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

In this paper the representation of $A_{\text{kk}}(2)$ 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 $A_{\text{kk}}(2)$ algebra, which is called $A_{\text{kk}}(d)$ algebra, is also obtained. The positivity condition of the energy spectrum is also investigated for $A_{\text{kk}}(2)$ algebra and $A_{\text{kk}}(d)$ algebra.