Quantum Superspace and Bloch Electron Systems with Zeeman Effects: *-Bracket Formalism for Super Curtright-Zachos Algebras

Abstract: We introduce supersymmetric extensions of the Hom-Lie deformation of the Virasoro algebra (super Curtright-Zachos algebra), as realized in the GL(1,1) quantum superspace, for Bloch electron systems under Zeeman effects. By examining the duality inherent in quantum superspace scaling operators, we establish a correspondence between quantum superspace and its physical realization through a novel operator mixing mechanism. For the continuous case, we construct super Curtright-Zachos algebra using magnetic translations and spin matrix bases, demonstrating explicit realizations for both N=1 and N=2 supersymmetric algebras with a natural N=2 decomposition. For the discrete case, we establish cyclic matrix representations in tight-binding models. We organize these structures through the *-bracket formalism with Z2-grading, revealing how the quantum superspace structure manifests in physical systems while preserving essential algebraic properties.