Adjoint action

of a Lie group

The linear action on the Lie algebra of the Lie group , denoted by , that is defined as follows: Each element of induces an inner automorphism of the Lie group by the formula , . Its differential, , gives an automorphism of the Lie algebra . The resulting linear representation is called the adjoint representation of (cf. also Adjoint representation of a Lie group).

The kernel of the adjoint representation, , contains the centre of (cf. also Centre of a group), and coincides with if is connected. The image is called the adjoint group; it is a Lie subgroup of , the group of all automorphisms of the Lie algebra .

The differential of the adjoint representation gives rise to a linear representation of the Lie algebra (cf. also Representation of a Lie algebra). It is given by the formula

and is called the adjoint representation of (cf. also Adjoint representation of a Lie group). The kernel, , coincides with the centre of the Lie algebra . The image, , forms a subalgebra of , the Lie algebra of all derivations of (cf. Derivation in a ring). If is a semi-simple Lie algebra (see Lie algebra, semi-simple), then and . An opposite extremal case is when is Abelian; in this case and .

If is a linear group acting on a vector space , then one can regard and the adjoint representation can be written in terms of matrix computation:

The adjoint orbit through (see Orbit) is defined to be ; it is a submanifold of .

Adjoint orbits for reductive Lie groups have been particularly studied. The adjoint orbit is called a semi-simple orbit (respectively, a nilpotent orbit) if is a semi-simple (respectively, nilpotent) endomorphism of . The set

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

If is compact, then all adjoint orbits are elliptic. For example, if (a unitary group), then each adjoint orbit is biholomorphic to a generalized flag variety , for some partition of , and vice versa (cf. also Flag space; Algebraic variety).

One writes 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 , for . If is a connected compact Lie group, then is bijective to , the set of all orbits in of the Weyl group , where is a maximal Abelian subspace of . 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 can be described in terms of invariants under the adjoint action. For example, the left-invariant measure on a connected Lie group is also right invariant if and only if for any . Such a Lie group is called unimodular (cf. Haar measure). This is the case if is nilpotent or reductive.

Let be the dual vector space of . The contragredient representation of the adjoint representation is called the co-adjoint representation. There is a close connection between irreducible unitary representations of 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)