# Difference between revisions of "Adjoint group"

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− | The linear group | + | 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|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==== | ||

− | + | {| | |

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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}} | |

− | + | |- | |

− | + | |valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys, "Linear algebraic groups", Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} | |

− | + | |- | |

− | + | |valign="top"|{{Ref|Po}}||valign="top"| L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} | |

− | + | |- | |

− | + | |valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} | |

+ | |- | ||

+ | |} |

## 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