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

TL;DR

This paper investigates deformations of the Virasoro and W-infinity algebras using Moyal products and Hom-Lie structures, connecting these mathematical frameworks to physical models of Bloch electrons.

Contribution

It introduces a scaled Curtright-Zachos algebra and derives two types of Hom-Lie deformations of the W-infinity algebra, linking algebraic structures to physical electron systems.

Findings

Scaled CZ algebra relates to magnetic translation operators

Deformed W-infinity algebra connects to Moyal product

Hamiltonian constructed for a tight binding model

Abstract

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-infinity 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-infinity 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.