Generalized super-$W_{1+\infty}$-$n$-algebra and Landau Problem
Fridolin Melong, Raimar Wulkenhaar
TL;DR
This work develops a unified deformation-theoretic framework for supersymmetric $n$-ary algebras based on the two-parameter $\mathcal{R}(p,q)$-calculus. It constructs the $\mathcal{R}(p,q)$-super $n$-bracket, analyzes generalized identities, and realizes a $\mathcal{R}(p,q)$-super $W_{1+\infty}$ $n$-algebra in the Supersymmetric Landau problem through the HB1 quantum algebra, providing explicit commutation relations and higher-order structures. It further derives $q$- and $(p,q)$-deformed special cases, including subalgebras and Witt-type structures, broadening the algebraic toolkit for deformed infinite-dimensional algebras. The results offer a pathway to applications in quantum Hall systems and supersymmetric quantum mechanics, where deformed multi-ary symmetries may play a role.
Abstract
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 $W_{1+\infty}$ $n$-algebra which obey the generalized super Jacobi identity (GSJI) for $n$ even. Also, we derive the $\mathcal{R}(p,q)$-super $W_{1+\infty}$ sub-$2n$-algebra and deduce particular cases induced by quantum algebras existing in the literature.