We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions in characteristic 2 that completely determines the isotropy behaviour of the involution. We also give an example of a non-total...

A formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4. As an application, an alternative proof of Shapiro’s “Pfister Factor Conjecture” is given for tensor products of at most five quaternion algebras. Throughout this paper, the characteristic of the base...

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R^3.

an involution or anti-involution is a self-inverse linear mapping. in this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. moreover, properties and geometrical meanings of these matrices will be given as reflections in r^3.

We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra. A quaternion algebra is a central simple algebra of dimension 4 over a field F . Generalizations of the notion of quaternion algebra to other commutative base rings R...

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 ...

We define an invariant of torsors under adjoint linear algebraic groups of type Cn—equivalently, central simple algebras of degree 2n with symplectic involution—for n divisible by 4 that takes values in H(k, μ2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple alg...

Using the ideas and techniques developed by Bayer-Fluckiger, Shapiro and Tignol about hyperbolic involutions of central simple algebras, criteria for the hyperbolicity of involutions of the form σ⊗ τ and σ⊗ ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A and ρ is an involution of a quaternion algebra are obt...

Let F be a Henselian valued field with char(F ) 6= 2, and let S be an inertially split F -central division algebra with involution σ∗ that is trivial on an inertial lift in S of the field Z(S). We prove necessary and sufficient conditions for S to contain a σ∗stable quaternion F -subalgebra, and for (S, σ∗) to decompose into a tensor product of quaternion algebras. These conditions are in terms...

A necessary and sufficient condition for a central simple algebra with involution over a field of characteristic two to be decomposable as a tensor product of quaternion algebras with involution, in terms of its Frobenius subalgebras, is given. It is also proved that a bilinear Pfister form, recently introduced by A. Dolphin, can classify totally decomposable central simple algebras of orthogon...