Division Algebras Research Articles

On the size of the genus of a division algebra

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) as the set of classes [D′] ∈ Br(K) in the Brauer group of K represented by central division algebras D′ of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.

Linearly Minimal Jordan Algebras of Characteristic Other Than 2

It is proved that every nontrivial linearly minimal Jordan algebra of characteristic other than 2 is a division algebra. Then Zel’manov’s classification of Jordan division algebras is applied to show that such an algebra is a field.

A note on the real division algebras with non-trivial derivations

We show that every 4-dimensional real division algebra having non-zero derivations is obtained from the real algebra C by a special kind of duplication process accompanied by an appropriate isotopy. Also this process produces new examples in dimension 8. Mathematics Subject Classification: 17A35, 17A36

Unifying Ancient and Modern Geometries Through Octonions

We show the first unified description of some of the oldest known geometries such as the Pappus’ theorem with more modern ones like Desargues' theorem, Monge's theorem and Ceva's theorem, through octonions, the highest normed division algebra in eight dimensions. We also show important applications in hadronic physics, giving a full description of the algebra of color applicable to quark physics, and comment on further applications.

A remark on certain filtrations on the inner automorphism groups of central division algebras over local number fields

A remark on certain filtrations on the inner automorphism groups of central division algebras over local number fields

Tensor products of nonassociative cyclic algebras

We study the tensor product of an associative and a nonassociative cyclic algebra. The condition for the tensor product to be a division algebra equals the classical one for the tensor product of two associative cyclic algebras by Albert or Jacobson, if the base field contains a suitable root of unity. Stronger conditions are obtained in special cases. Applications to space–time block coding are discussed.

On the images of multilinear maps of matrices over finite-dimensional division algebras

Let R be a central simple algebra finite-dimensional over its center F of characteristic 0. We will show that every element of reduced trace 0 in R can be expressed as [a,[c,b]]+λ[c,[a,b]] for some a,b,c∈R where λ≠0,−1. In addition, let D be a division algebra satisfying the conditions above. We will also show that the set of values of any nonzero multilinear polynomial of degree at most three, with coefficients from the center F of D, evaluated on Mk(D), k≥2, contains all matrices of reduced trace 0.

Essential dimension and error-correcting codes

One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GLn /μm, i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m. In this paper we study the essential dimension of groups of the form G = (GLn1 × · · · ×GLnr )/C , where C is a central subgroup of GLn1 × · · · ×GLnr . Equivalently, we are interested in the essential dimension of a generic r-tuple (A1, . . . , Ar) of central simple algebras such that deg(Ai) = ni and the Brauer classes of A1, . . . , Ar satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of C via the error-correcting code Code(C) which we naturally associate to C. We focus on the case where n1, . . . , nr are powers of the same prime. The upper and lower bounds on ed(G) we obtain are expressed in terms of coding-theoretic parameters of Code(C), such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r ≥ 3; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (GLn1 × · · · ×GLnr )/C. This description and its corollaries are used throughout the paper.

Excellence of function fields of conics

For every generalized quadratic form or hermitian form over a division algebra, the anisotropic kernel of the form obtained by scalar extension to the function field of a smooth projective conic is defined over the field of constants. Tjhe proof does not require any hypothesis on the characteristic.

Elliptical affine shape distributions for real normed division algebras

The statistical affine shape theory is set in this work in the context of real normed division algebras. The general densities apply for every field: real, complex, quaternion, octonion, and for any noncentral and non-isotropic elliptical distribution; then the separated published works about real and complex shape distributions can be obtained as corollaries by a suitable selection of the field parameter and univariate integrals involving the generator elliptical function. As a particular case, the complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients. • Statistical affine shape theory in the context of real normed division algebras is studied. • Affine shape density is obtained. • The complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients.

Matrices Over Nondivision Algebras Without Eigenvalues

We are concerned with matrices over nondivision algebras and show by an example from an \({\mathbb{R}^{4}}\) algebra that these matrices do not necessarily have eigenvalues, even if these matrices are invertible. The standard condition for eigenvectors \({\rm x \neq 0}\) will be replaced by the condition that x contains at least one invertible component which is the same as \({\rm x \neq 0}\) for division algebras. The topic is of principal interest, and leads to the question what qualifies a matrix over a nondivision algebra to have eigenvalues. And connected with this problem is the question, whether these matrices are diagonalizable or triangulizable and allow a Schur decomposition. There is a last section where the question whether a specific matrix A has eigenvalues is extended to all eight \({\mathbb{R}^{4}}\) algebras by applying numerical means. As a curiosity we found that the considered matrix A over the algebra of tessarines, which is a commutative algebra, introduced by Cockle (Phil Mag 35(3):434–437, 1849; http://www.oocities.org/cocklebio/), possesses eigenvalues.

Simple graded division algebras over the field of real numbers

We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.

Veronesean representations of Moufang planes

In 1901 Severi [18] proved that the complex quadric Veronese variety is determined by three algebraic/differential geometric properties. In 1984 Mazzocca and Me lone [10] obtained a combinatorial analogue of this result for finite quadric Veronese varieties. We make further abstraction of these properties to characterize Veronesean representations of all the Moufang projective planes defined over a quadratic alternative division algebra over an arbitrary field. In the process, new Veroneseans over a nonperfect field of characteristic 2 (related to purely inseparable field extensions) are found, and their corresponding projective representations of the associated groups studied. We show that these representations are indecomposable, but reducible, and determine their (irreducible) quotient and kernel.

UNITARY GRASSMANNIANS OF DIVISION ALGEBRAS

We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [17], where a similar issue for orthogonal and symplectic involutions has been treated, the present paper completes the study of Grassmannians of isotropic right ideals of division algebras.

Matric variate Pearson type II-Riesz distribution

The Pearson type II distribution is well known and is used in the general framework of real normed division algebras and Riesz distribution theory. Also, the so called Pearson type II-Riesz distribution, based on the Kotz–Riesz distribution, is presented in a unified way valid in the context of real, complex, quaternion and octonion random matrices. Specifically, the central nonsingular matric variate generalised Pearson type II-Riesz distribution and beta-Riesz type I distributions are derived in the addressed multiple numerical field settings.

Singular and Totally Singular Unitary Spaces

In this article, we present a decomposition theorem for unitary spaces over a central simple division algebra with involution in characteristic 2. This is a generalization of a decomposition result for quadratic forms in characteristic 2 from [4] and extends a generalization of the Witt decomposition theorem for nonsingular spaces in this setting to cover spaces that may be singular.

Homomorphisms and Involutions of Unramified Henselian Division Algebras

Let K be a Henselian field with a residue field $$ \overline{K} $$ , and let A1, A2 be finite-dimensional division unramified K-algebras with residue algebras Ā 1 and Ā 2 Further, let HomK(A1,A2) be the set of nonzero K-homomorphisms from A1 to A2. It is proved that there is a natural bijection between the set of nonzero $$ \overline{K} $$ -homomorphisms from Ā 1 to Ā 2 and the factor set of HomK and the factor set of HomK(A1,A2) under the equivalence relation: ϕ 1 ∼ ϕ 2 ⇔ there exists m ∈ 1 +MA2 such that ϕ2 = ϕ1 im, where im is the inner automorphism of A2 induced by m. A similar result is obtained for unramified algebras with involutions. Bibliography: 7 titles.

The ring of integer valued polynomials on [formula omitted] matrices and its integral closure

Let Mn(Z) denote the ring of n×n matrices with integer entries and IntQ(Mn(Z))⊆Q[x] the algebra of polynomials that preserve Mn(Z), i.e. polynomials for which f(M)∈Mn(Z) if M∈Mn(Z). The aim of this paper is to shed some light on this algebra by establishing two results. The first, general, result is to show that when this algebra is localized at a prime p, its integral closure can be identified with IntQ(Rn,p), the ring of polynomials in Q[x] preserving the maximal order in a division algebra of dimension n2 over Qp, the p-adic numbers. This is of interest computationally because one of the authors showed in an earlier paper that Bhargava's method of p-ordering can be extended to construct a regular basis for algebras of this type. Thus we have a method of computing an upper bound for the size of IntQ(Mn(Z)). Our second result is a computational method for constructing a p-local basis for IntQ(Mn(Z)) itself in the case n=2 in low degrees. These results together give an estimate of the difference in size between IntQ(M2(Z)) and its closure and, in particular, give constructions of polynomials in the latter but not in the former.

L ∞ norms of holomorphic modular forms in the case of compact quotient

Abstract We prove a sub-convex estimate for the sup-norm of L 2-normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over ℚ. More precisely we show that when the L 2 norm of an eigenfunction f is one, ∥ f ∥ ∞ ≪ ε k 1 2 - 1 33 + ε $ \Vert f \Vert _\infty \ll _\varepsilon k^{\frac{1}{2} - \frac{1}{33} +\varepsilon } $ for any ε > 0 ${\varepsilon >0}$ and for all k sufficiently large.

Semisimple Hopf actions on Weyl algebras

We study actions of semisimple Hopf algebras H on Weyl algebras A over an algebraically closed field of characteristic zero. We show that the action of H on A must factor through a group action; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.