Adjoint group

This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.

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

	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French) MR0682756 Template:ZBL
	J.E. Humphreys, "Linear algebraic groups", Springer (1975) MR0396773 Template:ZBL
	L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Template:ZBL
	J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French) MR0218496 Template:ZBL