View source for Adjoint action

''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]]).

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" />.

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

<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>

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" />.

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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<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>

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" />.

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

<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>

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.

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]]).

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]]).

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.

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]]).

====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) {{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>