Nonlinear Realizations of the $W_3^{(2)}$ Algebra

TL;DR

This paper explores nonlinear realizations of the classical $W_3^{(2)}$ algebra, deriving integrable equations like the Boussinesq and Toda lattice through coset and reduction methods.

Contribution

It introduces a novel application of coset space and covariant reduction techniques to realize $W_3^{(2)}$ algebra and derive related integrable systems.

Findings

Derived the Boussinesq equation with interchanged coordinates.

Obtained $sl(3,R)$ Toda lattice equations using extended algebra and additional space.

Demonstrated the effectiveness of coset and reduction methods in nonlinear algebra realizations.

Abstract

In this letter we consider the nonlinear realizations of the classical Polyakov's algebra $W_3^{(2)}$. The coset space method and the covariant reduction procedure allow us to deduce the Boussinesq equation with interchanged space and evolution coordinates. By adding one more space coordinate and introducing two copies of the $W_3^{(2)}$ algebra, the same method yields the $sl(3,R)$ Toda lattice equations.