Let K be a field (characteristic # 2), and let L be a quadratic extension of K. Then L = K{yja) for some aeK. Let D be the quaternion algebra (a, b/K) generated by elements i, j satisfying i = a, j 2 = b, ij = —ji. We will assume that D is a division algebra, i.e. that the quadratic form <1> ~> ~b, ab} is anisotropic. Let — denote the standard involution on L and on D so that y/a = —*Ja on L an...

The hermitian level of composition algebras with involution over a ring is studied. In particular, it is shown that the hermitian level of a composition algebra with standard involution over a semilocal ring, where two is invertible, is always a power of two when finite. Furthermore, any power of two can occur as the hermitian level of a composition algebra with nonstandard involution. Some bou...

in this paper, we define the notions of ultra and involution ideals in $bck$-algebras. then we get the relation among them and other ideals as (positive) implicative, associative, commutative and prime ideals. specially, we show that in a bounded implicative $bck$-algebra, any involution ideal is a positive implicative ideal and in a bounded positive implicative lower $bck$-semilattice, the not...

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 ...

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 ...

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 ...

in this paper we define the notions of ultra and involution ideals in bck-algebras. then we get the relation among them and other ideals as (positive) implicative, associative, commutative and prime ideals. specially, we show that in a bounded implicative bck-algebra, any involution ideal is a positive implicative ideal and in a bounded positive implicative lower bck-semilattice, the notions of...

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...