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''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108301.png" />''
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The linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108302.png" /> that is the image of the Lie group or algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108303.png" /> under the adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The adjoint group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108304.png" /> is contained in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108305.png" /> of automorphisms of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108306.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108307.png" />, and its Lie algebra coincides with the adjoint algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108308.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a0108309.png" />. A connected semi-simple group is a group of adjoint type (i.e. is isomorphic to its adjoint group) if and only if its roots generate the lattice of rational characters of the maximal torus; the centre of such a group is trivial. If the ground field has characteristic zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083010.png" /> is connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083011.png" /> is uniquely determined by the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083012.png" /> and is either called the adjoint group or the group of inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083013.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083014.png" /> is semi-simple, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083015.png" /> coincides with the connected component of the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010830/a01083016.png" />.
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The adjoint group of a linear group $G$'' is
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the linear group $\def\Ad{\textrm{Ad}\;} \Ad G$  which is the image of the Lie group or algebraic group $G$ under the adjoint representation (cf.
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[[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The adjoint group $\Ad G$ is contained in the group $\def\Aut{\textrm{Aut}\;} \def\g{\mathfrak g} \Aut \g $ of automorphisms of the Lie algebra $\g$ of $G$, and its Lie algebra coincides with the adjoint algebra $\Ad\g$ of $\g$. A connected semi-simple group is a group of adjoint type (i.e. is isomorphic to its adjoint group) if and only if its roots generate the lattice of rational characters of the maximal torus; the centre of such a group is trivial. If the ground field has characteristic zero and $G$ is connected, then $\Ad G$ is uniquely determined by the Lie algebra $\g$ and is either called the adjoint group or the group of inner automorphisms of $\g$. In particular, if $G$ is semi-simple, $\Ad G$ coincides with the connected component of the identity in $\Aut \g$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press (1958)  (Translated from Russian)   {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top"> J.E. Humphreys,  "Linear algebraic groups" , Springer (1975)  {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>
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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|Po}}||valign="top"| L.S. Pontryagin,  "Topological groups", Princeton Univ. Press (1958(Translated from Russian)    {{MR|0201557}} {{ZBL|0022.17104}}
====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|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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Revision as of 13:08, 26 March 2012

2010 Mathematics Subject Classification: Primary: 20XXX Secondary: 14LXX [MSN][ZBL]

The adjoint group of a linear group $G$ is the linear group $\def\Ad{\textrm{Ad}\;} \Ad G$ which is the image of the Lie group or algebraic group $G$ under the adjoint representation (cf. Adjoint representation of a Lie group). The adjoint group $\Ad G$ is contained in the group $\def\Aut{\textrm{Aut}\;} \def\g{\mathfrak g} \Aut \g $ of automorphisms of the Lie algebra $\g$ of $G$, and its Lie algebra coincides with the adjoint algebra $\Ad\g$ of $\g$. A connected semi-simple group is a group of adjoint type (i.e. is isomorphic to its adjoint group) if and only if its roots generate the lattice of rational characters of the maximal torus; the centre of such a group is trivial. If the ground field has characteristic zero and $G$ is connected, then $\Ad G$ is uniquely determined by the Lie algebra $\g$ and is either called the adjoint group or the group of inner automorphisms of $\g$. In particular, if $G$ is semi-simple, $\Ad G$ coincides with the connected component of the identity in $\Aut \g$.

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
[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 group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_group&oldid=21800
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article