Quantum Co-Adjoint Orbits of the Real Diamond Group

TL;DR

This paper provides explicit formulas for deformation quantization on co-adjoint orbits of the real diamond group, leading to new quantum geometric structures and their associated unitary representations.

Contribution

It introduces explicit Fedosov deformation quantization formulas for the real diamond group's co-adjoint orbits, connecting geometric quantization with representation theory.

Findings

Explicit formulas for quantum half-plans and hyperbolic cylinders

Construction of quantum hyperbolic paraboloids

Derivation of unitary representations from quantized orbits

Abstract

We present explicit formulas for deformation quantization on the co-adjoint orbits of the real diamond Lie group. From this we obtain quantum half-plans, quantum hyperbolic cylinders, quantum hyperbolic paraboloids via Fedosov deformation quantization and finally, the corresponding unitary representations of this group.