebra $\g$ into the module $\g$ acting by the formula

$$(\ad x)y = [x,y],\quad x,y\in \g,$$ where $[\;,\;]$ is the bracket operation in the algebra $\g$. The kernel $\ker \ad$ is the centre of the Lie algebra $\g$. The operators $\ad x$ are derivations of $\g$ and are called inner derivations. The image $\ad \g$ is called the adjoint linear Lie algebra and is an ideal in the Lie algebra $\def\Der{\textrm{Der}\;}\Der \g$ of all derivations of $\g$, moreover $\Der \g/\ad\g$ is the one-dimensional cohomology space $H^1(\g,\g)$ of $\g$, defined by the adjoint representation. In particular, $\ad \g = \Der\g$ if $\g$ is a semi-simple Lie algebra over a field of characteristic zero.

References

[Bo] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French) MR0682756 Zbl 0319.17002
[Hu] J.E. Humphreys, "Linear algebraic groups", Springer (1975) MR0396773 Zbl 0325.20039
[Ja] N. Jacobson, "Lie algebras", Interscience (1962) ((also: Dover, reprint, 1979)) MR0143793 Zbl 0121.27504
[Po] L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[Se] J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
How to Cite This Entry:
Adjoint representation of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_representation_of_a_Lie_group&oldid=21787
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article