In mathematics, the coadjoint representation
of a Lie group
is the
adjoint representation
. If
![{\displaystyle {\mathfrak {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
denotes the
Lie algebra of
![{\displaystyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
, the corresponding action of
![{\displaystyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
on
![{\displaystyle {\mathfrak {g}}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0acfc869f2ed2da044f5c582042afd3065e12bdb)
, the
dual space to
![{\displaystyle {\mathfrak {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
, is called the
coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant
1-forms
on
![{\displaystyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
.
The importance of the coadjoint representation was emphasised by work of
nilpotent Lie groups
![{\displaystyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
a basic role in their
representation theory is played by
coadjoint orbits.
In the Kirillov method of orbits, representations of
![{\displaystyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the
conjugacy classes of
![{\displaystyle G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
, which again may be complicated, while the orbits are relatively tractable.
Formal definition
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
![{\displaystyle {\mathfrak {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
.
Coadjoint orbit
A coadjoint orbit
for
in the dual space
of
may be defined either extrinsically, as the actual
orbit
![{\displaystyle \mathrm {Ad} _{G}^{*}\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b785018570eab7864550e8cf529b9be3be509cbb)
inside
![{\displaystyle {\mathfrak {g}}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0acfc869f2ed2da044f5c582042afd3065e12bdb)
, or intrinsically as the
homogeneous space ![{\displaystyle G/G_{\mu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc69378dd5beaf782c304147fe288f43f5f7933)
where
![{\displaystyle G_{\mu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4660b66db74aba7d5956e7505df87bfbf8f2981)
is the
stabilizer
of
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
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
![{\displaystyle \omega \in \Omega ^{2}({\mathcal {O}}_{\mu })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26743720d659cd39e2fd0cbd82d603cbb53a171a)
inherited from
![{\displaystyle {\mathfrak {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
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
2-form
![{\displaystyle \omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8)
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
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See also
References
External links