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 $\mathbb{H}({\beta }_1,{\beta }_2)$, denoting $E({\beta }_1,{\beta }_2)=\frac{1}{5}[1+{\beta }_1+2{\beta }_2+5{\beta }_1{\beta }_2+\alpha ({\beta }_1+3{\beta }_2+8{\beta }_1{\beta }_2)]$, if $E({\beta }_1,{\beta }_2)>0$, therefore the algebra A is split. If $E({\beta }_1,{\beta }_2)<0$, 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 $i:V\to C$ satisfying the property

$$i{(x)}^2=q(x)\cdot 1_C,\mathrm{\forall }x\in V,$$

such that for any K-algebra A and any K linear map $\gamma :V\to A$ with ${\gamma }^2(x)=q(x)\cdot 1_A$, $\mathrm{\forall }x\in V$, there exists a unique K-algebra morphism ${\gamma }^{\prime }:C\to A$ with $\gamma ={\gamma }^{\prime }\circ i$.

Such an algebra can be constructed using the tensor algebra associated to a vector space V. Let $T(V)=K\oplus V\oplus (V\otimes V)\oplus \cdots$ be the tensor algebra associated to the vector space V, and let $\mathcal{J}$ be the two-sided ideal of $T(V)$ generated by all elements of the form $x\otimes x-q(x)\cdot 1$ for all $x\in V$. The associated Clifford algebra is the factor algebra $C(V,q)=T(V)/\mathcal{J}$ (see [2, 3]).

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

If $\{e_1,e_2,\dots ,e_n\}$ is a basis of V, then the set $\{1,e_{j_1}{e}_{j_2}\cdots e_{j_s},1\le s\le n,1\le j_1<j_2<\cdots <j_s\le n\}$ 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:

$$q(x)=x_1^2+\cdots +x_r^2-x_{r+1}^2-\cdots -x_s^2,$$

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 ${\mathbb{R}}_{r,s}$ and the Clifford algebra on ${\mathbb{R}}_{r,s}$ is denoted by ${Cl}_{r,s}(\mathbb{R})$. For other details about Clifford algebras, the reader is referred to [4–6] and [7].

Example 1.2

• (i) For $p=q=0$, we have ${Cl}_{0,0}(K)\simeq K$.

• (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 ${Cl}_{0,1}(K)\simeq K(e_1)$. For $K=\mathbb{R}$, it follows that ${Cl}_{0,1}(\mathbb{R})\simeq \mathbb{C}$.

• (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 ℍ.

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

2 Preliminaries

Let $\mathbb{H}({\beta }_1,{\beta }_2)$ be a generalized real quaternion algebra, the algebra of the elements of the form $a=a_1\cdot 1+a_2{e}_2+a_3{e}_3+a_4{e}_4$, where $a_i\in \mathbb{R}$, $i\in \{1,2,3,4\}$, and the elements of the basis $\{1,e_2,e_3,e_4\}$ satisfy the following multiplication table:

We denote by $\mathbf{n}(a)$ the norm of a real quaternion a. The norm of a generalized quaternion has the following expression $\mathbf{n}(a)=a_1^2+{\beta }_1{a}_2^2+{\beta }_2{a}_3^2+{\beta }_1{\beta }_2{a}_4^2$. For ${\beta }_1={\beta }_2=1$, we obtain the real division algebra ℍ, 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 $\mathbb{H}({\beta }_1,{\beta }_2)$ is isomorphic to quaternion algebra $\mathbb{H}(x^2{\beta }_1,y^2{\beta }_2)$, where $x,y\in K^{\ast }$.

The quaternion algebra $\mathbb{H}({\beta }_1,{\beta }_2)$ with ${\beta }_1,{\beta }_2\in K^{\ast }$ is either a division algebra or is isomorphic to $\mathbb{H}(-1,-1)\simeq {\mathcal{M}}_2(K)$ [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:

$$0,1,1,2,3,5,8,13,21,\dots ,$$

with the n th term given by the formula

$$f_n=f_{n-1}+f_{n-2},n\ge 2,$$

where $f_0=0$, $f_1=1$.

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

$$h_n=h_{n-1}+h_{n-2},n\ge 2,$$

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:

$$F_n=f_n\cdot 1+f_{n+1}{e}_2+f_{n+2}{e}_3+f_{n+3}{e}_4,$$

for the n th Fibonacci quaternions and

$$H_n=h_n\cdot 1+h_{n+1}{e}_2+h_{n+2}{e}_3+h_{n+3}{e}_4=p{F}_n+q{F}_{n+1},$$

(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

$$f_n=\frac{1}{\sqrt{5}}[{\alpha }^n-{\beta }^n]=\frac{{\alpha }^n}{\sqrt{5}}[1-\frac{{\beta }^n}{{\alpha }^n}],$$

(2.3)

where $\alpha =\frac{1+\sqrt{5}}{2}$ and $\beta =\frac{1-\sqrt{5}}{2}$.

From the above, we obtain the following limit:

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}\mathbf{n}(F_n)& =\underset{n\to \mathrm{\infty }}{lim}(f_n^2+{\beta }_1{f}_{n+1}^2+{\beta }_2{f}_{n+2}^2+{\beta }_1{\beta }_2{f}_{n+3}^2)\\ =\underset{n\to \mathrm{\infty }}{lim}(\frac{{\alpha }^{2n}}{5}+{\beta }_1\frac{{\alpha }^{2n+2}}{5}+{\beta }_2\frac{{\alpha }^{2n+4}}{5}+{\beta }_1{\beta }_2\frac{{\alpha }^{2n+6}}{5})\\ =sgnE({\beta }_1,{\beta }_2)\cdot \mathrm{\infty },\end{array}$$

where $E({\beta }_1,{\beta }_2)=\frac{1}{5}[1+{\beta }_1+2{\beta }_2+5{\beta }_1{\beta }_2+\alpha ({\beta }_1+3{\beta }_2+8{\beta }_1{\beta }_2)]$, since ${\alpha }^2=\alpha +1$ (see [13]).

If $E({\beta }_1,{\beta }_2)>0$, there exists a number $n_1\in \mathbb{N}$ such that for all $n\ge n_1$, we have $\mathbf{n}(F_n)>0$. In the same way, if $E({\beta }_1,{\beta }_2)<0$, there exists a number $n_2\in \mathbb{N}$ such that for all $n\ge n_2$, we have $\mathbf{n}(F_n)<0$. Therefore, for all ${\beta }_1,{\beta }_2\in \mathbb{R}$ with $E({\beta }_1,{\beta }_2)\ne 0$, in the algebra $\mathbb{H}({\beta }_1,{\beta }_2)$ there is a natural number $n_0=max\{n_1,n_2\}$ such that $\mathbf{n}(F_n)\ne 0$. Hence $F_n$ is an invertible element for all $n\ge n_0$. Using the same arguments, we can compute the following limit:

$$\underset{n\to \mathrm{\infty }}{lim}\left(\mathbf{n}\left(H_n^{p,q}\right)\right)=\underset{n\to \mathrm{\infty }}{lim}(h_n^2+{\beta }_1{h}_{n+1}^2+{\beta }_2{h}_{n+2}^2+{\beta }_1{\beta }_2{h}_{n+3}^2)=sgn{E}^{\prime }({\beta }_1,{\beta }_2)\cdot \mathrm{\infty },$$

where $E^{\prime }({\beta }_1,{\beta }_2)=\frac{1}{5}{(p+\alpha q)}^{2}E({\beta }_1,{\beta }_2)$, if $E^{\prime }({\beta }_1,{\beta }_2)\ne 0$ (see [13]).

Therefore, for all ${\beta }_1,{\beta }_2\in \mathbb{R}$ with $E^{\prime }({\beta }_1,{\beta }_2)\ne 0$, in the algebra $\mathbb{H}({\beta }_1,{\beta }_2)$ there exists a natural number $n_0^{\prime }$ such that $\mathbf{n}(H_n^{p,q})\ne 0$, hence $H_n^{p,q}$ is an invertible element for all $n\ge n_0^{\prime }$.

Theorem 2.2 ([[13], Theorem 2.6])

For all ${\beta }_1,{\beta }_2\in \mathbb{R}$ with $E^{\prime }({\beta }_1,{\beta }_2)\ne 0$, there exists a natural number $n^{\prime }$ such that for all $n\ge n^{\prime }$, Fibonacci elements $F_n$ and generalized Fibonacci elements $H_n^{p,q}$ are invertible elements in the algebra $\mathbb{H}({\beta }_1,{\beta }_2)$.

Theorem 2.3 ([[13], Theorem 2.1])

The set ${\mathcal{H}}_n=\{H_n^{p,q}/p,q\in \mathbb{Z},n\ge m,m\in \mathbb{N}\}\cup \{0\}$ is a ℤ-module.

3 Main results

Remark 3.1 We remark that the ℤ-module from Theorem 2.3 is a free ℤ-module of rank two. Indeed, $\phi :\mathbb{Z}\times \mathbb{Z}\to {\mathcal{H}}_n$, $\phi ((p,q))=H_n^{p,q}$ is a ℤ-module isomorphism and $\{\phi (1,0)=F_n,\phi (0,1)=F_{n+1}\}$ is a basis in ${\mathcal{H}}_n$.

Remark 3.2 By extension of scalars, we obtain that $\mathbb{R}{\otimes }_{\mathbb{Z}}{\mathcal{H}}_n$ is an ℝ-vector space of dimension two. A basis is $\{{\overline{e}}_1=1\otimes F_n,{\overline{e}}_2=1\otimes F_{n+1}\}$. We have that $\mathbb{R}{\otimes }_{\mathbb{Z}}{\mathcal{H}}_n$ is an isomorphic with the ℝ-vector space ${\mathcal{H}}_n^{\mathbb{R}}=\{H_n^{p,q}/p,q\in \mathbb{R}\}\cup \{0\}$. A basis in ${\mathcal{H}}_n^{\mathbb{R}}$ is $\{F_n,F_{n+1}\}$.

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

Case 1: $\mathbb{H}({\beta }_1,{\beta }_2)$ is a division algebra

Remark 3.3 Since in this case $E({\beta }_1,{\beta }_2)>0$ for all $n\ge n^{\prime }$ (as in Theorem 2.2), then ${\mathcal{H}}_n^{\mathbb{R}}$ is an Euclidean vector space. Indeed, let $z,w\in {\mathcal{H}}_n^{\mathbb{R}}$, $z=x_1{F}_n+x_2{F}_{n+1}$, $w=y_1{F}_n+y_2{F}_{n+1}$, $x_1,x_2,y_1,y_2\in \mathbb{R}$. The inner product is defined as follows:

$$〈z,w〉=x_1{y}_1\mathbf{n}(F_n)+x_2{y}_2\mathbf{n}(F_{n+1}).$$

We remark that all properties of the inner product are fulfilled. Indeed, since for all $n\ge n^{\prime }$ we have $\mathbf{n}(F_n)>0$ and $\mathbf{n}(F_{n+1})>0$, it results that $〈z,z〉=x_1^2\mathbf{n}(F_n)+x_2^2\mathbf{n}(F_{n+1})=0$ if and only if $x_1=x_2=0$, therefore $z=0$.

On ${\mathcal{H}}_n^{\mathbb{R}}$ with the basis $\{F_n,F_{n+1}\}$, we define the following quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}:{\mathcal{H}}_n^{\mathbb{R}}\to \mathbb{R}$:

$$q_{{\mathcal{H}}_n^{\mathbb{R}}}(x_1{F}_n+x_2{F}_{n+1})=\mathbf{n}(F_n)x_1^2+\mathbf{n}(F_{n+1})x_2^2.$$

Let $Q_{{\mathcal{H}}_n^{\mathbb{R}}}$ be a bilinear form associated to the quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$,

$$\begin{array}{rl}{Q}_{{\mathcal{H}}_n^{\mathbb{R}}}(x,y)& =\frac{1}{2}(q_{{\mathcal{H}}_n^{\mathbb{R}}}(x+y)-q_{{\mathcal{H}}_n^{\mathbb{R}}}(x)-q_{{\mathcal{H}}_n^{\mathbb{R}}}(y))\\ =\mathbf{n}(F_n)x_1{y}_1+\mathbf{n}(F_{n+1})x_2{y}_2.\end{array}$$

The matrix associated to the quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is

$$A=\left(\begin{array}{cc}\mathbf{n}(F_n)& 0\\ 0& \mathbf{n}(F_{n+1})\end{array}\right).$$

We remark that $detA=\mathbf{n}(F_n)\mathbf{n}(F_{n+1})>0$ for all $n\ge n^{\prime }$. Since $E({\beta }_1,{\beta }_2)>0$, therefore $\mathbf{n}(F_n)>0$ for $n>n^{\prime }$. We obtain that the quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is positive definite and the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ 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 $\mathbb{H}({\beta }_1,{\beta }_2)$ is a division algebra, there is a natural number $n^{\prime }$ such that for all $n\ge n^{\prime }$, the Clifford algebra associated to the real vector space ${\mathcal{H}}_n^{\mathbb{R}}$ is isomorphic with the split quaternion algebra $\mathbb{H}(-1,-1)$.

Case 2: $\mathbb{H}({\beta }_1,{\beta }_2)$ is not a division algebra

Remark 3.5 (i) If $E({\beta }_1,{\beta }_2)>0$, then ${\mathcal{H}}_n^{\mathbb{R}}$ is an Euclidean vector space, for all $n\ge n^{\prime }$, as in Theorem 2.2. Indeed, let $z,w\in {\mathcal{H}}_n^{\mathbb{R}}$, $z=x_1{F}_n+x_2{F}_{n+1}$, $w=y_1{F}_n+y_2{F}_{n+1}$, $x_1,x_2,y_1,y_2\in \mathbb{R}$. The inner product is defined as follows:

$$〈z,w〉=x_1{y}_1\mathbf{n}(F_n)+x_2{y}_2\mathbf{n}(F_{n+1}).$$

• (ii) If $E({\beta }_1,{\beta }_2)<0$, then ${\mathcal{H}}_n^{\mathbb{R}}$ is also an Euclidean vector space, for all $n\ge n^{\prime }$, as in Theorem 2.2. Indeed, let $z,w\in {\mathcal{H}}_n^{\mathbb{R}}$, $z=x_1{F}_n+x_2{F}_{n+1}$, $w=y_1{F}_n+y_2{F}_{n+1}$, $x_1,x_2,y_1,y_2\in \mathbb{R}$. The inner product is defined as follows:

  $$〈z,w〉=-x_1{y}_1\mathbf{n}(F_n)-x_2{y}_2\mathbf{n}(F_{n+1}).$$

We have $〈z,z〉=-x_1^2\mathbf{n}(F_n)-x_2^2\mathbf{n}(F_{n+1})$, and since for all $n\ge n^{\prime }$ we have $\mathbf{n}(F_n)<0$ and $\mathbf{n}(F_{n+1})<0$, it results that $〈z,z〉=-x_1^2\mathbf{n}(F_n)-x_2^2\mathbf{n}(F_{n+1})=0$ if and only if $x_1=x_2=0$, therefore $z=0$.

On ${\mathcal{H}}_n^{\mathbb{R}}$ with the basis $\{F_n,F_{n+1}\}$,we define the following quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}:{\mathcal{H}}_n^{\mathbb{R}}\to \mathbb{R}$:

$$q_{{\mathcal{H}}_n^{\mathbb{R}}}(x_1{F}_n\text{+}{x}_2{F}_{n+1})=q_{{\mathcal{H}}_n^{\mathbb{R}}}(x_1{F}_n+x_2{F}_{n+1})=\mathbf{n}(F_n)x_1^2+\mathbf{n}(F_{n+1})x_2^2.$$

Let $Q_{{\mathcal{H}}_n^{\mathbb{R}}}$ be a bilinear form associated to the quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$,

$$\begin{array}{rcl}{Q}_{{\mathcal{H}}_n^{\mathbb{R}}}(x,y)& =& \frac{1}{2}(q_{{\mathcal{H}}_n^{\mathbb{R}}}(x+y)-q_{{\mathcal{H}}_n^{\mathbb{R}}}(x)-q_{{\mathcal{H}}_n^{\mathbb{R}}}(y))\\ =& \mathbf{n}(F_n)x_1{y}_1+\mathbf{n}(F_{n+1})x_2{y}_2.\end{array}$$

The matrix associated to the quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is

$$A=\left(\begin{array}{cc}\mathbf{n}(F_n)& 0\\ 0& \mathbf{n}(F_{n+1})\end{array}\right).$$

We remark that $detA=\mathbf{n}(F_n)\mathbf{n}(F_{n+1})>0$ for all $n\ge n^{\prime }$.

If $E({\beta }_1,{\beta }_2)>0$, therefore $\mathbf{n}(F_n)>0$ for $n>n^{\prime }$. We obtain that the quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is positive definite and the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic with ${Cl}_{2,0}(K)$ which is isomorphic to a split quaternion algebra.

If $E({\beta }_1,{\beta }_2)<0$, therefore $\mathbf{n}(F_n)<0$ for $n>n^{\prime }$. Then the quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is negative definite and the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic with ${Cl}_{0,2}(K)$ which is isomorphic to the quaternion division algebra ℍ.

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

Theorem 3.6 If $\mathbb{H}({\beta }_1,{\beta }_2)$ is not a division algebra, there is a natural number $n^{\prime }$ such that for all $n\ge n^{\prime }$, if $E({\beta }_1,{\beta }_2)>0$, then the Clifford algebra associated to the real vector space ${\mathcal{H}}_n^{\mathbb{R}}$ is isomorphic with the split quaternion algebra $\mathbb{H}(-1,-1)$. If $E({\beta }_1,{\beta }_2)<0$, then the Clifford algebra associated to the real vector space ${\mathcal{H}}_n^{\mathbb{R}}$ is isomorphic to the division quaternion algebra $\mathbb{H}(1,1)$.

Example 3.7 (1) For ${\beta }_1=1$, ${\beta }_2=-1$, we obtain the split quaternion algebra $\mathbb{H}(1,-1)$. In this case, we have $E({\beta }_1,{\beta }_2)=\frac{1}{5}[-5-10\alpha ]<0$ and, for $n^{\prime }=0$, we obtain $\mathbf{n}(F_n)=f_n^2+f_{n+1}^2-f_{n+2}^2-f_{n+3}^2<0$, $\mathbf{n}(F_{n+1})=f_{n+1}^2+f_{n+2}^2-f_{n+3}^2-f_{n+4}^2<0$ for all $n\ge 0$. The quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is negative definite, therefore the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic to ${Cl}_{0,2}(K)$ which is isomorphic to the quaternion division algebra $\mathbb{H}(1,1)$.

• (2) For ${\beta }_1=-2$, ${\beta }_2=-3$, we obtain the split quaternion algebra $\mathbb{H}(-2,-3)$. In this case, we have $E({\beta }_1,{\beta }_2)=\frac{1}{5}[23+43\alpha ]>0$. For $n^{\prime }=0$, we obtain $\mathbf{n}(F_n)=f_n^2-f_{n+1}^2-f_{n+2}^2+f_{n+3}^2>0$, $\mathbf{n}(F_{n+1})=f_{n+1}^2-f_{n+2}^2-f_{n+3}^2+f_{n+4}^2>0$ for all $n\ge 0$. The quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is positive definite, therefore the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic to ${Cl}_{2,0}(K)$ which is isomorphic to the split quaternion algebra $\mathbb{H}(-1,-1)$.

• (3) For ${\beta }_1=2$, ${\beta }_2=-3$, we obtain the split quaternion algebra $\mathbb{H}(2,-3)$. In this case, we have $E({\beta }_1,{\beta }_2)=\frac{1}{5}[-33-44\alpha ]<0$. For $n^{\prime }=0$, we obtain $\mathbf{n}(F_n)=f_n^2+2f_{n+1}^2-3f_{n+2}^2-6f_{n+3}^2<0$, $\mathbf{n}(F_{n+1})=f_{n+1}^2+2f_{n+2}^2-3f_{n+3}^2-6f_{n+4}^2>0$ for all $n\ge 0$. The quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is negative definite, therefore the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic to ${Cl}_{0,2}(K)$ which is isomorphic to the division quaternion algebra $\mathbb{H}(1,-1)$.

• (4) For ${\beta }_1={\beta }_2=-\frac{1}{2}$, we obtain the split quaternion algebra $\mathbb{H}(-\frac{1}{2},-\frac{1}{2})$. Therefore $E({\beta }_1,{\beta }_2)=\frac{3}{20}>0$, and for $n^{\prime }=1$ we obtain $\mathbf{n}(F_n)>0$ and $\mathbf{n}(F_{n+1})>0$. The quadratic form $q_{{\mathcal{H}}_n^{\mathbb{R}}}$ is positive definite, therefore the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic to ${Cl}_{2,0}(K)$ which is isomorphic to the split quaternion algebra $\mathbb{H}(-1,-1)$.

The algorithm

• (1) Let $\mathbb{H}({\beta }_1,{\beta }_2)$ be a quaternion algebra, $\alpha =\frac{1+\sqrt{5}}{2}$ and $E({\beta }_1,{\beta }_2)=\frac{1}{5}[1+{\beta }_1+2{\beta }_2+5{\beta }_1{\beta }_2+\alpha ({\beta }_1+3{\beta }_2+8{\beta }_1{\beta }_2)]$.

• (2) Let V be the ℝ-vector space ${\mathcal{H}}_n^{\mathbb{R}}=\{H_n^{p,q}/p,q\in \mathbb{R}\}\cup \{0\}$.

• (3) If $E({\beta }_1,{\beta }_2)>0$, then the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic to ${Cl}_{2,0}(K)$ which is isomorphic to the split quaternion algebra $\mathbb{H}(-1,-1)$.

• (4) If $E({\beta }_1,{\beta }_2)<0$, then the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic to ${Cl}_{0,2}(K)$ which is isomorphic to the division quaternion algebra $\mathbb{H}(1,1)$.

4 Conclusions

In this paper, we have extended the ℤ-module of the generalized Fibonacci quaternions to a real vector space ${\mathcal{H}}_n^{\mathbb{R}}$. We have proved that the Clifford algebra $C({\mathcal{H}}_n^{\mathbb{R}})$ associated to the tensor algebra $T({\mathcal{H}}_n^{\mathbb{R}})$ is isomorphic to a split quaternion algebra or to a division algebra if $E({\beta }_1,{\beta }_2)=\frac{1}{5}[1+{\beta }_1+2{\beta }_2+5{\beta }_1{\beta }_2+\alpha ({\beta }_1+3{\beta }_2+8{\beta }_1{\beta }_2)]$ 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.