Characterization of the $W_{1+\mathrm{\infty }}$-n-algebra and applications

In this paper, we construct the $W_{1+\mathrm{\infty }}$-n-algebras in the framework of the generalized quantum algebra. We characterize the $\mathcal{R}(p,q)$-multi-variable $W_{1+\mathrm{\infty }}$-algebra and derive its $n$-algebra which is the generalized Lie algebra for $n$ even. Furthermore, we investigate the $\mathcal{R}(p,q)$-elliptic hermitian matrix model and determine a toy model for the generalized quantum $W_{\mathrm{\infty }}$ constraints. Also, we deduce particular cases of our results.