Adjoint action

of a Lie group

The linear action on the Lie algebra $\frak g$ of the Lie group $G$, denoted by $\operatorname{Ad} : G \rightarrow \operatorname{GL} (\frak g )$, that is defined as follows: Each element $g$ of $G$ induces an inner automorphism of the Lie group $G$ by the formula $\operatorname { Int } ( g ) : G \rightarrow G$, $x \mapsto \operatorname { gxg } ^ { - 1 }$. Its differential, $\operatorname {Ad}( g ) : \mathfrak { g } \rightarrow \mathfrak { g }$, gives an automorphism of the Lie algebra $\frak g$. The resulting linear representation $\operatorname{Ad} : G \rightarrow \operatorname{GL} (\frak g )$ is called the adjoint representation of $G$ (cf. also Adjoint representation of a Lie group).

The kernel of the adjoint representation, $\operatorname { Ker } ( \operatorname{Ad} )$, contains the centre $Z _ { G }$ of $G$ (cf. also Centre of a group), and coincides with $Z _ { G }$ if $G$ is connected. The image $\text{Ad}( G )$ is called the adjoint group; it is a Lie subgroup of $\operatorname{Aut} ( \mathfrak{g} )$, the group of all automorphisms of the Lie algebra $\frak g$.

The differential of the adjoint representation $\operatorname{Ad} : G \rightarrow \operatorname{GL} (\frak g )$ gives rise to a linear representation $\operatorname {ad} : \mathfrak { g } \rightarrow \operatorname { End } ( \mathfrak { g } )$ of the Lie algebra $\frak g$ (cf. also Representation of a Lie algebra). It is given by the formula

\begin{equation*} ( \operatorname{ad} X ) ( Y ) = [ X , Y ] , X , Y \in \mathfrak { g }, \end{equation*}

and is called the adjoint representation of $\frak g$ (cf. also Adjoint representation of a Lie group). The kernel, $\operatorname { Ker } ( \text { ad } )$, coincides with the centre of the Lie algebra $\frak g$. The image, $\operatorname{ad}({\frak g} )$, forms a subalgebra of $\operatorname { Der } ( \mathfrak { g } )$, the Lie algebra of all derivations of $\frak g$ (cf. Derivation in a ring). If $\frak g$ is a semi-simple Lie algebra (see Lie algebra, semi-simple), then $\operatorname { Ker } ( \operatorname{ad} ) = \{ 0 \}$ and $\operatorname{ad}( \mathfrak{g} ) = \operatorname { Der } (\mathfrak{g} )$. An opposite extremal case is when $G$ is Abelian; in this case $\operatorname { Ker } ( \text { ad } ) = \mathfrak { g }$ and $\operatorname {ad} ( \mathfrak{g} ) = \{ 0 \}$.

If $G \subset \operatorname { GL } ( V )$ is a linear group acting on a vector space $V$, then one can regard $\mathfrak{g} \subset \text { End } ( V )$ and the adjoint representation can be written in terms of matrix computation:

\begin{equation*} \text{Ad}( g ) Y = g Y g ^ { - 1 } , ( \text { ad } X ) Y = X Y - Y X, \end{equation*}

\begin{equation*} g \in G , X , Y \in \mathfrak { g }. \end{equation*}

The adjoint orbit through $X \in \mathfrak g $ (see Orbit) is defined to be $\operatorname { Ad } ( G ) X = \{ \operatorname { Ad } ( g ) X : g \in G \}$; it is a submanifold of $\frak g$.

Adjoint orbits for reductive Lie groups $G$ have been particularly studied. The adjoint orbit $\operatorname{Ad}( G ) X$ is called a semi-simple orbit (respectively, a nilpotent orbit) if $\operatorname { ad } X$ is a semi-simple (respectively, nilpotent) endomorphism of $\frak g$. The set

\begin{equation*} \mathcal{N} = \{ X \in \mathfrak { g } : \operatorname{ad}X \, \text{is a nilpotent endomorphism of} \, \mathfrak { g } \} \end{equation*}

is an algebraic variety in $\frak g$, called the nilpotent variety. It is the union of a finite number of nilpotent orbits. On the other hand, $\operatorname{Ad}( G ) X$ is a closed set in $\frak g$ if and only if it is a semi-simple orbit. A semi-simple orbit $\operatorname{Ad}( G ) X$ is called an elliptic orbit (respectively, a hyperbolic orbit) if all eigenvalues of $\operatorname { ad } X$ are purely imaginary (respectively, real). Any elliptic orbit carries a $G$-invariant complex structure, while any hyperbolic obit carries a $G$-invariant paracomplex structure.

If $G$ is compact, then all adjoint orbits are elliptic. For example, if $G = U ( n )$ (a unitary group), then each adjoint orbit is biholomorphic to a generalized flag variety $U ( n ) / ( U ( n _ { 1 } ) \times \ldots \times U ( n _ { k } ) )$, for some partition $( n _ { 1 } , \dots , n _ { k } )$ of $n$, and vice versa (cf. also Flag space; Algebraic variety).

One writes $\mathfrak { g } / \operatorname{Ad}$ for the set of all adjoint orbits. The classical theory of the Jordan normal form of matrices (as well as the theory of other normal forms of matrices; cf. also Normal form) can be interpreted as the classification of $\mathfrak { g } / \operatorname{Ad}$, for $G = \operatorname{GL} ( n , \mathbf{C} )$. If $G$ is a connected compact Lie group, then $\mathfrak { g } / \operatorname{Ad}$ is bijective to $\mathfrak { a } / W$, the set of all orbits in $\frak a$ of the Weyl group $W$, where $\frak a$ is a maximal Abelian subspace of $\frak g$. This reduction is important in the Cartan–Weyl theory of the classification of irreducible representations, as well as their characters, for a compact Lie group (cf. also Lie group, compact; Irreducible representation).

Bi-invariant tensors on a Lie group $G$ can be described in terms of invariants under the adjoint action. For example, the left-invariant measure on a connected Lie group $G$ is also right invariant if and only if $\operatorname{det} \; \operatorname { Ad } ( g ) = 1$ for any $g \in G$. Such a Lie group is called unimodular (cf. Haar measure). This is the case if $G$ is nilpotent or reductive.

Let $\mathfrak{g} ^ { * }$ be the dual vector space of $\frak g$. The contragredient representation $\operatorname{Ad} ^ { * } : G \rightarrow \operatorname{GL} ( \mathfrak{g} ^ { * } )$ of the adjoint representation $( \operatorname{Ad} , \mathfrak{g} )$ is called the co-adjoint representation. There is a close connection between irreducible unitary representations of $G$ and co-adjoint orbits (see Orbit method).

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