A Clifford algebra associated to generalized Fibonacci quaternions

Cristina Flaut

doi:10.1186/1687-1847-2014-279

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.

Highlights

In, WK Clifford discovered Clifford algebras
The real vector space with this quadratic form is usually denoted by Rr,s and the Clifford algebra on Rr,s is denoted by Clr,s(R)
If H(β, β ) 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 HnR is isomorphic with the split quaternion algebra H(, – )

Summary

Introduction

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) = q(x) · C, ∀x ∈ V , such that for any K -algebra A and any K linear map γ : V → A with γ (x) = q(x) · 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. The most important Clifford algebras are those defined over real and complex vector spaces equipped with nondegenerate quadratic forms. The real vector space with this quadratic form is usually denoted by Rr,s and the Clifford algebra on Rr,s is denoted by Clr,s(R).

Preliminaries

Conclusions