MORE ON THE LINEARIZATION OF $W$-ALGEBRAS

TL;DR

This paper demonstrates that many complex W-(super)algebras can be embedded into simpler linear (super)conformal algebras, simplifying their structure and analysis, with a detailed example provided for the W_4 algebra.

Contribution

It introduces a general method for linearizing a broad class of W-(super)algebras by embedding them into linear (super)conformal algebras, including explicit examples.

Findings

W_N^{(N-1)} and related algebras can be linearized.

Embedding into linear algebras simplifies the structure of W-algebras.

Explicit example of W_4 algebra illustrates the method.

Abstract

We show that a wide class of $W$-(super)algebras, including $W_N^{(N-1)}$, $U(N)$-superconformal as well as $W_N$ nonlinear algebras, can be linearized by embedding them as subalgebras into some {\em linear} (super)conformal algebras with finite sets of currents. The general construction is illustrated by the example of $W_4$ algebra.