On the Casimir ${\cal{W}\cal{A}}_{\it{N}}$ algebras as the truncated $\cal{W}_{\infty}$ algebra

Abstract: The complete structure of the Casimir ${\cal{W}\cal{A}}{\it{N}}$ algebras are shown to exist in such a way that the Casimir ${\cal{W}\cal{A}}{\it{N}}$ algebra is a kind of truncated type of $\cal{W}{\infty}$ algebra both in the primary and in the quadratic basis, first using the associativity conditions in the basis of primary fields and second using the Miura basis coming from the free field realization as a different basis. Finally one can say that the Casimir ${\cal{W}\cal{A}}{\it{N}}$ algebra is a kind of truncated type of $\cal{W}{\infty}$algebra,so it is clear from any construction of $\cal{W}{\infty}$ algebra that by putting infinite number of fields $W_{s}$ with $s>N$ to zero we arrive at the Casimir ${\cal{W}\cal{A}}_{\it{N}}$ algebra.