标题： 关于卡西米尔${\cal{W}\cal{A}}_{\it{N}}$代数作为截断的$\cal{W}_{\infty}$代数 显示英文标题

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

摘要： 卡西米尔${\cal{W}\cal{A}}_{\it{N}}$代数的完整结构被证明以这样的方式存在，即卡西米尔${\cal{W}\cal{A}}_{\it{N}}$代数在主基和二次基下都是$\cal{W}_{\infty}$代数的一种截断类型，首先使用主场基中的结合性条件，其次使用来自自由场实现的米乌拉基作为不同的基。 最后可以说，Casimir ${\cal{W}\cal{A}}_{\it{N}}$ 代数是一种截断类型的 $\cal{W}_{\infty}$ 代数，因此从任何 $\cal{W}_{\infty}$ 代数的构造中可以看出，通过将无限数量的场 $W_{s}$ 和 $s>N$ 置为零，我们得到 Casimir ${\cal{W}\cal{A}}_{\it{N}}$ 代数。 显示英文摘要

摘要： 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.