# Difference between revisions of "Adjoint 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). 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 $\Aut \g$.