# Supersymmetric W-algebras

## TL;DR

This paper develops a comprehensive theory of supersymmetric W-algebras within vertex algebra frameworks, detailing their structure and providing explicit generators for specific Lie superalgebra cases.

## Contribution

It introduces a general approach to supersymmetric W-algebras and explicitly constructs generators for the case of the odd principal nilpotent element in gl(n+1|n).

## Key findings

- Structured description of W-algebras for odd nilpotent elements.
- Explicit free generators for the W-algebra of gl(n+1|n).
- Advances understanding of supersymmetric vertex algebra structures.

## Abstract

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As an application, we produce explicit free generators of the $W$-algebra associated with the odd principal nilpotent element of the Lie superalgebra $\mathfrak{gl}(n+1|n).$