On the representation of $A_{\kk}(2)$ algebra and $A_{\kk}(d)$ algebra

TL;DR

This paper investigates the representation of the $A_{\ ext{\kappa}}(2)$ algebra, introduces necessary new generators for consistency, extends it to multi-mode $A_{\kappa}(d)$ algebra, and examines the energy spectrum's positivity conditions.

Contribution

It introduces a consistent representation of $A_{\kappa}(2)$, extends it to $A_{\kappa}(d)$ algebra, and analyzes the energy spectrum positivity.

Findings

New generators are necessary for algebra consistency.

Multi-mode $A_{\kappa}(d)$ algebra is constructed.

Positivity conditions for the energy spectrum are analyzed.

Abstract

In this paper the representation of $A_{\kk}(2)$ algebra given by Daoud and Kibler [M.Daoud and M.Kibler, J.Phys.A 43 115303 (2010), 45 244036 (2012)] is investigated. It is shown that the new generators are necessary for consistency of the algebra. The multi-mode extension of $A_{\kk}(2)$ algebra, which is called $A_{\kk}(d)$ algebra, is also obtained. The positivity condition of the energy spectrum is also investigated for $A_{\kk}(2)$ algebra and $A_{\kk}(d)$ algebra.