Note on higher spins and holographic symmetry algebra

In this paper we discuss a higher spin extension of the holographic symmetry algebra for graviton and gluon. Our primary observation is that in the presence of higher spin particles the soft symmetry algebra has a subalgebra isomorphic to $w_{\mathrm{\infty }}$ which is generated by the $\mathit{\text{conformally}}\mathit{\text{soft}}\mathit{\text{higher}}\mathit{\text{spin}}\mathit{\text{particles}}$. This $w_{\mathrm{\infty }}$ subalgebra does not commute with the $w_{1+\mathrm{\infty }}$ subalegbra generated by the conformally soft gravitons. The same thing holds for the colored higher spin particles. One gets a subalgebra isomorphic to the $S$-algebra which is generated by the conformally soft colored higher spin particles. We further verify the soft algebra for colored higher spin particles using the (tree-level) $4$-point MHV amplitude of the higher spin Yang-Mills theory constructed in arXiv:2210.07130. At the end we also discuss the higher spin extension of the deformed holographic symmetry algebra for non-zero cosmological constant as constructed in arXiv:2312.00876.