Division Ring Research Articles

On locally solvable subgroups in division rings

Let D be a division ring with center F, and G a subnormal subgroup of D⁎. We show that if G is a locally solvable group such that a derived subgroup G(i) is algebraic over F, then G must be central. Also, if M is a non-abelian locally solvable maximal subgroup of G with M(i) algebraic over F, then D is a cyclic algebra of prime degree over F.

HOW A NONASSOCIATIVE ALGEBRA REFLECTS THE PROPERTIES OF A SKEW POLYNOMIAL

AbstractLet D be a unital associative division ring and D[t, σ, δ] be a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation. For each f ϵ D[t, σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f. These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f. In the classical literature, this nucleus appears as the eigenspace of f and is used to investigate the irreducible factors of f. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras Sf.

0-Cycles on Grassmannians as Representations of Projective Groups

Let F be an infinite division ring, V be a left F-vector space, \(r\ge 1\) be an integer. We study the structure of the representation of the linear group \(\mathrm {GL}_F(V)\) in the vector space of formal finite linear combinations of r-dimensional vector subspaces of V with coefficients in a field. This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth if F is locally compact and non-discrete.

Subgroups of division rings

We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders. Our results are highly dependent on the computer software package Magma, although alternatives would probably work fine. A computer approach is required because our results require a listing of all maximal orders; the sheer number of maximal orders grows exponentially quickly and determining them all is not obtainable in polynomial time. When discriminants are large enough, the determination of a single maximal order is also exponentially hard. We indicate the limits as these problems arise.

Quadratic D-forms with applications to hermitian forms

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.

Pre-Schauder Bases in Topological Vector Spaces

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

Isomorphism between endomorphism rings of modules over a semi simple ring

Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring. Any semi simple ring does not satisfy this property. We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.

A note on locally soluble almost subnormal subgroups in divsion rings

Let $D$ be a division ring with center $F$ and assume that $N$ is a locally soluble almost subnormal subgroup of the multiplicative group $D^*$ of $D$‎. ‎We prove that if $N$ is algebraic over $F$‎, ‎then $N$ is central‎. ‎This answers partially cite[Conjecture 1]{hai_13}‎.

Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

An abstract characterization of noncommutative projective lines

Let k be a field.We describe necessary and sufficient conditions for a k -linear abelian category to be a noncommutative projective line, i.e. a noncommutative \mathbb P^1 -bundle over a pair of division rings over k . As an application, we prove that \mathbb P^1_n , Piontkovski’s n th noncommutative projective line, is the noncommutative projectivization of an n -dimensional vector space.

Endomorphisms of the poset of idempotent matrices

Let D be a division ring, n≥3 an integer, and Pn(D) the poset of all n×n idempotent matrices over D with the partial order defined by P≤Q if PQ=QP=P. Let T∈Mn(D) be an invertible matrix and σ:D→D an endomorphism (τ:D→D an anti-endomorphism). For any P∈Pn(D) we denote by Pσ (Pτ) the idempotent matrix obtained from P by applying σ (τ) entrywise. The map ϕ:Pn(D)→Pn(D) defined by ϕ(P)=TPσT−1, P∈Pn(D) (ϕ(P)=Tt(Pτ)T−1, P∈Pn(D)) is an injective map preserving order in both directions. Every such map is called a standard map. It has been known before that if D is an EAS division ring, then every injective order preserving map ϕ:Pn(D)→Pn(D) is either standard or of a very special degenerate form. In this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the EAS assumption is indispensable. Then we define generalized standard maps and using them we describe the general form of injective order preservers on Pn(D) for an arbitrary division ring D. Our proof is shorter than the original one for the special case of EAS division rings. Under somewhat stronger assumptions we get two characterizations of standard maps. All the results are optimal as shown by counterexamples.

Free Group Algebras in Division Rings with Valuation II

AbstractWe apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$.Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$ contains a (noncommutative) free group algebra generated by symmetric elements.

Division rings with power commuting semi-linear additive maps

Division rings with power commuting semi-linear additive maps

Quaternionic representation of electromagnetism for material media

In this study, the electromagnetic fields are developed in the presence of both the electric and magnetic induction fields by quaternion algebra. In this sense, the polarization and magnetization effects, which are valid in the material media, gain much importance. Quaternions are one of the most convenient tools for representing electromagnetism with regard to having non-commutative but associative algebraic division ring. By defining the quaternion induction field, the quaternion source term has been obtained in basic and elegant notation for the first time. In addition, one type of Poynting theorem, named as the Minkowski form, has been presented including the permittivity and permeability constants by quaternions.

Multi-feature Multimodal Biometric Recognition Based on Quaternion Locality Preserving Projection

This paper proposed Quaternion locality preserving projection (QLPP) for multi-feature multimodal biometric recognition. Multi-features fill the real part or the three imaginary parts of quaternion to constitute the quaternion fusion features. In quaternion division ring, QLPP extracts the local information and finds essential manifold structure of the quaternion fusion features. Deferent from Quaternion principal component analysis (QPCA) and Quaternion fisher discriminant analysis (QFDA), QLPP takes advantage of the optimal linear approximations to find the nonlinear manifold structures. Two experiments are designed: one fuses four features from two biometric modalities, and the other fuses three features from three biometric modalities. The experimental results show the proposed algorithm achieves much better performance than the unimodal biometric algorithms, the traditional feature level fusion methods(weighted sum rule and series rule) and two quaternion representation methods(QPCA and QFDA).

Some results relevant to embeddability of rings (especially group algebras) in division rings

P.M. Cohn showed in 1971 that given a ring R, to describe, up to isomorphism, a division ring D generated by a homomorphic image of R is equivalent to specifying the set of square matrices over R which map to singular matrices over D, and he determined precisely the conditions that such a set of matrices must satisfy. The present author later developed another version of this data, in terms of closure operators on free R-modules.In this note, we examine the latter concept further, and show how an R-module M satisfying certain conditions can be made to induce such data. In an appendix we make some observations on Cohn's original construction, and note how the data it uses can similarly be induced by an appropriate sort of R-module.Our motivation is the longstanding question of whether, for G a right-orderable group and k a field, the group algebra kG must be embeddable in a division ring. Our hope is that the right kG-module M=k((G)) might induce a closure operator of the required sort. We re-prove a partial result in this direction due to N.I. Dubrovin, note a plausible generalization thereof which would give the desired embedding, and also sketch some thoughts on other ways of approaching the problem.

On division subrings normalized by almost subnormal subgroups in division rings

Let D be a division ring with infinite center, K a proper division subring of D and N an almost subnormal subgroup of the multiplicative group $$D^*$$ of D. The aim of this paper is to show that if K is N-invariant and N is non-central, then K is central. Some examples of almost subnormal subgroups in division rings that are not subnormal are also given.

English

Let D be a division ring finite dimensional over its center F. The goal of this paper is to prove that for any positive integer n there exists a ∈ D(n), the nth multiplicative derived subgroup such that F(a) is a maximal subfield of D. We also show that a single depth-n iterated additive commutator would generate a maximal subfield of D.

Groups with a nontrivial nonideal kernel

We classify finite groups G, such that the group algebra, QG (over the field of rational numbers Q), is the direct product of the group algebra Q[G/N] of a proper factor group G/N, and some division rings.

Maximal subgroups of SLn(D)

Given a non-commutative F-central division ring D, N a subnormal subgroup of GLn(D) and M a non-abelian maximal subgroup of N, if M is a non-abelian soluble maximal subgroup of N, then, n=1 and D is cyclic of prime degree p with a maximal cyclic subfield K/F such that the groups Gal(K/F) and M/(K⁎∩M) are isomorphic. Furthermore, for any x∈M∖K⁎, we have xp∈F⁎ and D=F[M]=⨁i=1pKxi.