3-Algebraic structures of the quantum Calogero-Moser model

We investigate the quantum Calogero-Moser model and reveal its hidden symmetries, i.e., the $W_{1+\mathrm{\infty }}$ and Virasoro-Witt 3-algebras. In the large $N$ limit, we note that these two infinite dimensional 3-algebras reduce to the $w_{\mathrm{\infty }}$ and special Virasoro-Witt 3-algebras which satisfy the fundamental identity condition, respectively.