Three Fibonacci-Chain Aperiodic Algebras

TL;DR

This paper introduces three new aperiodic algebras based on Fibonacci-chain quasicrystals, including a novel aperiodic Jordan algebra, expanding the mathematical tools for studying quasicrystals and their applications.

Contribution

It presents three Fibonacci-chain aperiodic algebras, notably the first aperiodic Jordan algebra, linking quasicrystal structures with algebraic frameworks.

Findings

Constructed a quasicrystal Lie algebra matching the original Fibonacci chain.

Developed the first aperiodic Jordan algebra.

Provided algebraic models for quasicrystal structures.

Abstract

Aperiodic algebras are infinite dimensional algebras with generators corresponding to an element of the aperiodic set. These algebras proved to be an useful tool in studying elementary excitations that can propagate in multilayered structures and in the construction of some integrable models in quantum mechanics. Starting from the works of Patera and Twarock we present three aperiodic algebras based on Fibonacci-chain quasicrystals: a quasicrystal Lie algebra, an aperiodic Witt algebra and, finally, an aperiodic Jordan algebra. While a quasicrystal Lie algebra was already constructed from a modification of the Fibonacci chain, we here present an aperiodic algebra that matches exactly the original quasicrystal. Moreover, this is the first time to our knowledge, that an aperiodic Jordan algebra is presented leaving room for both theoretical and applicative developments.