# A Clifford algebra associated to generalized Fibonacci quaternions

## Abstract

In this paper, using the construction of Clifford algebras, we associate to the set of generalized Fibonacci quaternions a quaternion algebra A (i.e., a Clifford algebra of dimension four). Indeed, for the generalized quaternion algebra $H( β 1 , β 2 )$, denoting $E( β 1 , β 2 )= 1 5 [1+ β 1 +2 β 2 +5 β 1 β 2 +α( β 1 +3 β 2 +8 β 1 β 2 )]$, if $E( β 1 , β 2 )>0$, therefore the algebra A is split. If $E( β 1 , β 2 )<0$, then the algebra A is a division algebra. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice versa.

MSC:11E88, 11B39.

## 1 Introduction

In 1878, WK Clifford discovered Clifford algebras. These algebras generalize the real numbers, complex numbers and quaternions (see [1]).

The theory of Clifford algebras is intimately connected with the theory of quadratic forms. In the following, we will consider K to be a field of characteristic not two. Let $(V,q)$ be a K-vector space equipped with a nondegenerate quadratic form over the field K. A Clifford algebra for $(V,q)$ is a K-algebra C with a linear map $i:V→C$ satisfying the property

$i ( x ) 2 =q(x)⋅ 1 C ,∀x∈V,$

such that for any K-algebra A and any K linear map $γ:V→A$ with $γ 2 (x)=q(x)⋅ 1 A$, $∀x∈V$, there exists a unique K-algebra morphism $γ ′ :C→A$ with $γ= γ ′ ∘i$.

Such an algebra can be constructed using the tensor algebra associated to a vector space V. Let $T(V)=K⊕V⊕(V⊗V)⊕⋯$ be the tensor algebra associated to the vector space V, and let $J$ be the two-sided ideal of $T(V)$ generated by all elements of the form $x⊗x−q(x)⋅1$ for all $x∈V$. The associated Clifford algebra is the factor algebra $C(V,q)=T(V)/J$ (see [2, 3]).

Theorem 1.1 (Poincaré-Birkhoff-Witt [[2], p.44])

If ${ e 1 , e 2 ,…, e n }$ is a basis of V, then the set ${1, e j 1 e j 2 ⋯ e j s ,1≤s≤n,1≤ j 1 < j 2 <⋯< j s ≤n}$ is a basis in $C(V,q)$.

The most important Clifford algebras are those defined over real and complex vector spaces equipped with nondegenerate quadratic forms. Every nondegenerate quadratic form over a real vector space is equivalent to the following standard diagonal form:

$q(x)= x 1 2 +⋯+ x r 2 − x r + 1 2 −⋯− x s 2 ,$

where $n=r+s$ is the dimension of the vector space. The pair of integers $(r,s)$ is called the signature of the quadratic form. The real vector space with this quadratic form is usually denoted by $R r , s$ and the Clifford algebra on $R r , s$ is denoted by $Cl r , s (R)$. For other details about Clifford algebras, the reader is referred to [46] and [7].

Example 1.2

1. (i)

For $p=q=0$, we have $Cl 0 , 0 (K)≃K$.

2. (ii)

For $p=0$, $q=1$, it results that $Cl 0 , 1 (K)$ is a two-dimensional algebra generated by a single vector $e 1$ such that $e 1 2 =−1$, and therefore $Cl 0 , 1 (K)≃K( e 1 )$. For $K=R$, it follows that $Cl 0 , 1 (R)≃C$.

3. (iii)

For $p=0$, $q=2$, the algebra $Cl 0 , 2 (K)$ is a four-dimensional algebra spanned by the set ${1, e 1 , e 2 , e 1 e 2 }$. Since $e 1 2 = e 2 2 = ( e 1 e 2 ) 2 =−1$ and $e 1 e 2 =− e 2 e 1$, we obtain that this algebra is isomorphic to the division quaternions algebra .

4. (iv)

For $p=1$, $q=1$ or $p=2$, $q=0$, we obtain the algebra $Cl 1 , 1 (K)≃ Cl 2 , 0 (K)$ which is isomorphic with a split (i.e., nondivision) quaternion algebra [8].

## 2 Preliminaries

Let $H( β 1 , β 2 )$ be a generalized real quaternion algebra, the algebra of the elements of the form $a= a 1 ⋅1+ a 2 e 2 + a 3 e 3 + a 4 e 4$, where $a i ∈R$, $i∈{1,2,3,4}$, and the elements of the basis ${1, e 2 , e 3 , e 4 }$ satisfy the following multiplication table:

We denote by $n(a)$ the norm of a real quaternion a. The norm of a generalized quaternion has the following expression $n(a)= a 1 2 + β 1 a 2 2 + β 2 a 3 2 + β 1 β 2 a 4 2$. For $β 1 = β 2 =1$, we obtain the real division algebra , with the basis ${1,i,j,k}$, where $i 2 = j 2 = k 2 =−1$ and $ij=−ji$, $ik=−ki$, $jk=−kj$.

Proposition 2.1 ([[3], Proposition 1.1])

The quaternion algebra $H( β 1 , β 2 )$ is isomorphic to quaternion algebra $H( x 2 β 1 , y 2 β 2 )$, where $x,y∈ K ∗$.

The quaternion algebra $H( β 1 , β 2 )$ with $β 1 , β 2 ∈ K ∗$ is either a division algebra or is isomorphic to $H(−1,−1)≃ M 2 (K)$ [3].

For other details about the quaternions, the reader is referred, for example, to [3, 9, 10].

The Fibonacci numbers were introduced by Leonardo of Pisa (1170-1240) in his book Liber abbaci, book published in 1202 AD (see [[11], pp.1, 3]). This name is attached to the following sequence of numbers:

$0,1,1,2,3,5,8,13,21,…,$

with the n th term given by the formula

$f n = f n − 1 + f n − 2 ,n≥2,$

where $f 0 =0$, $f 1 =1$.

In [12], the author generalized Fibonacci numbers and gave many properties of them:

$h n = h n − 1 + h n − 2 ,n≥2,$

where $h 0 =p$, $h 1 =q$, with p, q being arbitrary integers. In the same paper [[12], relation (7)], the following relation between Fibonacci numbers and generalized Fibonacci numbers was obtained:

$h n + 1 =p f n +q f n + 1 .$
(2.1)

For the generalized real quaternion algebra, the Fibonacci quaternions and generalized Fibonacci quaternions are defined in the same way:

$F n = f n ⋅1+ f n + 1 e 2 + f n + 2 e 3 + f n + 3 e 4 ,$

for the n th Fibonacci quaternions and

$H n = h n ⋅1+ h n + 1 e 2 + h n + 2 e 3 + h n + 3 e 4 =p F n +q F n + 1 ,$
(2.2)

for the n th generalized Fibonacci quaternions.

In the following, we will denote the n th generalized Fibonacci number and the n th generalized Fibonacci quaternion element by $h n p , q$, respectively $H n p , q$. In this way, we emphasize the starting integers p and q.

It is known that the expression for the n th term of a Fibonacci element is

$f n = 1 5 [ α n − β n ] = α n 5 [ 1 − β n α n ] ,$
(2.3)

where $α= 1 + 5 2$ and $β= 1 − 5 2$.

From the above, we obtain the following limit:

$lim n → ∞ n ( F n ) = lim n → ∞ ( f n 2 + β 1 f n + 1 2 + β 2 f n + 2 2 + β 1 β 2 f n + 3 2 ) = lim n → ∞ ( α 2 n 5 + β 1 α 2 n + 2 5 + β 2 α 2 n + 4 5 + β 1 β 2 α 2 n + 6 5 ) = sgn E ( β 1 , β 2 ) ⋅ ∞ ,$

where $E( β 1 , β 2 )= 1 5 [1+ β 1 +2 β 2 +5 β 1 β 2 +α( β 1 +3 β 2 +8 β 1 β 2 )]$, since $α 2 =α+1$ (see [13]).

If $E( β 1 , β 2 )>0$, there exists a number $n 1 ∈N$ such that for all $n≥ n 1$, we have $n( F n )>0$. In the same way, if $E( β 1 , β 2 )<0$, there exists a number $n 2 ∈N$ such that for all $n≥ n 2$, we have $n( F n )<0$. Therefore, for all $β 1 , β 2 ∈R$ with $E( β 1 , β 2 )≠0$, in the algebra $H( β 1 , β 2 )$ there is a natural number $n 0 =max{ n 1 , n 2 }$ such that $n( F n )≠0$. Hence $F n$ is an invertible element for all $n≥ n 0$. Using the same arguments, we can compute the following limit:

$lim n → ∞ ( n ( H n p , q ) ) = lim n → ∞ ( h n 2 + β 1 h n + 1 2 + β 2 h n + 2 2 + β 1 β 2 h n + 3 2 ) =sgn E ′ ( β 1 , β 2 )⋅∞,$

where $E ′ ( β 1 , β 2 )= 1 5 ( p + α q ) 2 E( β 1 , β 2 )$, if $E ′ ( β 1 , β 2 )≠0$ (see [13]).

Therefore, for all $β 1 , β 2 ∈R$ with $E ′ ( β 1 , β 2 )≠0$, in the algebra $H( β 1 , β 2 )$ there exists a natural number $n 0 ′$ such that $n( H n p , q )≠0$, hence $H n p , q$ is an invertible element for all $n≥ n 0 ′$.

Theorem 2.2 ([[13], Theorem 2.6])

For all $β 1 , β 2 ∈R$ with $E ′ ( β 1 , β 2 )≠0$, there exists a natural number $n ′$ such that for all $n≥ n ′$, Fibonacci elements $F n$ and generalized Fibonacci elements $H n p , q$ are invertible elements in the algebra $H( β 1 , β 2 )$.

Theorem 2.3 ([[13], Theorem 2.1])

The set $H n ={ H n p , q /p,q∈Z,n≥m,m∈N}∪{0}$ is a -module.

## 3 Main results

Remark 3.1 We remark that the -module from Theorem 2.3 is a free -module of rank two. Indeed, $φ:Z× Z → H n$, $φ((p,q))= H n p , q$ is a -module isomorphism and ${φ(1,0)= F n ,φ(0,1)= F n + 1 }$ is a basis in $H n$.

Remark 3.2 By extension of scalars, we obtain that $R ⊗ Z H n$ is an -vector space of dimension two. A basis is ${ e ¯ 1 =1⊗ F n , e ¯ 2 =1⊗ F n + 1 }$. We have that $R ⊗ Z H n$ is an isomorphic with the -vector space $H n R ={ H n p , q /p,q∈R}∪{0}$. A basis in $H n R$ is ${ F n , F n + 1 }$.

Let $T( H n R )$ be the tensor algebra associated to the -vector space $H n R$, and let $C( H n R )$ be the Clifford algebra associated to the tensor algebra $T( H n R )$. From Theorem 1.1, it results that this algebra has dimension four.

### Case 1: $H( β 1 , β 2 )$ is a division algebra

Remark 3.3 Since in this case $E( β 1 , β 2 )>0$ for all $n≥ n ′$ (as in Theorem 2.2), then $H n R$ is an Euclidean vector space. Indeed, let $z,w∈ H n R$, $z= x 1 F n + x 2 F n + 1$, $w= y 1 F n + y 2 F n + 1$, $x 1 , x 2 , y 1 , y 2 ∈R$. The inner product is defined as follows:

$〈z,w〉= x 1 y 1 n( F n )+ x 2 y 2 n( F n + 1 ).$

We remark that all properties of the inner product are fulfilled. Indeed, since for all $n≥ n ′$ we have $n( F n )>0$ and $n( F n + 1 )>0$, it results that $〈z,z〉= x 1 2 n( F n )+ x 2 2 n( F n + 1 )=0$ if and only if $x 1 = x 2 =0$, therefore $z=0$.

On $H n R$ with the basis ${ F n , F n + 1 }$, we define the following quadratic form $q H n R : H n R →R$:

$q H n R ( x 1 F n + x 2 F n + 1 )=n( F n ) x 1 2 +n( F n + 1 ) x 2 2 .$

Let $Q H n R$ be a bilinear form associated to the quadratic form $q H n R$,

$Q H n R ( x , y ) = 1 2 ( q H n R ( x + y ) − q H n R ( x ) − q H n R ( y ) ) = n ( F n ) x 1 y 1 + n ( F n + 1 ) x 2 y 2 .$

The matrix associated to the quadratic form $q H n R$ is

$A= ( n ( F n ) 0 0 n ( F n + 1 ) ) .$

We remark that $detA=n( F n )n( F n + 1 )>0$ for all $n≥ n ′$. Since $E( β 1 , β 2 )>0$, therefore $n( F n )>0$ for $n> n ′$. We obtain that the quadratic form $q H n R$ is positive definite and the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to $Cl 2 , 0 (K)$ which is isomorphic to a split quaternion algebra.

From the above results and using Proposition 2.1, we obtain the following theorem.

Theorem 3.4 If $H( β 1 , β 2 )$ is a division algebra, there is a natural number $n ′$ such that for all $n≥ n ′$, the Clifford algebra associated to the real vector space $H n R$ is isomorphic with the split quaternion algebra $H(−1,−1)$.

### Case 2: $H( β 1 , β 2 )$ is not a division algebra

Remark 3.5 (i) If $E( β 1 , β 2 )>0$, then $H n R$ is an Euclidean vector space, for all $n≥ n ′$, as in Theorem 2.2. Indeed, let $z,w∈ H n R$, $z= x 1 F n + x 2 F n + 1$, $w= y 1 F n + y 2 F n + 1$, $x 1 , x 2 , y 1 , y 2 ∈R$. The inner product is defined as follows:

$〈z,w〉= x 1 y 1 n( F n )+ x 2 y 2 n( F n + 1 ).$
1. (ii)

If $E( β 1 , β 2 )<0$, then $H n R$ is also an Euclidean vector space, for all $n≥ n ′$, as in Theorem 2.2. Indeed, let $z,w∈ H n R$, $z= x 1 F n + x 2 F n + 1$, $w= y 1 F n + y 2 F n + 1$, $x 1 , x 2 , y 1 , y 2 ∈R$. The inner product is defined as follows:

$〈z,w〉=− x 1 y 1 n( F n )− x 2 y 2 n( F n + 1 ).$

We have $〈z,z〉=− x 1 2 n( F n )− x 2 2 n( F n + 1 )$, and since for all $n≥ n ′$ we have $n( F n )<0$ and $n( F n + 1 )<0$, it results that $〈z,z〉=− x 1 2 n( F n )− x 2 2 n( F n + 1 )=0$ if and only if $x 1 = x 2 =0$, therefore $z=0$.

On $H n R$ with the basis ${ F n , F n + 1 }$,we define the following quadratic form $q H n R : H n R →R$:

$q H n R ( x 1 F n + x 2 F n + 1 )= q H n R ( x 1 F n + x 2 F n + 1 )=n( F n ) x 1 2 +n( F n + 1 ) x 2 2 .$

Let $Q H n R$ be a bilinear form associated to the quadratic form $q H n R$,

$Q H n R ( x , y ) = 1 2 ( q H n R ( x + y ) − q H n R ( x ) − q H n R ( y ) ) = n ( F n ) x 1 y 1 + n ( F n + 1 ) x 2 y 2 .$

The matrix associated to the quadratic form $q H n R$ is

$A= ( n ( F n ) 0 0 n ( F n + 1 ) ) .$

We remark that $detA=n( F n )n( F n + 1 )>0$ for all $n≥ n ′$.

If $E( β 1 , β 2 )>0$, therefore $n( F n )>0$ for $n> n ′$. We obtain that the quadratic form $q H n R$ is positive definite and the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic with $Cl 2 , 0 (K)$ which is isomorphic to a split quaternion algebra.

If $E( β 1 , β 2 )<0$, therefore $n( F n )<0$ for $n> n ′$. Then the quadratic form $q H n R$ is negative definite and the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic with $Cl 0 , 2 (K)$ which is isomorphic to the quaternion division algebra .

From the above results and using Proposition 2.1, we obtain the following theorem.

Theorem 3.6 If $H( β 1 , β 2 )$ is not a division algebra, there is a natural number $n ′$ such that for all $n≥ n ′$, if $E( β 1 , β 2 )>0$, then the Clifford algebra associated to the real vector space $H n R$ is isomorphic with the split quaternion algebra $H(−1,−1)$. If $E( β 1 , β 2 )<0$, then the Clifford algebra associated to the real vector space $H n R$ is isomorphic to the division quaternion algebra $H(1,1)$.

Example 3.7 (1) For $β 1 =1$, $β 2 =−1$, we obtain the split quaternion algebra $H(1,−1)$. In this case, we have $E( β 1 , β 2 )= 1 5 [−5−10α]<0$ and, for $n ′ =0$, we obtain $n( F n )= f n 2 + f n + 1 2 − f n + 2 2 − f n + 3 2 <0$, $n( F n + 1 )= f n + 1 2 + f n + 2 2 − f n + 3 2 − f n + 4 2 <0$ for all $n≥0$. The quadratic form $q H n R$ is negative definite, therefore the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to $Cl 0 , 2 (K)$ which is isomorphic to the quaternion division algebra $H(1,1)$.

1. (2)

For $β 1 =−2$, $β 2 =−3$, we obtain the split quaternion algebra $H(−2,−3)$. In this case, we have $E( β 1 , β 2 )= 1 5 [23+43α]>0$. For $n ′ =0$, we obtain $n( F n )= f n 2 − f n + 1 2 − f n + 2 2 + f n + 3 2 >0$, $n( F n + 1 )= f n + 1 2 − f n + 2 2 − f n + 3 2 + f n + 4 2 >0$ for all $n≥0$. The quadratic form $q H n R$ is positive definite, therefore the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to $Cl 2 , 0 (K)$ which is isomorphic to the split quaternion algebra $H(−1,−1)$.

2. (3)

For $β 1 =2$, $β 2 =−3$, we obtain the split quaternion algebra $H(2,−3)$. In this case, we have $E( β 1 , β 2 )= 1 5 [−33−44α]<0$. For $n ′ =0$, we obtain $n( F n )= f n 2 +2 f n + 1 2 −3 f n + 2 2 −6 f n + 3 2 <0$, $n( F n + 1 )= f n + 1 2 +2 f n + 2 2 −3 f n + 3 2 −6 f n + 4 2 >0$ for all $n≥0$. The quadratic form $q H n R$ is negative definite, therefore the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to $Cl 0 , 2 (K)$ which is isomorphic to the division quaternion algebra $H(1,−1)$.

3. (4)

For $β 1 = β 2 =− 1 2$, we obtain the split quaternion algebra $H(− 1 2 ,− 1 2 )$. Therefore $E( β 1 , β 2 )= 3 20 >0$, and for $n ′ =1$ we obtain $n( F n )>0$ and $n( F n + 1 )>0$. The quadratic form $q H n R$ is positive definite, therefore the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to $Cl 2 , 0 (K)$ which is isomorphic to the split quaternion algebra $H(−1,−1)$.

The algorithm

1. (1)

Let $H( β 1 , β 2 )$ be a quaternion algebra, $α= 1 + 5 2$ and $E( β 1 , β 2 )= 1 5 [1+ β 1 +2 β 2 +5 β 1 β 2 +α( β 1 +3 β 2 +8 β 1 β 2 )]$.

2. (2)

Let V be the -vector space $H n R ={ H n p , q /p,q∈R}∪{0}$.

3. (3)

If $E( β 1 , β 2 )>0$, then the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to $Cl 2 , 0 (K)$ which is isomorphic to the split quaternion algebra $H(−1,−1)$.

4. (4)

If $E( β 1 , β 2 )<0$, then the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to $Cl 0 , 2 (K)$ which is isomorphic to the division quaternion algebra $H(1,1)$.

## 4 Conclusions

In this paper, we have extended the -module of the generalized Fibonacci quaternions to a real vector space $H n R$. We have proved that the Clifford algebra $C( H n R )$ associated to the tensor algebra $T( H n R )$ is isomorphic to a split quaternion algebra or to a division algebra if $E( β 1 , β 2 )= 1 5 [1+ β 1 +2 β 2 +5 β 1 β 2 +α( β 1 +3 β 2 +8 β 1 β 2 )]$ is positive or negative. We also have given an algorithm which allows us to find a division quaternion algebra starting from a split quaternion algebra and vice versa.

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## Acknowledgements

I want thank the anonymous referees for their remarkable comments, suggestions and ideas which helped me to improve this paper. The author also thanks Professor Ravi P Agarwal and members of the Springer Open Team for their support.

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Flaut, C. A Clifford algebra associated to generalized Fibonacci quaternions. Adv Differ Equ 2014, 279 (2014). https://doi.org/10.1186/1687-1847-2014-279