Moscow International School of Physics 2024

It was demonstrated recently that the $W_{1+\infty}$ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In our work, we realize that every element of such a ray is associated with a $\widetilde W$ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the $\widetilde W$ algebras studied earlier. We suggest a definition of the $\widetilde W$ algebra based on the matrix realization of the $W_{1+\infty}$ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the $W_{1+\infty}$ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with $\widetilde W$ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero model describes the hypergeometric $\tau$-functions corresponding to the completed cycles.