Coadjoint representation

Adjoint and Coadjoint Representations, Semisimplicity, the System of Roots... 

We shall now deal with the orbits of coadjoint representation. On these orbits, there is a natural symplectic structure and any homogeneous symplectic manifold may be realized in the from of the orbit of the representation Ad in a certain Lie algebra. 

Hence, the orbits are organised as shown in Fig. 10. Let us now investigate the orbits of coadjoint representation. In the algebra G, we choose the basis... 

If on the algebra G there exists a symmetric nondegenerate scalar product (X, y) such that (Adj,X,Ad y) = (X, y), then the adjoint and coadjoint representations are equivalent, that is, they have, in particular, identical orbits. 

It is known that on the orbits of coadjoint representation of any Lie group, there exists a natural symplectic structure (the structure of Kirillov) invariant with respect to the coadjoint representation. 

In particular, the dimension of each orbit of coadjoint representation is even. It is easily seen that the symplectic form oj is invariant under the coadjoint action that is, Ad a = a , h G 0. It also turns out that du = 0. 

Besides the energy integral Ii = H Eqs. (3) always possess the integrals I2 = = P which generate the annihilator of the Poisson bracket and fix the orbit of the coadjoint representation = iWo, P = Iq. On this orbit we have a... 

Theorem 4.4.8 (Bolsinov). The system of Euler equations (1) is completely Liouville integrable on orbits of general position of the coadjoint representation Ad (n(G) . 

Lebedev, D. R., and Manin, Yu. I. The Hamiltonian operator of GePfand-Dikii and coadjoint representation of the Volterra group. Funkts, Analiz i yego Prilozhen, 18 (1980), No. 4, 40-46. 

Belyaev, A. V. "On the motion of an n-dimensional rigid body with the group of symmetries SO (A ) SO (AT — /) in a field with linear potential. Invariants of coadjoint representation of some Lie algebras. Dokl. Akad. Nauk SSSR, 282 (1985) No. 5, 1038-1042. 

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