QUANTUM AND CLASSICAL LIE SYSTEMS FOR EX EXTENDED SYMPLECTIC GROUPS

In the framework of Lie systems, we study the affine symplectic group G(AS) and the Jacobi group G(J). We construct canonical bases for the irreducible unitary representations of G(J) consisting of K-J-vectors, where K-J is the maximal compact subgroup of G(J). We study the quantum Lie systems based on the extended Poincare disk M diffeomorphic to the maximal elliptic coadjoint orbit of G(J). We establish the quasienergy operator reduced to M and the corresponding Wei-Norman equations. Moreover, we obtain the solutions of the time-dependent Schrodinger equation with the initial states represented by K-J-vectors. Finally, we obtain a Poisson algebra isomorphism between the quantum and classical Lie systems based on the Poisson manifold M.