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 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.

Formal definition

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Let   be a Lie group and   be its Lie algebra. Let   denote the adjoint representation of  . Then the coadjoint representation   is defined by

  for  

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  

where   is the adjoint representation of the Lie algebra  .

Coadjoint orbit

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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

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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

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References

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  • Kirillov, A.A., Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Society, ISBN 0821835300, ISBN 978-0821835302
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