Tensor Product Of Quaternion Algebras Research Articles

Hyperquaternionic Representations of Conic Sections

In this paper, by means of an isomorphism, we express the Clifford algebra Cl5,3 as hyperquaternion algebra H ⊗ H ⊗ H ⊗ H (a four-fold tensor product of quaternion algebras) and we provide the hyperquaternionic approach to the inner product null space (IPNS) representation of conic sections.

Hyperquaternion Conformal Groups

The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as $$O(p+1,q+1)$$ isomorphic to the nD conformal group with $$n=p+q.$$ The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory.

Dual Hyperquaternion Poincaré Groups

A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators. An extended multivector calculus is introduced (allowing a possible sign change of the metric or of the exterior product). The Poincaré groups are formulated as a dual extension of hyperquaternion pseudo-orthogonal groups. The canonical decomposition and the invariants are discussed. As concrete example, the 4D Poincaré group is examined together with a numerical application. Finally, the hyperquaternion representation is compared to the quantum mechanical one. Potential applications include in particular, moving reference frames and computer graphics.

On the possible decompositions of totally decomposable orthogonal involutions in characteristic two

In characteristic two, we determine all possible total decompositions of a totally decomposable algebra with orthogonal involution into tensor products of quaternion algebras with involution.

Involutions on tensor products of quaternion algebras

We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra , endowed with an orthogonal involution. We construct examples of algebras isomorphic to a tensor product of quaternion algebras with k split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with k factors with involution. This extends an earlier result of Sivatski where the algebra considered is of degree 8 and index 4, and endowed with some orthogonal involution.

Hyperquaternions: A New Tool for Physics

A hyperquaternion formulation of Clifford algebras in n dimensions is presented. The hyperquaternion algebra is defined as a tensor product of quaternion algebras $$\mathbb {H}$$ (or a subalgebra thereof). An advantage of this formulation is that the hyperquaternion product is defined independently of the choice of the generators. The paper gives an explicit expression of the generators and develops a generalized multivector calculus. Due to the isomorphism $$\mathbb {H}\otimes \mathbb {H}\simeq m(4, \mathbb {R})$$ , hyperquaternions yield all real, complex and quaternion square matrices. A hyperconjugation is introduced which generalizes the concepts of transposition, adjunction and transpose quaternion conjugate. As applications, simple expressions of the unitary and unitary symplectic groups are obtained. Finally, the hyperquaternions are compared, in the context of physical applications, to another algebraic structure based on octonions which has been proposed recently.

Splitting fields of central simple algebras of exponent two

By Merkurjev's Theorem every central simple algebra of exponent two is Brauer equivalent to a tensor product of quaternion algebras. In particular, if every quaternion algebra over a given field is split, then there exists no central simple algebra of exponent two over this field. This note provides an independent elementary proof for the latter fact.

Totally decomposable quadratic pairs

In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic pair is adjoint to a quadratic Pfister form. This result is new in characteristic two, otherwise it is equivalent to the Pfister Factor Conjecture proven in Becher (Invent Math 173:1–6, 2008).

On totally decomposable algebras with involution in characteristic two

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 orthogonal type.

Square-central and Artin–Schreier elements in division algebras

We study the behavior of square-central elements and Artin–Schreier elements in division algebras of exponent 2 and degree a power of 2. We provide chain lemmas for such elements in division algebras over 2-fields F of cohomological 2-dimension \({{\rm cd}_2(F) \leq 2}\) and deduce a common slot lemma for tensor products of quaternion algebras over such fields. We also extend to characteristic 2 a theorem proven by Merkurjev for characteristic not 2 on the decomposition of any central simple algebra of exponent 2 and degree a power of 2 over a field F with \({{\rm cd}_2(F) \leq 2}\) as a tensor product of quaternion algebras.

Addendum: "Chain equivalences for symplectic bases, quadratic forms and tensor products of quaternion algebras"

We show how the chain lemma for quaternion algebras over fields of F ≠ 2 and [Formula: see text] can be strengthened so that up to two slots are changed at a time.

Chain equivalences for symplectic bases, quadratic forms and tensor products of quaternion algebras

We present a set of generators for the symplectic group which is different from the well-known set of transvections, from which the chain equivalence for quadratic forms in characteristic 2 is an immediate result. Based on the chain equivalences for quadratic forms, both in characteristic 2 and not 2, we provide chain equivalences for tensor products of quaternion algebras over fields with no nontrivial 3-fold Pfister forms. The chain equivalence for biquaternion algebras in characteristic 2 is also obtained in this process, without any assumption on the base-field.

Decomposable and indecomposable algebras of degree $$8$$ 8 and exponent 2

We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let B be a biquaternion algebra over F(√a) with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether B has a descent to F. This invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2-cohomological dimension 3 and over a field M(t) where the u-invariant of M is 8 and t is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by Merkurjev in “Appendix”.

On split products of quaternion algebras with involution in characteristic two

The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution is shown to have an affirmative answer.

On the u-invariant of hermitian forms

Let K be a field of characteristic not 2 and A a central simple algebra with an involution σ. A result of Mahmoudi provides an upper bound for the u-invariants of hermitian forms and skew-hermitian forms over (A, σ) in terms of the u-invariant of K. In this paper we give a different upper bound when A is a tensor product of quaternion algebras and σ is a the tensor product of canonical involutions. We also show that our bounds are sharper than those of Mahmoudi.

Algebras with involution that become hyperbolic over the function field of a conic

We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain (Q,bar), even though it contains Q and is totally decomposable into a tensor product of quaternion algebras

Discriminant of Symplectic Involutions

We define an invariant of torsors under adjoint linear algebraic groups of type C_n-equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that takes values in H^3(F, mu_2). 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 algebra of degree 8 with symplectic involution can be decomposed as a tensor product of quaternion algebras with involution.

Pfister involutions

The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discussed.

QUADRATIC CORESTRICTION, -EMBEDDING PROBLEMS, AND EXPLICIT CONSTRUCTION

ABSTRACT Let be a quadratic extension of fields of characteristic not 2. We give a natural correspondence between cup-products of square classes of and certain Galois embedding problems over . Under this correspondence, the obstruction to the embedding problem associated to is the corestriction from to of the quaternion algebra over given by . We reduce the explicit construction problem for such embedding problems to the construction of an -algebra isomorphism from an algebra expressed as a tensor product of quaternion algebras to a matrix ring. We apply these results to determine the obstructions to, and a method for explicit construction for, all -embedding problems extending or , where the subdirect products are taken over a common factor group .

Embedding Obstructions for the Dihedral, Semidihedral, and Quaternion 2-Groups

For each of the dihedral, semidihedral, and quaternion 2-groups, we represent the obstructions to certain Brauer problems as tensor products of quaternion algebras. Then we reduce various embedding problems with cyclic 2-kernels into two Brauer problems, thus finding the obstructions in some specific cases.