Casimir WA_N algebras as the truncated W_∞ algebra

The complete structure of the Casimir WA_N algebras is shown to exist in such a way that the Casimir WA_N algebra is a kind of truncated type of ∞ 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. We can conclude that the Casimir WA_N algebra is a kind of truncated type of W_∞ algebra, so it is clear from any construction of W_∞ algebra that by putting infinite number of fields Ws with W_s > N to zero, we arrive at the Casimir WA_N algebra. We concentrated in this work only for the particular case of 5 algebra since this example gives us explicitly a method on how to deal with the general case N.