标题： $W_{1+\infty}$和$\widetilde W$代数，以及 Ward 恒等式 显示英文标题

标题： $W_{1+\infty}$ and $\widetilde W$ algebras, and Ward identities

摘要： 最近证明了 $W_{1+\infty}$ 代数包含了与所有整数斜率的射线（包括垂直射线）相关的交换子代数。 本文指出，此类射线中的每个元素都与广义 $\widetilde W$ 代数相关联。 特别是，与有理Calogero哈密顿量相关的最简单的交换子代数与之前研究过的 $\widetilde W$ 代数相关。 我们基于 $W_{1+\infty}$ 代数的矩阵实现，建议将广义 $\widetilde W$ 代数定义为变量 $p_k$ 上的微分算子，并且还提出了一种无歧义的递归定义，但该定义涉及比其交换子代数所含更多的 $W_{1+\infty}$ 代数元素。 正整数射线与形成WLZZ矩阵模型的Ward恒等式集的$\widetilde W$代数相关联，而与三角Calogero-Sutherland模型相关的垂直射线描述了对应于完整循环的超几何$\tau$函数。 显示英文摘要

摘要： It was demonstrated recently that the $W_{1+\infty}$ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized $\widetilde W$ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the $\widetilde W$ algebras studied earlier. We suggest a definition of the generalized $\widetilde W$ algebra as differential operators in variables $p_k$ basing on the matrix realization of the $W_{1+\infty}$ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the $W_{1+\infty}$ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with $\widetilde W$ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric $\tau$-functions corresponding to the completed cycles.