# ${\cal W}$ algebras are L$_\infty$ algebras

## TL;DR

The paper demonstrates that classical ${
m W}$ algebras naturally form L$_ ext{infinity}$ algebras, providing new insights into their structure and symmetry transformations, with explicit application to the ${
m W}_3$ algebra.

## Contribution

It establishes a formal connection between ${
m W}$ algebras and L$_ ext{infinity}$ algebras, expanding the understanding of their algebraic structure and symmetries.

## Key findings

- ${
m W}$ algebras form L$_ ext{infinity}$ algebras with field-dependent parameters
- Develops a general formalism for the ${
m W}$-L$_ ext{infinity}$ correspondence
- Explicitly applies the formalism to the classical ${
m W}_3$ algebra

## Abstract

It is shown that the closure of the infinitesimal symmetry transformations underlying classical ${\cal W}$ algebras give rise to L$_\infty$ algebras with in general field dependent gauge parameters. Therefore, the class of well understood ${\cal W}$ algebras provides highly non-trivial examples of such strong homotopy Lie-algebras. We develop the general formalism for this correspondence and apply it explicitly to the classical ${\cal W}_3$ algebra.