# Higher Spins and Yangian Symmetries

## TL;DR

This paper establishes detailed relations between higher spin ${
m W}_
_ ext{infinity}[ ext{lambda}]$ algebras, affine Yangians, and SH$^c$ algebras, providing explicit formulas and mappings that connect these structures and explore their integrability implications.

## Contribution

It provides explicit expressions and identifications between ${
m W}_
_ ext{infinity}[ ext{lambda}]$, affine Yangian, and SH$^c$ algebras, including free field cases and the lambda-parameter relation.

## Key findings

- Explicit formulas for ${
m W}_
_ ext{infinity}[ ext{lambda}]$ modes in terms of affine Yangian generators.
- Identification of the lambda parameter with affine Yangian parameters.
- Explicit dictionary between affine Yangian and SH$^c$ generators.

## Abstract

The relation between the bosonic higher spin ${\cal W}_\infty[\lambda]$ algebra, the affine Yangian of $\mathfrak{gl}_{1}$, and the SH$^c$ algebra is established in detail. For generic $\lambda$ we find explicit expressions for the low-lying ${\cal W}_\infty[\lambda]$ modes in terms of the affine Yangian generators, and deduce from this the precise identification between $\lambda$ and the parameters of the affine Yangian. Furthermore, for the free field cases corresponding to $\lambda=0$ and $\lambda=1$ we give closed-form expressions for the affine Yangian generators in terms of the free fields. Interestingly, the relation between the ${\cal W}_\infty$ modes and those of the affine Yangian is a non-local one, in general. We also establish the explicit dictionary between the affine Yangian and the SH$^c$ generators. Given that Yangian algebras are the hallmark of integrability, these identifications should pave the way towards uncovering the relation between the integrable and the higher spin symmetries.