Lie algebras associated with one-dimensional aperiodic point sets

TL;DR

This paper introduces a novel class of infinite-dimensional Lie algebras derived from one-dimensional aperiodic point sets, revealing new algebraic structures related to quasicrystals and model sets.

Contribution

It defines Lie algebras over aperiodic point sets using a multiplicative property and a graded composition law, a new approach in the study of quasicrystal structures.

Findings

Lie algebras are associated with aperiodic point sets

The algebras are infinite-dimensional and semi-direct products

New algebraic structures related to quasicrystals are identified

Abstract

The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive graded composition law and permits the introduction of Lie algebras over such aperiodic point sets. These infinite dimensional Lie algebras are shown to be representatives of a new type of semi-direct product induced Lie algebras.