Linear Hamiltonians on homogeneous K\"ahler manifolds of coherent states

Abstract

Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equations of motion with holomorphic polynomials coefficients. For hermitian symmetric manifolds the equations of motion are matrix Riccati equations. It is presented the simplest example of the non-symmetric space $SU(3)/S(U(1)\times U(1)\times U(1))$ where the polynomials describing the equations of motion have the maximum degree 3.