In mathematics, the coadjoint representation
of a Lie group
is the dual of the adjoint representation. If
denotes the Lie algebra of
, the corresponding action of
on
, the dual space to
, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on
.
The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups
a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of
are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of
, which again may be complicated, while the orbits are relatively tractable.
Let
be a Lie group and
be its Lie algebra. Let
denote the adjoint representation of
. Then the coadjoint representation
is defined by
for ![{\displaystyle g\in G,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8f1db98bd258bd88891d6d7526943f25b3e9c8)
where
denotes the value of the linear functional
on the vector
.
Let
denote the representation of the Lie algebra
on
induced by the coadjoint representation of the Lie group
. Then the infinitesimal version of the defining equation for
reads:
for ![{\displaystyle X,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02822e78fbd33e840391494e864fa34509d102c0)
where
is the adjoint representation of the Lie algebra
.
Coadjoint orbit
A coadjoint orbit
for
in the dual space
of
may be defined either extrinsically, as the actual orbit
inside
, or intrinsically as the homogeneous space
where
is the stabilizer of
with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated.
The coadjoint orbits are submanifolds of
and carry a natural symplectic structure. On each orbit
, there is a closed non-degenerate
-invariant 2-form
inherited from
in the following manner:
.
The well-definedness, non-degeneracy, and
-invariance of
follow from the following facts:
(i) The tangent space
may be identified with
, where
is the Lie algebra of
.
(ii) The kernel of the map
is exactly
.
(iii) The bilinear form
on
is invariant under
.
is also closed. The canonical 2-form
is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.
Properties of coadjoint orbits
The coadjoint action on a coadjoint orbit
is a Hamiltonian
-action with momentum map given by the inclusion
.
Examples
See also
References
External links