Realization of coherent state Lie algebras by differential operators

TL;DR

This paper presents a method to realize coherent state Lie algebras using first-order differential operators with polynomial coefficients on Kähler orbits, providing explicit formulas involving Bernoulli numbers and structure constants.

Contribution

It introduces a novel realization of coherent state Lie algebras through differential operators with explicit formulas, advancing the mathematical understanding of Lie algebra representations.

Findings

Explicit formulas involving Bernoulli numbers are derived.

Realization applies to semisimple Lie groups on Kähler orbits.

The approach enhances the mathematical framework for coherent states.

Abstract

A realization of coherent state Lie algebras by first-order differential operators with holomorphic polynomial coefficients on K\"ahler coherent state orbits is presented. Explicit formulas involving the Bernoulli numbers and the structure constants for the semisimple Lie groups are proved.