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 $\ast$-bracket formalism with $Z_2$-grading, revealing how the quantum superspace structure manifests in physical systems while preserving essential algebraic properties.