Abstract
Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of these results to the algebraic equations and De Moivre’s formula in generalized quaternion algebras and in generalized octonion division algebras.
Highlights
Let γ1, γ2 ∈ R \ {0}, let H (γ1, γ2) be the generalized quaternion algebra with basis {1, e1, e2, e3} and H(1, 1) be the quaternion division algebra with basis {1, i, j, k}
Let O(α, β, γ) be a generalized octonion algebra over R, with basis {1, f1, ..., f7} and the multiplication given in the following table:
Due to Proposition 1, in the present paper we reduced the study of an algebraic equation in an arbitrary algebra H (γ1, γ2) with γ1, γ2 ∈ R \ {0} to study of the corresponding algebraic equation in one of the following two algebras: division quaternion algebra or coquaternion algebra
Summary
Let γ1, γ2 ∈ R \ {0}, let H (γ1, γ2) be the generalized quaternion algebra with basis {1, e1, e2, e3} and H(1, 1) be the quaternion division algebra with basis {1, i, j, k}. Let O(α, β, γ) be a generalized octonion algebra over R, with basis {1, f1, ..., f7} and the multiplication given in the following table:. De Moivre’s formula and Euler’s formula in generalized quaternion algebras, founded in [11], was proved using this new method, for γ1, γ2 > 0 With this technique, the above mentioned results were obtained for the octonions. Since the spaces H (γ1, γ2) and H(1, 1) are normed spaces, the continuity of A is equivalent with the boundedness of A, i.e. there is a real constant c such that for all x ∈ H (γ1, γ2) , we have In this situation, the algebra H(γ1, γ2) is isomorphic with H(1, −1). Since each algebra H(γ1, γ2) is isomorphic with algebra of quaternions or coquaternions, it results that the above operators provide us a simple way to generalize known results in these two algebras to generalized quaternion algebra
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