Three Fibonacci-Chain Aperiodic Algebras

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.