On the $W$-algebra in the Calegero-Sutherland model using the Exchange operators

TL;DR

This paper investigates the $W_$ algebra within the Calogero-Sutherland model using exchange operators, revealing all sub-algebras and discussing nonlinear terms and potential links to nonlinear $W_$ algebra.

Contribution

It provides a simplified proof of the $W_$ algebra in the model and explores the presence of nonlinear terms and their implications.

Findings

All sub-algebras of $W_$ are present in the model.

The algebra includes nonlinear terms in general.

Discussion of possible connections to nonlinear $W_$ algebra.

Abstract

We study the $W_\infty$ algebra in the Calegero-Sutherland model using the exchange operators. The presence of all the sub-algebras of $W_\infty$ is shown in this model. A simplified proof for this algebra, in the symmetric ordered basics, is given. It is pointed out that the algebra contains in general, nonlinear terms. Possible connection to the nonlinear $W_\infty$ is discussed.