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
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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:
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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
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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) |
[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) |
[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 |
[a4] | F. Warner, "Foundations of differentiable manifolds and Lie groups" , Springer (1983) |
[a5] | G. Warner, "Harmonic analysis on semisimple Lie groups I" , Springer (1972) |
Adjoint action. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_action&oldid=24361