Discrete Heisenberg-Weyl Group and Modular Group

TL;DR

This paper explores the algebraic structure generated by discrete Heisenberg-Weyl groups with irrational rotations and their connection to the modular group, with implications for lattice regularization and duality principles.

Contribution

It demonstrates that generators of two specific discrete Heisenberg-Weyl groups generate the entire algebra of bounded operators on L2(R), revealing a natural modular group action.

Findings

Generators produce the full algebra of bounded operators

Modular group action is naturally implied

Applications to lattice regularization and duality principles

Abstract

It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers $\theta$ and $-1/ \theta$ generate the whole algebra $\cal B$ of bounded operators on $L_2(\bf R)$. The natural action of the modular group in $\cal B$ is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed.