Generalized $\tilde{W}$ algebras

Recently, a new generalized family of infinite-dimensional $\tilde{W}$ algebras, each associated with a particular element of a commutative subalgebra of the $W_{1+\mathrm{\infty }}$ algebra, was described. This paper provides a comprehensive account of the aforementioned association, accompanied by the requisite proofs and illustrative examples. This approach allows a derivation of Ward identities for selected WLZZ matrix models and the expansion of corresponding $W$-operators in terms of an infinite set of variables $p_k$.