Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models

We discuss the Curtright-Zachos (CZ) deformation of the Virasoro algebra and its extentions in terms of magnetic translation (MT) group in a discrete Bloch electron system, so-called the tight binding model (TBM), as well as in its continuous system. We verify that the CZ generators are essentially composed of a specific combination of MT operators representing deformed and undeformed $U(1)$ translational groups, which determine phase factors for a $\ast$-bracket commutator. The phase factors can be formulated as a $\ast$-ordered product of the commutable $U(1)$ operators by interpreting the AB phase factor of discrete MT action as fluctuation parameter $q$ of a quantum plane. We also show that some sequences of TBM Hamiltonians are described by the CZ generators.