A Holomorphic Representation of the Semidirect Sum of Symplectic and Heisenberg Lie Algebras

Abstract

Abstract.A representation of the Jacobi algebra by first order differential operators with polynomial coefficients on a Kähler manifold which as set is the product of the complex multidimensional plane times the Siegel ball is presented.

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

“…In [12] we have underlined that the homogeneous metric corresponding to ω D J 1 is a balanced metric [1,31]. In the present paper it is emphasized that the metric corresponding to ω D J n [6,7,9] is the balanced metric. We also point out several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.…”

“…In [6,7,9] we have attached coherent states (CS) [64] to the Jacobi group G J n with support on the Siegel-Jacobi ball D J n . The particular case of coherent states attached to the Jacobi group G J 1 defined on the Siegel-Jacobi disk D J 1 has been investigated in [5,10].…”

On Reducible Verma Modules over Jacobi Algebra

“…H n is an ideal in G n , i.e., [H n , G n ] = H n , determined by the commutation relations (following the notation of [6]):…”

On the Geometry of Siegel–jacobi Domains

“…Remark 4.3 In the case n = 1 and 8πm = 1, the expansion (4.21) was obtained in [3], using the coherent state method.…”

“…In the case n = 1 and 8πm = 1, the expansion (4.21) was obtained in[3], using the coherent state method.Remark 4.4 We now discuss the unitary representations of Jacobi groups based on Siegel-Jacobi domains in the language of coherent states [18]. Let Q(H) be the set of all one-dimensional projections of the Hilbert space H. Let P [ψ] denote the one-dimensional projection determined by ψ ∈ H\{0}.…”