A HOLOMORPHIC REPRESENTATION OF LIE ALGEBRAS SEMIDIRECT SUM OF SEMISIMPLE AND HEISENBERG ALGEBRAS

摘要

The algebra semidirect sum of the real three-dimensional Heisenberg algebra and the su(1, 1) algebra admits a realization by first order differential operators with polynomial coefficients. In order to construct such a representation we use coherent state vectors based on the Kaehler manifold which as set is the product of the complex plane and the unit disk. We present the Hilbert space of holomorphic functions on which the differential operators act: the reproducing kernel, the group invariant measure, the base of orthonormal polynomials.