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 , denoting , if , therefore the algebra A is split. If , 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.
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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 be a K-vector space equipped with a nondegenerate quadratic form over the field K. A Clifford algebra for is a K-algebra C with a linear map satisfying the property
such that for any K-algebra A and any K linear map with , , there exists a unique K-algebra morphism with .
Such an algebra can be constructed using the tensor algebra associated to a vector space V. Let be the tensor algebra associated to the vector space V, and let be the two-sided ideal of generated by all elements of the form for all . The associated Clifford algebra is the factor algebra (see [2, 3]).
Theorem 1.1 (Poincaré-Birkhoff-Witt [[2], p.44])
If is a basis of V, then the set is a basis in .
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:
where is the dimension of the vector space. The pair of integers is called the signature of the quadratic form. The real vector space with this quadratic form is usually denoted by and the Clifford algebra on is denoted by . For other details about Clifford algebras, the reader is referred to [4–6] and [7].
Example 1.2
-
(i)
For , we have .
-
(ii)
For , , it results that is a two-dimensional algebra generated by a single vector such that , and therefore . For , it follows that .
-
(iii)
For , , the algebra is a four-dimensional algebra spanned by the set . Since and , we obtain that this algebra is isomorphic to the division quaternions algebra ℍ.
-
(iv)
For , or , , we obtain the algebra which is isomorphic with a split (i.e., nondivision) quaternion algebra [8].
2 Preliminaries
Let be a generalized real quaternion algebra, the algebra of the elements of the form , where , , and the elements of the basis satisfy the following multiplication table:
We denote by the norm of a real quaternion a. The norm of a generalized quaternion has the following expression . For , we obtain the real division algebra ℍ, with the basis , where and , , .
Proposition 2.1 ([[3], Proposition 1.1])
The quaternion algebra is isomorphic to quaternion algebra , where .
The quaternion algebra with is either a division algebra or is isomorphic to [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:
with the n th term given by the formula
where , .
In [12], the author generalized Fibonacci numbers and gave many properties of them:
where , , 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:
For the generalized real quaternion algebra, the Fibonacci quaternions and generalized Fibonacci quaternions are defined in the same way:
for the n th Fibonacci quaternions and
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 , respectively . 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
where and .
From the above, we obtain the following limit:
where , since (see [13]).
If , there exists a number such that for all , we have . In the same way, if , there exists a number such that for all , we have . Therefore, for all with , in the algebra there is a natural number such that . Hence is an invertible element for all . Using the same arguments, we can compute the following limit:
where , if (see [13]).
Therefore, for all with , in the algebra there exists a natural number such that , hence is an invertible element for all .
Theorem 2.2 ([[13], Theorem 2.6])
For all with , there exists a natural number such that for all , Fibonacci elements and generalized Fibonacci elements are invertible elements in the algebra .
Theorem 2.3 ([[13], Theorem 2.1])
The set 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, , is a ℤ-module isomorphism and is a basis in .
Remark 3.2 By extension of scalars, we obtain that is an ℝ-vector space of dimension two. A basis is . We have that is an isomorphic with the ℝ-vector space . A basis in is .
Let be the tensor algebra associated to the ℝ-vector space , and let be the Clifford algebra associated to the tensor algebra . From Theorem 1.1, it results that this algebra has dimension four.
Case 1: is a division algebra
Remark 3.3 Since in this case for all (as in Theorem 2.2), then is an Euclidean vector space. Indeed, let , , , . The inner product is defined as follows:
We remark that all properties of the inner product are fulfilled. Indeed, since for all we have and , it results that if and only if , therefore .
On with the basis , we define the following quadratic form :
Let be a bilinear form associated to the quadratic form ,
The matrix associated to the quadratic form is
We remark that for all . Since , therefore for . We obtain that the quadratic form is positive definite and the Clifford algebra associated to the tensor algebra is isomorphic to 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 is a division algebra, there is a natural number such that for all , the Clifford algebra associated to the real vector space is isomorphic with the split quaternion algebra .
Case 2: is not a division algebra
Remark 3.5 (i) If , then is an Euclidean vector space, for all , as in Theorem 2.2. Indeed, let , , , . The inner product is defined as follows:
-
(ii)
If , then is also an Euclidean vector space, for all , as in Theorem 2.2. Indeed, let , , , . The inner product is defined as follows:
We have , and since for all we have and , it results that if and only if , therefore .
On with the basis ,we define the following quadratic form :
Let be a bilinear form associated to the quadratic form ,
The matrix associated to the quadratic form is
We remark that for all .
If , therefore for . We obtain that the quadratic form is positive definite and the Clifford algebra associated to the tensor algebra is isomorphic with which is isomorphic to a split quaternion algebra.
If , therefore for . Then the quadratic form is negative definite and the Clifford algebra associated to the tensor algebra is isomorphic with 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 is not a division algebra, there is a natural number such that for all , if , then the Clifford algebra associated to the real vector space is isomorphic with the split quaternion algebra . If , then the Clifford algebra associated to the real vector space is isomorphic to the division quaternion algebra .
Example 3.7 (1) For , , we obtain the split quaternion algebra . In this case, we have and, for , we obtain , for all . The quadratic form is negative definite, therefore the Clifford algebra associated to the tensor algebra is isomorphic to which is isomorphic to the quaternion division algebra .
-
(2)
For , , we obtain the split quaternion algebra . In this case, we have . For , we obtain , for all . The quadratic form is positive definite, therefore the Clifford algebra associated to the tensor algebra is isomorphic to which is isomorphic to the split quaternion algebra .
-
(3)
For , , we obtain the split quaternion algebra . In this case, we have . For , we obtain , for all . The quadratic form is negative definite, therefore the Clifford algebra associated to the tensor algebra is isomorphic to which is isomorphic to the division quaternion algebra .
-
(4)
For , we obtain the split quaternion algebra . Therefore , and for we obtain and . The quadratic form is positive definite, therefore the Clifford algebra associated to the tensor algebra is isomorphic to which is isomorphic to the split quaternion algebra .
The algorithm
-
(1)
Let be a quaternion algebra, and .
-
(2)
Let V be the ℝ-vector space .
-
(3)
If , then the Clifford algebra associated to the tensor algebra is isomorphic to which is isomorphic to the split quaternion algebra .
-
(4)
If , then the Clifford algebra associated to the tensor algebra is isomorphic to which is isomorphic to the division quaternion algebra .
4 Conclusions
In this paper, we have extended the ℤ-module of the generalized Fibonacci quaternions to a real vector space . We have proved that the Clifford algebra associated to the tensor algebra is isomorphic to a split quaternion algebra or to a division algebra if 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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DOI: https://doi.org/10.1186/1687-1847-2014-279