The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse concept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion algebra problems can be transformed into complex algebra problems to express ...

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several ...

In 1988, J.R. Faulkner has given a procedure to construct an octonion algebra on a finite dimensional unitary alternative algebra of degree three over a field K. Here we use a similar procedure to get a quaternion algebra. Then we obtain some conditions for these octonion and quaternion algebras to be split or division algebras. Then we consider the implications of the found conditions to the u...

1. The statement This brief note is devoted to a simple (and well-known) result in noncommutative algebra, which is not deep but nevertheless subtler than it appears. It concerns the so-called quaternion algebras: Definition 1.1. Let k be a commutative ring1. Let a ∈ k and b ∈ k. The quaternion algebra Ha,b is defined to be the k-algebra with generators i and j and relations i2 = a, j2 = b, ij ...

The subject of this paper is the Brauer group of a nonsingular complex projective variety. More specifically, we study the question of whether a 2-torsion element of the cohomological Brauer group is representable by a quaternion algebra over the generic point. Using intersection theory – on schemes and on algebraic stacks – we are able to describe an obstruction to such a representation, and t...

1. The statement This brief note is devoted to a simple (and well-known) result in noncommutative algebra, which is not deep but nevertheless subtler than it appears. It concerns the so-called quaternion algebras: Definition 1.1. Let k be a commutative ring1. Let a ∈ k and b ∈ k. The quaternion algebra Ha,b is defined to be the k-algebra with generators i and j and relations i2 = a, j2 = b, ij ...

The construction of a class of associative composition algebras qn on R 4 generalizing the wellknown quaternions Q provides an explicit representation of the universal enveloping algebra of the real three-dimensional Lie algebras having tracefree adjoint representations (class A Bianchi type Lie algebras). The identity components of the four-dimensional Lie groups GL(qn,l) Cqn (general linear g...

In terms of generators and defining relations, a description is given of the Hochschild cohomology algebra for one of the series of local algebras of quaternion type. As a corollary, the Hochschild cohomology algebra is described for the group algebras of generalized quaternion groups over algebraically closed fields of characteristic 2. Introduction Let R be a finite-dimensional algebra over a...

In this chapter, the problem of quaternion beamformer based on linear and widely linear hypercomplex processing is investigated in scenarios, where there exist one signal and one interference that are uncorrelated. First, we introduce brief information about the quaternion algebra and a quaternion model of linear symmetric array with twocomponent electromagnetic (EM) vector-sensors is presented...