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

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