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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:VC satisfying the property

i ( x ) 2 =q(x) 1 C ,xV,

such that for any K-algebra A and any K linear map γ:VA with γ 2 (x)=q(x) 1 A , xV, there exists a unique K-algebra morphism γ :CA with γ= γ i.

Such an algebra can be constructed using the tensor algebra associated to a vector space V. Let T(V)=KV(VV) 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 xxq(x)1 for all xV. 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 ,1sn,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 ,n2,

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 ,n2,

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,qZ,nm,mN}{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,qR}{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 [510α]<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 n0. 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 n0. 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 [3344α]<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 n0. 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,qR}{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

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Keywords

  • Clifford algebras
  • generalized Fibonacci quaternions