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.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

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] 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 = tψn+1 + tψn−1 + λVnψn,

(1.1)

where λ measures the strength potential, and the potential sequence Vn 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.

MONTH YEAR

1

arXiv:2302.04044v1 [math.RA] 8 Feb 2023

THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

Figure 1. A one-dimensional quasicrystal F obtained through the cut-and-

project scheme. The one-dimensional quasicrystal is obtained intersecting an

integral lattice Z2 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.

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 τ2, 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 S0 = B, which

then yields to the following sequence

B, A, AB, ABA, ABAAB, ...

(2.1)

2

VOLUME, NUMBER

THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

n

· · ·

-4

-3

-2

-1

0

1

2

3

4

· · ·

F1,0 (n)

· · · −2− 4τ −1− 3τ −1− 2τ

−τ

1 1 + τ 2 + 2τ 2 + 3τ 3 + 4τ

· · ·

F1/2,0 (n)

· · · −2− 4τ −2− 3τ −1− 2τ −1− τ 0 1 + τ 1 + 2τ 2 + 3τ 2 + 4τ

· · ·

F0,0 (n)

· · · −3− 4τ −2− 3τ −2− 2τ −1− τ 0

τ

1 + 2τ 1 + 3τ 2 + 4τ

· · ·

Table 1. Elements of the Fibonacci-chain Quasicrystal Fα,β (n) for α ∈

{

1, 12 , 0

}

and β = 0.

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 real-

isations of the Fibonacci chain F in one-dimensional quasicrystals starting from a cut-and-

project scheme. As illustrated in Fig. (1) we will consider an integral lattice Z2 ⊂ R2 and a re-

gion, 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 win-

dow 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

Z2 ⊂ R2 be the integral lattice and Z [τ ] = Z + τZ the Dirichlet ring. Notice that the two

roots of the equation x2 = 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

n = n1 + n2τ −→ n∗ = n1 + (1− τ)n2.

(2.2)

The Galois automorphism in (2.2) is suitable to be the star map of our cut-and-project qua-

sicrystal, 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

F (Ω) = {n ∈ Z [τ ] ;n∗ ∈ Ω} ,

(2.3)

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 χΩ (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 Fα,β containing the Dirichlet numbers that are image of

Fα,β (m) =

⌊m

τ

+ α

⌋

+mτ + β,

(2.4)

where m,β ∈ Z, α ∈ R, and bxc 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 Fibonacci-

chain quasicrystal is given by the set

{

F1/2,0 (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

MONTH YEAR

3

THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

x ` y

y −1− 3τ −1− 2τ

−τ

1

1 + τ

2 + 2τ

2 + 3τ

· · ·

x

...

...

...

...

...

...

...

−1− 3τ

· · ·

−1− 3τ −2− 4τ −3− 6τ −4− 8τ −5− 9τ −6− 11τ −7− 12τ

· · ·

−1− 2τ

· · ·

−τ

−1− 2τ −2− 4τ −3− 6τ −4− 7τ −5− 9τ −6− 11τ

· · ·

−τ

· · ·

2 + 2τ

1 + τ

−τ

−1− 3τ −2− 4τ −3− 6τ −4− 7τ

· · ·

1

· · ·

4 + 5τ

3 + 4τ

2 + 2τ

1

−τ

−1− 3τ −2− 4τ

· · ·

1 + τ

· · ·

5 + 7τ

4 + 6τ

3 + 4τ

2 + 2τ

1 + τ

−τ

−1− 2τ

· · ·

2 + 2τ

· · ·

7 + 11τ

6 + 9τ

5 + 7τ

4 + 5τ

3 + 4τ

2 + 2τ

1 + τ

· · ·

2 + 3τ

· · ·

8 + 12τ

7 + 11τ

6 + 9τ

5 + 7τ

4 + 6τ

3 + 4τ

2 + 3τ

· · ·

...

...

...

...

...

...

...

...

Table 2. Quasiaddition of x ` y where x and y belong to the Fibonacci-chain

Quasicrystal F (n).

that Fα,β identifies the set F (Ω) with Ω = (−1 + α+ β, α+ β]. We thus have that the

characteristic function χα,β for the quasicrystal Fα,β is given by

χα,β (n) =

{

1

if − 1 + α+ β < n ≤ α+ β

0

else

,

(2.5)

where n ∈ Z [τ ].

2.3. Quasiaddition. An important symmetry of the Fibonacci-chain quasicrystals is ex-

pressed through a binary operation called quasiaddition [MP93] and defined as

n ` m = τ2n− τm,

(2.6)

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

n ` n = n,

(2.7)

n ` (n ` m) = m ` n,

(2.8)

(n+ p) ` (m+ p) = (n ` m) + p,

(2.9)

(n ` m) + (m ` n) = n+m,

(2.10)

(n ` m)− (m ` n) = (n−m) ` (m− n) ,

(2.11)

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

(n ` m)∗ = (1− τ)2 n− (1− τ)m,

(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.

4

VOLUME, NUMBER

THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

[Lx, Ly]

L0

L1

L1+τ

L2+2τ

L2+3τ

L−2−4τ

(2 + 4τ)L−2−4τ

0

(3 + 5τ)L−1−3τ

0

(4 + 7τ)L−τ

L−1−3τ

(1 + 3τ)L−1−3τ

0

0

0

(3 + 6τ)L1

L−1−2τ

(1 + 2τ)L−1−2τ

0

(2 + 3τ)L−τ

(3 + 4τ)L1

(3 + 5τ)L1+τ

L−τ

τL−τ

0

(1 + 2τ)L1

0

(2 + 4τ)L2+2τ

L0

0

L1

(1 + τ)L1+τ

(2 + 2τ)L2+2τ

(2 + 3τ)L2+3τ

L1

−L1

0

0

0

0

L1+τ

(−1− τ)L1+τ

0

0

0

(1 + 2τ)L3+4τ

L2+2τ

(−2− 2τ)L2+2τ

0

0

0

τL4+5τ

L2+3τ

(−2− 3τ)L2+3τ

0

(−1− 2τ)L3+4τ

−τL4+5τ

0

Table 3. Commutation relations for a sample of generators of L (F (Ω)) that

are given for various x, y ∈ F (Ω).

3. 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 τ2. The coordinates of this chain are those of F1,0 (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 qua-

sicrystal L (F (Ω)) as the infinite dimensional vector space spanned by {Lx}x∈F(Ω) with the

bilinear product defined by

[Lx, Ly] = (y − x)χΩ (x∗ + y∗)Lx+y,

(3.2)

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

[Lx, [Ly, Lz]] = (z − y) (y + z − x)χΩ (x∗ + y∗ + z∗)χΩ (y∗ + z∗)Lx+y+z,

(3.3)

[Ly, [Lz, Lx]] = (x− z) (z + x− y)χΩ (x∗ + y∗ + z∗)χΩ (z∗ + x∗)Lx+y+z,

(3.4)

[Lz, [Lx, Ly]] = (y − x) (x+ y + z)χΩ (x∗ + y∗ + z∗)χΩ (x∗ + y∗)Lx+y+z,

(3.5)

and since χΩ (x

∗ + y∗ + z∗) = 1 implies χΩ (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

[Lx, [Ly, Lz]] + [Ly, [Lz, Lx]] + [Lz, [Lx, Ly]] = 0,

(3.6)

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 [Lx, Ly] with x, y ∈ F (Ω) are explicitly given

in Tab. 3.

Observing (3.2), few remarks are easily spotted. 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 χΞ (x

∗ + y∗) is tautologically zero for every x, y ∈ F (Ξ). Moreover, the

MONTH YEAR

5

THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

[Lm, Ln]

L0

L1

L2

L3

L4

L5

L6

L7

L−4

−4L−4

0

−6L−2

0

0

−9L1

0

−11L3

L−3

−3L−3 −4L−2 −5L−1

0

−7L1 −8L2 −9L3 −10L4

L−2

−2L−2

0

0

0

0

−7L3

0

0

L−1

−L−1

0

−3L1

0

−5L3 −6L4

0

−8L6

L0

0

−L1

−2L2 −3L3 −4L4 −5L5 −6L6 −7L7

L1

L1

0

−L3

0

0

−4L6

0

−6L8

L2

2L2

L3

0

0

−2L6 −3L7 −4L8 −5L9

L3

3L3

0

0

0

0

−2L8

0

0

L4

4L4

0

2L6

0

0

−L9

0

−3L11

Table 4. Commutation relations for a sample of generators of the aperiodic

Witt algebras W (F0,0).

[Lm, Ln] L0

L1

L2

L3

L4

L5

L6

L7

L−4

0 −5L−3

0

−7L−1 −8L0

0 −10L2

0

L−3

0

0

0

−6L0

0

0

0

0

L−2

0 −3L−1 −4L0 −5L1 −6L2

0 −8L4 −9L5

L−1

0 −2L0

0

−4L2

0

0 −7L5

0

L0

0

0

0

0

0

0

0

0

L1

0

0

0

−2L4 −3L5

0 −5L7

0

L2

0

0

0

−L5

0

0

0

0

L3

0

2L4

L5

0

−L7

0 −3L9 −4L10

L4

0

3L5

0

L7

0

0 −2L10

0

Table 5. Commutation relations for a sample of generators of the aperiodic

Witt algebras W (F1,0).

subalgebra L (F (Ξ)) is an ideal of L (F (Ω)) if Ξ = [c, 1] where 0 < c . This implies that

L (F (Ω)) are never semisimple Lie algebras.

4. 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

{Ln}n∈Z with integer index, instead of over the base {Lx}x∈Fα,β which was indexed by points

of the Fibonacci-chain quasicrystal. First of all we have to notice from (2.4) that Fα,β is of

the form

Fα,β (n) = n′ + τn,

(4.1)

for some n′ ∈ Z. Thus, the correspondence between {Ln}n∈Z and {Lx}x∈Fα,β is easily obtained

by setting Fα,β (n) −→ n ∈ Z. Therefore we define the aperiodic Witt algebra W (Fα,β) as

the vector space spanned by {Ln}n∈Z, equipped with the following bilinear product

[Ln, Lm] = (n−m)χΩ (Fα,β (n)∗ + Fα,β (m)∗)Ln+m.

(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

6

VOLUME, NUMBER

THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

[Lm, Ln]

L0

L1

L2

L3

L4

L5

L6

L7

L−4

−4L−4

0

−6L−2

0

5C −9L1

0

−11L3

L−3

−3L−3 −4L−2 −5L−1

2C −7L1 −8L2 −9L3 −10L4

L−2

−2L−2

0

1

2C

0

0

−7L3

0

0

L−1

−L−1

0

−3L1

0

−5L3 −6L4

0

−8L6

L0

0

−L1

−2L2 −3L3 −4L4 −5L5 −6L6 −7L7

L1

L1

0

−L3

0

0

−4L6

0

−6L8

L2

2L2

L3

0

0

−2L6 −3L7 −4L8 −5L9

L3

3L3

0

0

0

0

−2L8

0

0

L4

4L4

0

2L6

0

0

−L9

0

−3L11

Table 6. Commutation relations for a sample of generators of the aperiodic

Virasoro algebra V (F0,0 (Ω)).

[Lm, Ln] L0

L1

L2

L3

L4

L5

L6

L7

L−4

0 −5L−3

0

−7L−1

5C − 8L0

0 −10L2

0

L−3

0

0

0

2C − 6L0

0

0

0

0

L−2

0 −3L−1

1

2C − 4L0

−5L1

−6L2

0 −8L4 −9L5

L−1

0 −2L0

0

−4L2

0

0 −7L5

0

L0

0

0

0

0

0

0

0

0

L1

0

0

0

−2L4

−3L5

0 −5L7

0

L2

0

0

0

−L5

0

0

0

0

L3

0

2L4

L5

0

−L7

0 −3L9 −4L10

L4

0

3L5

0

L7

0

0 −2L10

0

Table 7. Commutation relations for a sample of generators of the aperiodic

Virasoro algebra V (F1,0 (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 (F0,0) are shown in Table

(4) while those for α = 1 and β = 0, i.e. W (F1,0), 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 qua-

sicrystals Fα,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 alge-

bra W (F (Ω)) does. To achieve that central extension in our case, i.e. an aperiodic Virasoro

algebra for the Fibonacci-chain V (Fα,0 (Ω)), we just follow [TW00a] adapting it to our special

case in our notation. We then define V (Fα,0 (Ω)) as the vector space spanned by C∪{Ln}n∈Z,

where C is the central generator of the extension, with a bilinear product given by [Ln, C] = 0

for all Ln ∈ {Ln}n∈Z and

[Ln, Lm] = (n−m) χΩ (Fα,0 (n)∗ + Fα,0 (m)∗)Ln+m + 112n

(

n2 − 1

)

δn,−mC,

(4.3)

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.

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THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

5. 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 {Lx}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 {Lx}x∈F(Ω) with bilinear product given by

Lx ◦ Ly =

1

2

(Lx`y + Ly`x) .

(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.

Lx ◦ Ly = Ly ◦ Lx,

(5.2)

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

(Lx ◦ Ly) ◦ (Lx ◦ Lx) = (Lx ◦ Ly) ◦ Lx

(5.3)

=

1

2

(Lx`y ◦ Lx + Ly`x ◦ Lx)

(5.4)

=

1

2

(Lx ◦ Lx`y + Lx ◦ Ly`x)

(5.5)

= Lx ◦ (Ly ◦ Lx)

(5.6)

= Lx ◦ (Ly ◦ (Lx ◦ Lx)) ,

(5.7)

so that the Jordan identity

(Lx ◦ Ly) ◦ (Lx ◦ Lx) = Lx ◦ (Ly ◦ (Lx ◦ Lx)) ,

(5.8)

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 {Lx}x∈F1/2

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′+τn and F (m) = m′+τm, and then consider the quasiaddition

of two elements, i.e.

F (n) ` F (m) = τ2

(

n′ + τn

)

− τ

(

m′ + τm

)

.

(5.9)

A straightforward calculation shows that

F (n) ` F (m) = F

(

n′ −m′ + 2n−m

)

,

(5.10)

while, on the other hand, switching the two addends we obtain

F (m) ` F (n) = F

(

m′ − n′ + 2m− n

)

.

(5.11)

We now have an equivalent definition of J (F) on the vector space spanned by {Ln}n∈Z with

bilinear product defined as

Ln ◦ Lm =

1

2

(Ln′−m′+2n−m + Lm′−n′+2m−n) ,

(5.12)

where n′,m′ ∈ Z and n′ = τn−F (n) and m′ = τm−F (m).

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THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

La ◦ Lb

L−2

L−1

L0

L1

L2

L−4

1

2 (L1 + L−7)

1

2 (L4 + L−9)

1

2 (L7 + L−11)

1

2 (L9 + L−12)

1

2 (L12 + L−14)

L−3

1

2 (L−1 + L−4)

1

2 (L2 + L−6)

1

2 (L5 + L−8)

1

2 (L7 + L−9)

1

2 (L10 + L−11)

L−2

L−2

1

2 (L1 + L−4)

1

2 (L4 + L−6)

1

2 (L6 + L−7)

1

2 (L9 + L−9)

L−1

1

2 (L1 + L−4)

L−1

1

2 (L−3 + L2)

1

2 (L4 + L−4)

1

2 (L7 + L−6)

L0

1

2 (L4 + L−6)

1

2 (L−3 + L2)

L0

1

2 (L2 + L−1)

1

2 (L5 + L−3)

L1

1

2 (L6 + L−7)

1

2 (L4 + L−4)

1

2 (L2 + L−1)

L1

1

2 (L4 + L−1)

L2

1

2 (L9 + L−9)

1

2 (L7 + L−6)

1

2 (L5 + L−3)

1

2 (L4 + L−1)

L2

L3

1

2 (L11 + L−10)

1

2 (L9 + L−7)

1

2 (L7 + L−4)

1

2 (L6 + L−2)

1

2 (L4 + L1)

L4

1

2 (L14 + L−12)

1

2 (L12 + L−9)

1

2 (L10 + L−6)

1

2 (L9 + L−4)

1

2 (L7 + L−1)

Table 8. Multiplication table for a sample of generators of the aperiodic Jor-

dan algebra J (F1,0).

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

Ln ◦ Lm =

1

2

(Lp + Lq) .

(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 (F1,0). 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 (F1,0) is non-unital. Indeed, suppose

it exists an identity element I. This would imply that I ◦ L0 = L0 ◦ I = L0. We would then

have

L0 ◦ Ln1 + L0 ◦ Ln2 + ...+ L0 ◦ Lnm + ... = L0

(5.15)

which means that at least one element Lr would give L0 ◦ Lr = L0. But from (5.14), the

only possibility is for L0 = Lr. Repeating the argument for all Ln we have that the identity

element must be of the form I = ...+ L−1 + L0 + L1 + L2 + ... . But, if so, then consider

L0 ◦ (...+ L−1 + L0 + L1 + L2 + ...) .

(5.16)

Since (5.12) the functions 0 ` n and n ` 0 are monotone functions in n then L−1 resulting

from L0 ◦ L1 (see Tab.(8)) cannot be obtained by any other L0 ◦ Ln 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 = {L0 ◦ x : x ∈ J (F1,0)} is a proper ideal of

J (F1,0) since L1 /∈ k. Thus J (F1,0) is not a simple algebra.

6. 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 F1,0, the second one is an aperiodic Witt algebra, i.e.

W (F1,0), that exactly matches such quasicrystal and that can be extended to a Virasoro

algebra V (F1,0). 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

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THREE FIBONACCI-CHAIN APERIODIC ALGEBRAS

quasicrystal, i.e. J (F1,0). 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 solv-

able 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 al-

gebra. 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.

uit = u

i

xxx − 6aijkujukx,

(6.1)

possesses nondegenerate generalised symmetries or conservation laws if and only if

{

aijk

}

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 (F1,0) 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

{

aijk

}

are

those of J (F1,0) might arise in the study of solitons propagating onto parallel lines departing

from points of a F1,0 Fibonacci-chain quasicrystal.

7. Acknowledgments

This work was funded by the Quantum Gravity Research institute. Authors would like

to thank Fang Fang, Marcelo Amaral, Dugan Hammock, and Richard Clawson for insightful

discussions and suggestions.

References

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[MT03] V. Mazorchuk and R. Twarock, Virasoro-type algebras associated with a penrose tiling, Journal of

Physics A: Mathematical and General, 36.15 (2003):4363-4373.

[MP93] Moody R. V. and Patera J. Quasicrystals and icosians. Journal of Physics A: Mathematical and

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(1998):209-213.

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A. Sen and C. Castro Perelman, A Hamiltonian model of the Fibonacci quasicrystal using non-local

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[Se95]

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[Sv91]

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[Sv92]

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[Sv93]

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[TW00a] R. Twarock, Aperiodic Virasoro algebra. Journal of Mathematical Physics, 41.7 (2000):5088-5106.

[TW00b] R. Twarock. An exactly solvable Calogero model for a non-integrally laced group. Physics Letters A,

275.3 (2000):169172.

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K Dobrev, J-D Hennig and W Lucke, Singapore: World Scientific, 1999, pp 530–4.

MSC2020: 11B39, 52C23, 17B65, 17C50

Departimento de Matematica, Universidade do Algarve, Campus de Gambelas,, Faro, PT

Email address: d.corradetti@gmail.com

Quantum Gravity Research, Los Angeles, California, CA 90290, USA,

Email address: DavidC@QuantumGravityResearch.org

Quantum Gravity Research, Los Angeles, California, CA 90290, USA,

Email address: Raymond@QuantumGravityResearch.org

Quantum Gravity Research, Los Angeles, California, CA 90290, USA,

Email address: Klee@QuantumGravityResearch.org

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11