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''of a Lie group''
 
''of a Lie group''
  
The linear action on the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201501.png" /> of the [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201502.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201503.png" />, that is defined as follows: Each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201504.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201505.png" /> induces an inner [[Automorphism|automorphism]] of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201506.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201508.png" />. Its differential, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a1201509.png" />, gives an automorphism of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015010.png" />. The resulting linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015011.png" /> is called the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015012.png" /> (cf. also [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]).
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The linear action on the [[Lie algebra|Lie algebra]] $\frak g$ of the [[Lie group|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|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|Adjoint representation of a Lie group]]).
  
The kernel of the adjoint representation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015013.png" />, contains the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015015.png" /> (cf. also [[Centre of a group|Centre of a group]]), and coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015016.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015017.png" /> is connected. The image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015018.png" /> is called the [[Adjoint group|adjoint group]]; it is a Lie subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015019.png" />, the group of all automorphisms of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015020.png" />.
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The kernel of the adjoint representation, $\operatorname { Ker } ( \operatorname{Ad} )$, contains the centre $Z _ { G }$ of $G$ (cf. also [[Centre of a group|Centre of a group]]), and coincides with $Z _ { G }$ if $G$ is connected. The image $\text{Ad}( G )$ is called the [[Adjoint group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015021.png" /> gives rise to a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015022.png" /> of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015023.png" /> (cf. also [[Representation of a Lie algebra|Representation of a Lie algebra]]). It is given by the formula
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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|Representation of a Lie algebra]]). It is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015024.png" /></td> </tr></table>
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\begin{equation*} ( \operatorname{ad} X ) ( Y ) = [ X , Y ] , X , Y \in \mathfrak { g }, \end{equation*}
  
and is called the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015025.png" /> (cf. also [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The kernel, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015026.png" />, coincides with the centre of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015027.png" />. The image, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015028.png" />, forms a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015029.png" />, the Lie algebra of all derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015030.png" /> (cf. [[Derivation in a ring|Derivation in a ring]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015031.png" /> is a semi-simple Lie algebra (see [[Lie algebra, semi-simple|Lie algebra, semi-simple]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015033.png" />. An opposite extremal case is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015034.png" /> is Abelian; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015036.png" />.
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and is called the adjoint representation of $\frak g$ (cf. also [[Adjoint representation of a Lie group|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|Derivation in a ring]]). If $\frak g$ is a semi-simple Lie algebra (see [[Lie algebra, semi-simple|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015037.png" /> is a [[Linear group|linear group]] acting on a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015038.png" />, then one can regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015039.png" /> and the adjoint representation can be written in terms of matrix computation:
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If $G \subset \operatorname { GL } ( V )$ is a [[Linear group|linear group]] acting on a [[Vector space|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015040.png" /></td> </tr></table>
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\begin{equation*} \text{Ad}( g ) Y = g Y g ^ { - 1 } , ( \text { ad } X ) Y = X Y - Y X, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015041.png" /></td> </tr></table>
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\begin{equation*} g \in G , X , Y \in \mathfrak { g }. \end{equation*}
  
The adjoint orbit through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015042.png" /> (see [[Orbit|Orbit]]) is defined to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015043.png" />; it is a submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015044.png" />.
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The adjoint orbit through $X \in \mathfrak g $ (see [[Orbit|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015045.png" /> have been particularly studied. The adjoint orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015046.png" /> is called a semi-simple orbit (respectively, a nilpotent orbit) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015047.png" /> is a semi-simple (respectively, nilpotent) [[Endomorphism|endomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015048.png" />. The set
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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|endomorphism]] of $\frak g$. The set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015049.png" /></td> </tr></table>
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\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|algebraic variety]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015050.png" />, called the nilpotent variety. It is the union of a finite number of nilpotent orbits. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015051.png" /> is a closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015052.png" /> if and only if it is a semi-simple orbit. A semi-simple orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015053.png" /> is called an elliptic orbit (respectively, a hyperbolic orbit) if all eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015054.png" /> are purely imaginary (respectively, real). Any elliptic orbit carries a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015056.png" />-invariant complex structure, while any hyperbolic obit carries a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015058.png" />-invariant paracomplex structure.
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is an [[Algebraic variety|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015059.png" /> is compact, then all adjoint orbits are elliptic. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015060.png" /> (a [[Unitary group|unitary group]]), then each adjoint orbit is biholomorphic to a generalized flag variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015061.png" />, for some partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015063.png" />, and vice versa (cf. also [[Flag space|Flag space]]; [[Algebraic variety|Algebraic variety]]).
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If $G$ is compact, then all adjoint orbits are elliptic. For example, if $G = U ( n )$ (a [[Unitary group|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|Flag space]]; [[Algebraic variety|Algebraic variety]]).
  
One writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015064.png" /> 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|Normal form]]) can be interpreted as the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015065.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015066.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015067.png" /> is a connected compact Lie group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015068.png" /> is bijective to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015069.png" />, the set of all orbits in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015070.png" /> of the [[Weyl group|Weyl group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015072.png" /> is a maximal Abelian subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015073.png" />. 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|Lie group, compact]]; [[Irreducible representation|Irreducible representation]]).
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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|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|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|Lie group, compact]]; [[Irreducible representation|Irreducible representation]]).
  
Bi-invariant tensors on a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015074.png" /> can be described in terms of invariants under the adjoint action. For example, the left-invariant measure on a connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015075.png" /> is also right invariant if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015076.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015077.png" />. Such a Lie group is called unimodular (cf. [[Haar measure|Haar measure]]). This is the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015078.png" /> is nilpotent or reductive.
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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|Haar measure]]). This is the case if $G$ is nilpotent or reductive.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015079.png" /> be the dual vector space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015080.png" />. The [[Contragredient representation|contragredient representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015081.png" /> of the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015082.png" /> is called the co-adjoint representation. There is a close connection between irreducible unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120150/a12015083.png" /> and co-adjoint orbits (see [[Orbit method|Orbit method]]).
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Let $\mathfrak{g} ^ { * }$ be the dual vector space of $\frak g$. The [[Contragredient representation|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|Orbit method]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics, Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chap. 2–3 (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> "Théorie des algèbres de Lie. Topologie des groupes de Lie" , ''Sém. Sophus Lie de l'Ecole Norm. Sup. 1954/55'' , Secr. Math. Paris (1955) (0)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> T. Kobayashi,   "Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory" , ''Transl. Ser. II'' , '''183''' , Amer. Math. Soc. (1998) pp. 1–31</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Warner,   "Foundations of differentiable manifolds and Lie groups" , Springer (1983)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. Warner,   "Harmonic analysis on semisimple Lie groups I" , Springer (1972)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> N. Bourbaki, "Elements of mathematics, Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chap. 2–3 (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> "Théorie des algèbres de Lie. Topologie des groupes de Lie" , ''Sém. Sophus Lie de l'Ecole Norm. Sup. 1954/55'' , Secr. Math. Paris (1955) (0) {{MR|}} {{ZBL|0068.02102}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> T. Kobayashi, "Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory" , ''Transl. Ser. II'' , '''183''' , Amer. Math. Soc. (1998) pp. 1–31 {{MR|1615135}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> F. Warner, "Foundations of differentiable manifolds and Lie groups" , Springer (1983) {{MR|0722297}} {{ZBL|0516.58001}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> G. Warner, "Harmonic analysis on semisimple Lie groups I" , Springer (1972)</td></tr></table>

Latest revision as of 16:59, 1 July 2020

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).

References

[a1] N. Bourbaki, "Elements of mathematics, Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chap. 2–3 (Translated from French) MR0682756 Zbl 0319.17002
[a2] "Théorie des algèbres de Lie. Topologie des groupes de Lie" , Sém. Sophus Lie de l'Ecole Norm. Sup. 1954/55 , Secr. Math. Paris (1955) (0) Zbl 0068.02102
[a3] T. Kobayashi, "Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory" , Transl. Ser. II , 183 , Amer. Math. Soc. (1998) pp. 1–31 MR1615135
[a4] F. Warner, "Foundations of differentiable manifolds and Lie groups" , Springer (1983) MR0722297 Zbl 0516.58001
[a5] G. Warner, "Harmonic analysis on semisimple Lie groups I" , Springer (1972)
How to Cite This Entry:
Adjoint action. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_action&oldid=11210
This article was adapted from an original article by Toshiyuki Kobayashi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article