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.


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 corresponding 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 Virasoro symmetry in [Tw99a] and the construction of exactly solvable models in [TW00b].In fact, this is not surprising since important results were obtained studying elementary excitations that can propagate in multilayered structures with constituents arranged in a quasiperiodic fashion.These excitations include plasmon-polaritons, spin waves, light waves and electrons, among others [AC03].In this context, relevant physical properties are analysed in terms of Hamiltonians [Ja21,SCP] which in the case of nearest-neighbour, tight-binding models are of the form (Hψ) n = tψ n+1 + tψ n−1 + λV n ψ n , (1.1) where λ measures the strength potential, and the potential sequence V n is generated according to some aperiodic substitution rule [Ma] such as Fibonacci, Thue-Morse, Period-doubling, Triadic Cantor, etc.It is therefore with renewed interest that we look to aperiodic algebras that match exactly those chains and especially the Fibonacci chains.
In this work, we introduce three aperiodic algebras for a class of Fibonacci-chain quasicrystals [LS86]: the first one is a quasicrystal Lie algebra, the second is an aperiodic Witt algebra with a Virasoro extension and, finally, we present an aperiodic Jordan algebra.
Research supported by Quantum Gravity Research fundings.The present work is structured as follows.In sec.2, we briefly review the 1-dimensional Fibonacci-chain quasicrystals in a general setting following [LS86].We then define the binary operation of quasiaddition which encode a geometrical invariance of this class of quasicrystals.In sec.3, 4 and 5, we construct three different Fibonacci-chain aperiodic algebras.The first algebra, treated in sec.3, is a quasicrystal Lie algebra and is a review of the original works of Twarock and Patera [PPT98,TW00a], which constitutes our starting point.As for the second aperiodic algebra, i.e. the aperiodic Witt algebra and its Virasoro extension treated in sec.4, we used a slightly different approach than [TW00a].Indeed, by focusing on the analytic definition of the Fibonacci-chain quasicrystal, we defined the aperiodic Witt algebra following an index scheme based on integers rather than aperiodic quasicrystal coordinates.In the definition of this algebra, we avoided the point defect introduced by [PPT98, TW00a], so that the quasicrystal, on which our aperiodic algebra is based, contains only tiles that have length τ and τ 2 , and matches exactly a Fibonacci-chain quasicrystal.Finally, in sec.5 we present the third aperiodic algebra which is a Jordan algebra realized exploiting the property of invariance of quasiaddition over Fibonacci-chain quasicrystals.To our knowledge, this is the first time that an aperiodic Jordan algebra is presented in literature.
2. The Fibonacci-chain Quasicrystal 2.1.Definition.A Fibonacci chain F is a one-dimensional aperiodic sequence that can be obtained by the substitution rules A −→ AB, B −→ A, with starting points S 0 = B, which then yields to the following sequence B, A, AB, ABA, ABAAB, ... (2.1) It is easy to note that the number of blocks in (2.1) increases following the Fibonacci sequence 1, 1, 2, 3, 5... and that the ratio between A and B tends to the golden mean τ = 1 + √ 5 /2.

2.2.
Realisations of the Fibonacci-chain Quasicrystal.We will now give different realisations of the Fibonacci chain F in one-dimensional quasicrystals starting from a cut-andproject scheme.As illustrated in Fig.
(1) we will consider an integral lattice Z 2 ⊂ R 2 and a region, called acceptance window, bounded by two lines with same irrational slope 1/τ ≈ 0.618... and separated by an interval that we will suppose here to be of length 1. Points of the lattice that fall into the acceptance window will be then projected to the lower line thus realising a one-dimensional Fibonacci-chain quasicrystal.Obviously translations of the acceptance window will correspond to different realisation of the same Fibonacci-chain quasicrystal.
In order to make this construction theoretically precise we here define a cut-and-project scheme, but we also give an explicit formula for the coordinates of the quasicrystal.Let Z 2 ⊂ R 2 be the integral lattice and Z [τ ] = Z + τ Z the Dirichlet ring.Notice that the two roots of the equation x 2 = x + 1 are τ ≈ 1.618... and 1 − τ ≈ −0.618..., thus naturally defines a Galois automorphism * that for every number n ∈ Z [τ ] corresponds (2.2) The Galois automorphism in (2.2) is suitable to be the star map of our cut-and-project quasicrystal, i.e. the involution that sends points from the parallel space to the perpendicular space and vice versa [BG17].We then have that a Fibonacci-chain quasicrystal is given by 3) where the acceptance window Ω is the segment (0, 1]. It is useful to remark that, by virtue of (2.3), a Dirichlet integer n ∈ Z [τ ] belongs to the Fibonacci-chain quasicrystal if and only if the characteristic function χ Ω (n * ) = 1.This notation will come in handy in order to define the Witt aperiodic algebra in the next section.
A more explicit way to define the previous Fibonacci-chain quasicrystal F is to consider a direct formulation of the coordinates as in [LS86], where a general class of Fibonacci-chain quasicrystals is defined as the set F α,β containing the Dirichlet numbers that are image of where m, β ∈ Z, α ∈ R, and x is the floor function of x.Since the term β acts as a translation, we focus mostly on β = 0 and our primary interest will be in α = 0, 1/2 or 1.Indeed, it is straightforward to see that for α = 1 and β = 0, then the set is equal to F (Ω) with Ω = (0, 1] for which a list of explicit values are given in the first row of Tab. 1.Another notable Fibonaccichain quasicrystal is given by the set F 1/2,0 (n) n∈Z which is also called the palindrome Fibonacci-chain quasicrystal and whose elements can be also be found in Tab. 1.Such quasicrystal has 180-degree rotational symmetry about the origin and is sometimes convenient for generalizations to higher dimensions.More generally, a straightforward calculation shows where n ∈ Z [τ ].

2.3.
Quasiaddition.An important symmetry of the Fibonacci-chain quasicrystals is expressed through a binary operation called quasiaddition [MP93] and defined as for every n, m ∈ Z √ 5 .Obviously quasiaddition is not commutative nor associative but is flexible, i.e. n (m n) = (n m) n,and enjoys the following properties for every n, m, p ∈ Z √ 5 .An important feature of this operation is that given any two points n, m of a Fibonacci-chain quasicrystal F (Ω) then n m still belongs to F (Ω). Indeed, if the image of the star map n * , m * in ((2.2)) belongs to Ω, then also (2.12) belongs to Ω since the set is convex.Thus a Fibonacci-chain quasicrystal is closed under quasiaddition.A sample of the multiplication table of the quasiaddition is given in Tab. 2.
Table 3. Commutation relations for a sample of generators of L (F (Ω)) that are given for various x, y ∈ F (Ω).

A Fibonacci-Chain Quasicrystal Lie Algebras
In the following sections we present three aperiodic algebras with generators in one to one correspondence with the Fibonacci-chain quasicrystal previously defined.The first quasicrystal Lie algebra is just a review of that presented in [PPT98] for a one-dimensional quasicrystal originated by a modification of the Fibonacci-chain and that enjoys a reflection symmetry at x = 1/2.The tile at the origin is a defect with length 1, while all other tiles have length τ and τ 2 .The coordinates of this chain are those of F 1,0 (n) (see Tab. (1)) with the addition of the 0 element, i.e. ..., −1 − 3τ, −1 − 2τ, −τ, 0, 1, 1 + τ, 2 + 2τ, ... (3.1) Following [PPT98], we now define the quasicrystal Lie algebra for the Fibonacci-chain quasicrystal L (F (Ω)) as the infinite dimensional vector space spanned by {L x } x∈F (Ω) with the bilinear product defined by where x, y ∈ F (Ω) and Ω = [0, 1] is the acceptance window of the quasicrystal.Since (3.2) it is obviously antisymmetric, we only have to show that it satisfies the Jacobi identity.In fact, this is straightforward since and since χ Ω (x * + y * + z * ) = 1 implies χ Ω (x * + y * ) = 1 for every x * , y * , z * ∈ Ω = [0, 1] and, more generally for Ω = [a, b] with ab ≥ 0. From previous equations, we have that First of all, we notice that if we consider an arbitrary interval for the acceptance window Ξ = [a, 1], then if a ≥ 1/2 the Lie algebra is abelian since χ Ξ (x * + y * ) is tautologically zero for every x, y ∈ F (Ξ).Moreover, the Table 4. Commutation relations for a sample of generators of the aperiodic Witt algebras W (F 0,0 ).

An aperiodic Witt algebra and its Virasoro extensions
We now define an aperiodic Witt algebra and a Virasoro extension of it.Such algebras were first introduced in [TW00a], but are here presented in an equivalent way over the base {L n } n∈Z with integer index, instead of over the base {L x } x∈F α,β which was indexed by points of the Fibonacci-chain quasicrystal.First of all we have to notice from (2.4) that F α,β is of the form F α,β (n) = n + τ n, (4.1) for some n ∈ Z.Thus, the correspondence between {L n } n∈Z and {L x } x∈F α,β is easily obtained by setting F α,β (n) −→ n ∈ Z. Therefore we define the aperiodic Witt algebra W (F α,β ) as the vector space spanned by {L n } n∈Z , equipped with the following bilinear product (4.2) A straightforward calculation, similar to the one presented in the previous section, shows that this is indeed a Lie algebra if (−1 + α + β) (α + β) ≥ 0. This means that in the special case of the palindrome Fibonacci-chain, where α = 1/2 and β = 0, we do not have an aperiodic Witt algebra.On the other hand, leaving β = 0, we have that α = 0 and α = 1 give rise Table 6.Commutation relations for a sample of generators of the aperiodic Virasoro algebra V (F 0,0 (Ω)).7. Commutation relations for a sample of generators of the aperiodic Virasoro algebra V (F 1,0 (Ω)).
to two different aperiodic Witt algebras that are non-abelian and, therefore, interesting cases.Starting with α = 0 and β = 0, explicit commutation relations for W (F 0,0 ) are shown in Table (4) while those for α = 1 and β = 0, i.e.W (F 1,0 ), are shown in Table (5).The structure constants here are found to be in terms of Dirichlet integers, which is in general irrational.4.1.An aperiodic Virasoro extension.We will now focus on the Fibonacci-chain quasicrystals F α,0 with α = 0, 1.While Fibonacci-chain quasicrystal Lie algebras such as L (F (Ω)) do not allow for a central extension (cfr.[PT99]), the previously defined aperiodic Witt algebra W (F (Ω)) does.To achieve that central extension in our case, i.e. an aperiodic Virasoro algebra for the Fibonacci-chain V (F α,0 (Ω)), we just follow [TW00a] adapting it to our special case in our notation.We then define V (F α,0 (Ω)) as the vector space spanned by C ∪ {L n } n∈Z , where C is the central generator of the extension, with a bilinear product given by [L n , C] = 0 for all L n ∈ {L n } n∈Z and for every n, m ∈ Z.Such extension might not be unique, but is an interesting algebra for its possible applications in theoretical physics (see a similar algebra in [Tw99a]) and it was then worth mentioning it.

A Fibonacci-chain Jordan Algebra
In this section we define for the first time an aperiodic Jordan algebra, which is an infinite dimensional Jordan algebra J (F), whose generators {L x } x∈F are in one to one correspondence with the Fibonacci-chain quasicrystal F. Since our construction is valid for all α and β we will drop such notation and write just F unless is needed.We then define J (F) as the real vector space spanned by {L x } x∈F (Ω) with bilinear product given by (5.1) The product in (5.1) is obviously well-defined since for every x, y ∈ F then x y, y x ∈ F as shown in (2.12).Moreover, it is clearly commutative, i.e.
so, in order to prove that is a Jordan algebra it is sufficient to show that it satisfies the Jordan identity.Indeed, since x x = x, we have that so that the Jordan identity is fulfilled.Jordan algebras are notorious for an abundance of idempotent elements and this makes no exception.Indeed, we note that since x x = x all elements of the basis {L x } x∈F 1/2 are idempotent elements.An alternative definition of the algebra can be given on a basis whose elements are indexed by integer.Indeed, let F (n) = n +τ n and F (m) = m +τ m, and then consider the quasiaddition of two elements, i.e. (5.9) A straightforward calculation shows that while, on the other hand, switching the two addends we obtain (5.11) We now have an equivalent definition of J (F) on the vector space spanned by {L n } n∈Z with bilinear product defined as where n , m ∈ Z and n = τ n − F (n) and m = τ m − F (m).
Table 8.Multiplication table for a sample of generators of the aperiodic Jordan algebra J (F 1,0 ).
A simple analysis of (5.12) shows an useful property of the above Jordanian product.Let n, m ∈ Z and call p, q ∈ Z the indices of the generators such that (5.13) We thus have from (5.12) and a straighforward calculation that p + q = n + m. (5.14) In order to give concreteness to our construction we will now focus on a specific aperiodic Jordan algebra, i.e.J (F 1,0 ).For such algebra a sample of its multiplication table is in Tab. 8.An implication of (5.14), together with the idempotency of the generators, and the multiplication relations in Tab. 8, is that the algebra J (F 1,0 ) is non-unital.Indeed, suppose it exists an identity element I.This would imply that I • L 0 = L 0 • I = L 0 .We would then have which means that at least one element L r would give L 0 • L r = L 0 .But from (5.14), the only possibility is for L 0 = L r .Repeating the argument for all L n we have that the identity element must be of the form I = ...
(5.16) Since (5.12) the functions 0 n and n 0 are monotone functions in n then L −1 resulting from L 0 • L 1 (see Tab.(8)) cannot be obtained by any other L 0 • L n with n ∈ Z.Therefore, the L −1 component is non zero and, thus, in contraddiction with I being the identity element.
A similar argument shows that the ideal k ) is not a simple algebra.

Conclusions and Developments
In this paper we presented three aperiodic algebras that originate quite naturally from a special class of Fibonacci-chain quasicrystals.While the first one is a review of the one introduced in [PPT98] and is a quasicrystal Lie algebra obtained with a one-point defect from the Fibonacci-chain quasicrystal F 1,0 , the second one is an aperiodic Witt algebra, i.e.W (F 1,0 ), that exactly matches such quasicrystal and that can be extended to a Virasoro algebra V (F 1,0 ).Finally, we introduced a completely new class of aperiodic algebras, i.e. the aperiodic Jordan algebras, and presented a special case for the same Fibonacci-chain quasicrystal, i.e.J (F 1,0 ).Such aperiodic Jordan algebras were made possible by exploiting an important symmetry of Fibonacci-chain quasicrystals which is encoded by quasiaddition and holds also for higher dimensional quasicrystals.The definition of such algebras is already an interesting subject from a mathematical point of view, but from the physical side we think the three class of algebras we presented here are an indispensable tools to everyone who is interested in physical aperiodic structures that can be modeled by Fibonacci-chain quasicrystals.
Indeed, aperiodic Virasoro algebras can be used to study deformations of exactly solvable models of Calogero-Sutherland type [TW00b].More specifically, perturbations of the Hamiltonian describing a many-body quantum mechanical system on a circle with n identical particles of mass m can be expressed in terms of the generators of an aperiodic Virasoro algebra.On the other hand, as for the aperiodic Jordan algebra, it is well known a one-to-one correspondence between finite dimensional Jordan algebras and multifield Korteweg-de Vries equations.Indeed, in the early '90, Svinolupov [Sv91] and Sokolov showed that a generalisation of the KdV equation, i.e.
possesses nondegenerate generalised symmetries or conservation laws if and only if a i jk are constants of structure of a finite dimensional Jordan algebra[Sv93, Thm 1.1].Similar results hold for the modified KdV equation [Sv93], for the Sine-Gordon equation and for generalisations of the non-linear Schroedinger equation [Sv92].While, the aperiodic Jordan algebra J (F 1,0 ) is infinite dimensional, nevertheless it can be used to define finite dimensional algebras limiting the aperiodic set of the quasicrystal to a specific set consecutive elements and imposing the product to be 0 when the quasiaddition of two elements falls outside the selected portion of the quasicrystal.With this modification, commutativity and the Jordan identity still hold, so that the resulting algebra is finite dimensional, falling into Svinolupov theorem's hypothesis.In fact, we expect that systems of the type (6.1)where the constants of structure a i jk are those of J (F 1,0 ) might arise in the study of solitons propagating onto parallel lines departing from points of a F 1,0 Fibonacci-chain quasicrystal.

Figure 1 .
Figure1.A one-dimensional quasicrystal F obtained through the cut-andproject scheme.The one-dimensional quasicrystal is obtained intersecting an integral lattice Z 2 with the acceptance window Ω here represented by a region bounded by two lines of irrational slope 1/τ .Points that fall into the acceptance window are projected onto the lower line on which lies quasicrystal.
.6) thus fulfilling the Jacobi identity.Therefore, definition (3.2) is a Lie algebra for every Ω = [a, b] with ab ≥ 0. A sample of commutation relations [L x , L y ] with x, y ∈ F (Ω) are explicitly given in Tab. 3. Observing (3.2), few remarks are easily spotted.

Table 2 .
Quasiaddition of x y where x and y belong to the Fibonacci-chain Quasicrystal F (n).