Coadjoint representation

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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 orbit. 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.

Formal definition[edit]

Let be a Lie group and be its Lie algebra. Let denote the adjoint representation of . Then the coadjoint representation is defined as . More explicitly,

for

where denotes the value of a linear functional on a vector .

Let denote the representation of the Lie algebra on induced by the coadjoint representation of the Lie group . Then for where is the adjoint representation of the Lie algebra . One may make this observation from the infinitesimal version of the defining equation for above, which is as follows :

for . .

Coadjoint orbit[edit]

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. Let be an antisymmetric bilinear form on defined by,

Then one may define by

.

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 is exactly .

(iii) is invariant under .

is also closed. The canonical 2-form is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form on the coadjoint orbit.

Properties of coadjoint orbits[edit]

The coadjoint action on a coadjoint orbit is a Hamiltonian -action with moment map given by .

Examples[edit]

See also[edit]

References[edit]

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