Casimir 𝒲𝒜N algebras as the truncated 𝒲∞ algebra

Abstract

The complete structure of the Casimir [Formula: see text] algebras is shown to exist in such a way that the Casimir [Formula: see text] algebra is a kind of truncated type of [Formula: see text] 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 [Formula: see text] algebra is a kind of truncated type of [Formula: see text] algebra, so it is clear from any construction of [Formula: see text] algebra that by putting infinite number of fields [Formula: see text] with [Formula: see text] to zero, we arrive at the Casimir [Formula: see text] algebra. We concentrated in this work only for the particular case of [Formula: see text] algebra since this example gives us explicitly a method on how to deal with the general case [Formula: see text].