Non-commutative Division Algebra Research Articles

Division algebras and MRD codes from skew polynomials

AbstractLet $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\;\sigma,\delta ]$ a skew polynomial ring.Using skew polynomials $f\in R$ , we construct division algebras and maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.

A Generalization of Quaternions and Their Applications

There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the 4×4 matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by 4×4 permutation matrices of the C2×C2 group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical tunneling through an arbitrary one-dimensional (1D) conduction band energy profile. This demonstrates that six different spinors (4×4 matrices) can be used to represent the amplitudes of the left and right propagating waves in a 1D device.

Quaternion rings over local rings

In 1843, Hamilton (1805–1865) discovered the 4-dimensional division algebra [Formula: see text] over the field [Formula: see text] of real numbers. Hamilton’s big discovery is the following beautiful multiplications for the basis [Formula: see text] :[Formula: see text] [Formula: see text] contains the field [Formula: see text] of complex numbers; therefore [Formula: see text]. Frobenius (1849–1917) showed the following outstanding theorem. Theorem (Frobenius). Up to isomorphism, the only finite-dimensional non-commutative division algebra over [Formula: see text] is [Formula: see text]. Starting from given any ring [Formula: see text] and a free right [Formula: see text]-module [Formula: see text], the quaternion ring [Formula: see text] is canonically defined by the multiplications [Formula: see text]. For a commutative field [Formula: see text] with [Formula: see text], [Formula: see text] is a division ring or isomorphic to the ring of [Formula: see text] matrices over [Formula: see text]. This is a classical theorem. For nonzero [Formula: see text] in the center of a ring [Formula: see text], the generalized quaternion ring [Formula: see text] is defined. [Formula: see text] is [Formula: see text]. In [I. Kikumasa, K. Koike and K. Oshiro, Complex rings and quaternion rings, East-West J. Math. 21 (2019) 1–19; I. Kikumasa, G. Lee and K. Oshiro, Complex Rings, Quaternion Rings and Octonion Rings (Lambert Academic Publishing, 2020); G. Lee and K. Oshiro, Quaternion rings and octonion rings, Front. Math. China 12(1) (2017) 143–155], quaternion rings and generalized quaternion rings over division rings or other rings are studied. Now, in this paper, from ring theoretic viewpoints, we study quaternion rings [Formula: see text] and generalized quaternion rings [Formula: see text] over local rings [Formula: see text]. From our results, we can clearly look over several classical results on [Formula: see text] and [Formula: see text] over commutative fields [Formula: see text].

Узагальнений аналіз алгебр з діленням розмірності два

The mathematical properties of division algebras of dimension 2 are investigated on the basis of the analysis of possible values of the parameters introduced into the laws of composition of basic elements. Generalized expressions for calculating the inverse and neutral elements of the indicated algebras are given. The relations of the parameters defining the normalized division algebras are determined. Possibilities of application of linear orthogonal transformations for the analysis of isomorphism of such algebras are considered. The concept of an exponential function is introduced to represent the elements of the considered non-commutative division algebra in exponential form.

Free subgroups in maximal subgroups of GLn(D)

ABSTRACTLet D be a noncommutative finite dimensional F-central division algebra and M a noncommutative maximal subgroup of GLn(D). It is shown that either M contains a noncyclic free subgroup or M is absolutely irreducible and there exists a unique maximal subfield K of Mn(D) such that K*M, K∕F is Galois with Gal(K∕F)≅M∕K* and Gal(K∕F) is a finite simple group.

Algebra and geometry of Hamilton’s quaternions

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three-dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.

On Linear Groups of Degree 2 Over a Noncommutative Division Algebra

ABSTRACTLet k be a field of characteristic different from 2 and K a noncommutative division k-algebra of finite dimension over k. In the paper, subgroups of the general linear group GL2(K) that contain a subgroup which is conjugate in GL2(K) to the group of matrices with r ∈ k are described.

Existence of Nonabelian Free Subgroups in the Maximal Subgroups of GLn(D)

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GLn(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLn(D)(K*)=M, K* ◁ M, K/F is Galois with Gal (K/F) ≅ M/K*, and F[M]=Mn(D). In particular, when F is global or local, it is proved that if ([D:F], Char (F))=1, then every non-abelian maximal subgroup of GL1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5.

Central units of integral group rings

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\Z G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $\mathcal{Z} (\U (\Z G))$ of the unit group $\U (\Z G)$ in case $G$ is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in $\mathcal{Z}(\U(\Z G))$ for all finite strongly monomial groups $G$. We call these units generalized Bass units. Finally, we show that the commutator group $\U(\Z G)/\U(\Z G)'$ and $\mathcal{Z}(\U(\Z G))$ have the same rank if $G$ is a finite group such that $\Q G$ has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra, or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of $\Q$. This allows us to prove that in this case the natural images of the Bass units of $\Z G$ generate a subgroup of finite index in $\U(\Z G)/\U(\Z G)'$.

Discrete subgroups acting transitively on vertices of a Bruhat–Tits building

We describe all the discrete subgroups of Ad(G0)(F ) o Aut(F ) which act transitively on the set of vertices of B = B(F;G0) the Bruhat-Tits building of a pair (F;G0) of a characteristic zero non-archimedean local eld and a simply-connected absolutely almost simple F -group if B is of dimension at least 4. In fact, we classify all the maximal such subgroups. We show that there are exactly eleven families of such subgroups and explicitly construct them. Moreover, we show that four of these families act simply transitively on the vertices. In particular, we show that there is no such actions if either the dimension of the building is larger than 7, F is not isomorphic to Qp for some prime p, or the building is associated to SLn;D0 where D0 is a non-commutative division algebra. Along the way we also give a new proof of Siegel-Klingen theorem on the rationality of certain Dedekind zeta functions and L-functions. 1

Division Algebras with Radicable Multiplicative Groups

Given a divisible finite field extension K/F, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) ≅ ℤ2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups D* is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield K/F, and (3) D is the ordinary quaternion division algebra and is divisible.

From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups

AbstractThe topic of this paper is the construction of a finite set of generators for a subgroup of finite index in the unit group u(ℤG) of the integral group ring of a finite group G. The present paper is a continuation of earlier research by Bass and Milnor, Jespers and Leal, and Ritter and Sehgal who constructed such generators provided that the group G does not have a non-abelian fixed-point free epimorphic image and the rational group algebra ℚG does not have simple epimorphic images that are two-by-two matrices over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra. In this paper we allow simple images of the type M2(ℚ). We will do so by introducing new additional generators using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of PSL2(ℤ). Furthermore, for each simple Wedderburn component M2(ℚ) of ℚG, the new generators give a free subgroup that is embedded in M2(ℤ).

Nontriviality of certain quotients of [formula omitted] groups of division algebras

For a division algebra D finite dimensional over its center Z ( D ) = F , it is a conjecture that CK 1 ( D ) : = Coker ( K 1 F → K 1 D ) is trivial if and only if D ≅ ( − 1 , − 1 F ) with F a formally real Pythagorean field. Since CK 1 ( D ) is very difficult to work with, we consider here NK 1 ( D ) : = Nrd D ( D ∗ ) / F ∗ ind ( D ) , which is a homomorphic image of CK 1 ( D ) . A field E is said to be NKNT if for every noncommutative division algebra D finite dimensional over E ⊆ Z ( D ) , NK 1 ( D ) is nontrivial. It is proved that if E is finitely generated but not algebraic over some subfield then E is NKNT. As a consequence, if Z ( D ) is finitely generated over its prime subfield or over an algebraically closed field, then CK 1 ( D ) is nontrivial.

Finite Generation of Units in Alternative Loop Rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z L of L over the ring of rational integers is finitely generated.

Tits alternative for maximal subgroups of GLn( D)

Let D be a noncommutative division algebra of finite dimension over its centre F. Given a maximal subgroup M of GL n ( D) with n⩾1, it is proved that either M contains a noncyclic free subgroup or there exists a finite family { K i } r 1 of fields properly containing F with K ∗ i⊂M for all 1⩽ i⩽ r such that M/ A is finite if Char F=0 and M/ A is locally finite if Char F= p>0, where A=K ∗ 1×⋯×K ∗ r .

Presentations of the unit group of an order in a non-split quaternion algebra

We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra H(K) over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of H( Z[ 1+ −7 2 ]) . As a consequence a presentation is discovered for the orthogonal group SO 3( Z[ 1+ −7 2 ]) . These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra.

Linear equations and heights over division algebras

We make a start on the investigation of “small” solutions to systems of homogeneous linear equations over non-commutative division algebras. In this paper we prove some upper and lower bounds for the sizes of solutions to such systems. To measure solutions and coefficient matrices we define heights which satisfy natural invariance and finiteness properties.

Full-diversity, high-rate space-time block codes from division algebras

We present some general techniques for constructing full-rank, minimal-delay, rate at least one space-time block codes (STBCs) over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras (field extensions) as well as using noncommutative division algebras of the rational field /spl Qopf/ embedded in matrix rings. The first half of the paper deals with constructions using field extensions of /spl Qopf/. Working with cyclotomic field extensions, we construct several families of STBCs over a wide range of signal sets that are of full rank, minimal delay, and rate at least one appropriate for any number of transmit antennas. We study the coding gain and capacity of these codes. Using transcendental extensions we construct arbitrary rate codes that are full rank for arbitrary number of antennas. We also present a method of constructing STBCs using noncyclotomic field extensions. In the later half of the paper, we discuss two ways of embedding noncommutative division algebras into matrices: left regular representation, and representation over maximal cyclic subfields. The 4/spl times/4 real orthogonal design is obtained by the left regular representation of quaternions. Alamouti's (1998) code is just a special case of the construction using representation over maximal cyclic subfields and we observe certain algebraic uniqueness characteristics of it. Also, we discuss a general principle for constructing cyclic division algebras using the nth root of a transcendental element and study the capacity of the STBCs obtained from this construction. Another family of cyclic division algebras discovered by Brauer (1933) is discussed and several examples of STBCs derived from each of these constructions are presented.

Heisenberg algebras and coefficient rings*

In the noncommutative theory of local rings, the existence of coefficient fields is not always granted. We study a counter-example constructed using an enveloping algebra of a Heisenberg algebra to see how to describe a good coeficient ring for a non-commutative local ring, with commutative residue field. Fhrther, dealing with the case of a noncommutative residue division algebra, we use a theorem of Hochschild on the Brauer group to describe a canonical coefficient ring in the case when the exponent of the residue division algebra is prime to the characteristic of the residue field.*

On Wedderburn's Division Algebra Theorem of 1914

We give a self-contained proof of Wedderburn's theorem which constructs a large class of non-commutative division algebras generalising Hamilton's quaternions.