Generalized super-$W_{1+\mathrm{\infty }}$-$n$-algebra and Landau Problem

We investigate the $\mathcal{R}(p,q)$-super $n$-bracket and study their properties such that the generalized super Jacobi identity (GJSI). Furthermore, from the $\mathcal{R}(p,q)$-operators in a Supersymmetric Landau problem, we furnish the $\mathcal{R}(p,q)$-super $W_{1+\mathrm{\infty }}$ $n$-algebra which obey the generalized super Jacobi identity (GSJI) for $n$ even. Also, we derive the $\mathcal{R}(p,q)$-super $W_{1+\mathrm{\infty }}$ sub-$2n$-algebra and deduce particular cases induced by quantum algebras existing in the literature.