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.
THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS
DANIELE CORRADETTI, DAVID CHESTER, RAYMOND ASCHHEIM, AND KLEE IRWIN
Abstract. Aperiodic algebras are infinite dimensional algebras with generators correspond-
ing 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 construc-
tion 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.
1. Introduction
Crystallographic Coxeter groups are an essential tool in Lie theory being in one-to-one
correspondence with semisimple finite-dimensional Lie algebras over the complex number field
and, thus, playing a fundamental role in physics. On the other hand, non-crystallographic
Coxeter groups are deeply connected with icosahedral quasicrystals and numerous aperiodic
structures [MP93, KF15]. It is then natural, in such context, to look at algebras that are
invariant by non-crystallographic symmetries and, more specifically, at aperiodic Lie algebras.
Aperiodic algebras are a class of infinite dimensional algebras with each generator corre-
sponding to an element of the aperiodic set. A family of aperiodic Lie algebras was firstly
introduced by Patera, Pelantova and Twarock [PPT98], then generalized and extended in
[PT99, TW00a] and, later on, studied by Mazorchuk [Ma02, MT03]. These algebras turned
out to be suitable for physical applications and theoretical models such as the breaking of Vira-
soro symmetry in [Tw99a] an