Moyal product and generalized Hom-Lie-Virasoro symmetries in Bloch electron systems

We explore two variations of the Curtright-Zachos (CZ) deformation of the Virasoro algebra. Firstly, we introduce a scaled CZ algebra that inherits the scaling structure found in the differential operator representation of the magnetic translation (MT) operators. We then linearly decompose the scaled CZ generators to derive two types of Hom-Lie deformations of the $W_{\infty }$ algebra. We discuss ⁎-bracket formulations of these algebras and their connection to the Moyal product. We show that the ⁎-bracket form of the scaled CZ algebra arises from the Moyal product, while we obtain the second type of deformed $W_{\infty }$ through a coordinate transformation of the first type of Moyal operators. From a physical point of view, we construct the Hamiltonian of a tight binding model (TBM) using the Wyle matrix representation of the scaled CZ algebra. We note that the integer powers of q are linked to the quantum fluctuations that are inherent in the Moyal product.