Adjoint group

2020 Mathematics Subject Classification: Primary: 20GXX Secondary: 14LXX [MSN][ZBL]

The adjoint group of a linear group $G$ is the linear group $\def\Ad{\mathop{\textrm{Ad}}} \Ad G$ 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{\mathop{\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