The Complete structure of the nonlinear $W_4$ and $W_5$ algebras from quantum Miura transformation

TL;DR

This paper explicitly computes the operator product expansions for certain quantum algebras derived from the quantum Miura transformation, revealing the full structure of the nonlinear $W_4$ and $W_5$ algebras.

Contribution

It provides the explicit structure of the nonlinear $W_4$ and $W_5$ algebras using primary fields and OPEs from quantum Miura transformation for general $n$.

Findings

Explicit OPEs for $n=3$ and 4

Complete structure of nonlinear $W_4$ and $W_5$ algebras

Identification of primary fields with spins 3, 4, and 5

Abstract

Starting from the well-known quantum Miura transformation for the Lie algebra $A_n$, we compute explicitly the OPEs for $n=3$ and 4. The primary fields with spin 3, 4 and 5 are found (for general $n$). By using these primary fields and the OPEs from quantum Miura transformation, we derive the complete structure of the nonlinear $W_4$ and $W_5$ algebras.