Open Access

A Clifford algebra associated to generalized Fibonacci quaternions

Advances in Difference Equations20142014:279

https://doi.org/10.1186/1687-1847-2014-279

Received: 9 July 2014

Accepted: 14 October 2014

Published: 31 October 2014

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.

Keywords

Clifford algebrasgeneralized Fibonacci quaternions

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 i j = j i , i k = k i , j k = k j .

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 det A = 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 det A = 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 10 α ] < 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 + 43 α ] > 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 44 α ] < 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.

Declarations

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.

Authors’ Affiliations

(1)
Faculty of Mathematics and Computer Science, Ovidius University, Constanta, Romania

References

  1. Lewis DW: Quaternion algebras and the algebraic legacy of Hamilton’s quaternions. Ir. Math. Soc. Bull. 2006, 57: 41-64.MATHGoogle Scholar
  2. Knus MA: Quadratic Forms, Clifford Algebras and Spinors. Univ. Estadual de Campinas, Sao Paulo; 1988.MATHGoogle Scholar
  3. Lam TY: Quadratic Forms over Fields. Am. Math. Soc., Providence; 2004.Google Scholar
  4. El Kinani EH, Ouarab A:The embedding of U q ( s l ( 2 ) ) and sine algebras in generalized Clifford algebras. Adv. Appl. Clifford Algebras 1999, 9(1):103-108. 10.1007/BF03041942MathSciNetView ArticleMATHGoogle Scholar
  5. Koç C: C -Lattices and decompositions of generalized Clifford algebras. Adv. Appl. Clifford Algebras 2010, 20(2):313-320. 10.1007/s00006-009-0178-zView ArticleMathSciNetMATHGoogle Scholar
  6. O’Meara OT: Introduction to Quadratic Forms. Springer, Berlin; 1962.MATHGoogle Scholar
  7. Smith TL: Decomposition of generalized Clifford algebras. Q. J. Math. 1991, 42: 105-112. 10.1093/qmath/42.1.105View ArticleMathSciNetMATHGoogle Scholar
  8. Flaut C, Shpakivskyi V: Some identities in algebras obtained by the Cayley-Dickson process. Adv. Appl. Clifford Algebras 2013, 23(1):63-76. 10.1007/s00006-012-0344-6MathSciNetView ArticleMATHGoogle Scholar
  9. Schafer RD: An Introduction to Nonassociative Algebras. Academic Press, New York; 1966.MATHGoogle Scholar
  10. Tărnăuceanu M: A characterization of the quaternion group. An. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 2013, 21(1):209-214.MathSciNetGoogle Scholar
  11. Koshy T: Fibonacci and Lucas Numbers with Applications. Wiley-Interscience, New York; 2001.View ArticleMATHGoogle Scholar
  12. Horadam AF: A generalized Fibonacci sequence. Am. Math. Mon. 1961, 68: 455-459. 10.2307/2311099MathSciNetView ArticleMATHGoogle Scholar
  13. Flaut C, Shpakivskyi V: On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. Adv. Appl. Clifford Algebras 2013, 23(3):673-688. 10.1007/s00006-013-0388-2MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Flaut; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.