Linearization of W algebras and W superalgebras

TL;DR

This paper extends the linearization technique of W-algebras to W-superalgebras, enabling simpler calculations and expressions of generators, with examples including W_n, W(sl(3|1),sl(3)), and W(osp(1|4),sp(4)).

Contribution

It introduces a method to linearize W-superalgebras, generalizing previous techniques for W-algebras, and provides explicit examples and calculations.

Findings

Extended linearization technique to W-superalgebras.

Simplified calculation of W(G,H) algebra structures.

Explicit examples of linearized W-superalgebras.

Abstract

In a recent paper, the authors have shown that the secondary reduction of W-algebras provides a natural framework for the linearization of W-algebras. In particular, it allows in a very simple way the calculation of the linear algebra $W(G,H)_{\geq0}$ associated to a wide class of W(G,H) algebras, as well as the expression of the W generators of W(G,H) in terms of the generators of $W(G,H)_{\geq0}$. In this paper, we present the extension of the above technique to W-superalgebras, i.e. W-algebras containing fermions and bosons of arbitrary (positive) spins. To be self-contained the paper recall the linearization of W-algebras. We include also examples such as the linearization of W_n algebras; W(sl(3|1),sl(3)) and W(osp(1|4),sp(4)) = WB_2 superalgebras.