Generalized Analysis of Division Algebras of Dimension 2

2021 ◽  
Vol 43 (4) ◽  
pp. 5-21
Author(s):  
S.I. Klipkov ◽  
Keyword(s):  
Division Algebra ◽  
Exponential Function ◽  
Division Algebras ◽  
Exponential Form ◽  
Mathematical Properties ◽  
Dimension 2 ◽  
Commutative Division Algebra

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.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

scholarly journals On the Automorphisms of the Four-dimensional Real Division Algebras

2017 ◽  
Vol 9 (2) ◽  
pp. 95
Author(s):  
Andre S. Diabang ◽  
Alassane Diouf ◽  
Mankagna A. Diompy ◽  
Alhousseynou Ba
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Automorphisms Groups

In this paper, we study partially the automorphisms groups of four-dimensional division algebra. We have proved that there is an equivalence between Der(A)=su(2) and Aut(A)=SO(3). For an unitary four-dimensional real division algebra, there is an equivalence between dim(Der(A))=1 and Aut(A)=SO(2).

A Note on Subnormal Subgroups of Division Algebras

1978 ◽  
Vol 30 (01) ◽  
pp. 161-163 ◽  
Author(s):  
Gary R. Greenfield
Keyword(s):  
Division Algebra ◽  
Multiplicative Group ◽  
Subnormal Subgroup ◽  
Local Fields ◽  
Division Algebras ◽  
Subnormal Subgroups

Let D be a division algebra and let D* denote the multiplicative group of nonzero elements of D. In [3] Herstein and Scott asked whether any subnormal subgroup of D* must be normal in D*. Our purpose here is to show that division algebras over certain p-local fields do not satisfy such a “subnormal property”.

scholarly journals Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

2016 ◽  
Vol 2016 (715) ◽  
Author(s):  
Bart De Bruyn ◽  
Hendrik Van Maldeghem
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Polar Spaces ◽  
Dual Polar Spaces

AbstractSuppose 𝕆 is an alternative division algebra that is quadratic over some subfield 𝕂 of its center

scholarly journals MAL'CEV–NEUMANN RINGS AND NONCROSSED PRODUCT DIVISION ALGEBRAS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250052 ◽  
Author(s):  
CÉCILE COYETTE
Keyword(s):  
Division Algebra ◽  
Laurent Series ◽  
Elementary Proof ◽  
Formal Series ◽  
Direct Approach ◽  
Division Algebras ◽  
Sufficient Condition ◽  
Series Ring

The first section of this paper yields a sufficient condition for a Mal'cev–Neumann ring of formal series to be a noncrossed product division algebra. This result is used in Sec. 2 to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p2 for p any odd prime). The arguments are based on those of Hanke in [A direct approach to noncrossed product division algebras, thesis dissertation, Postdam (2001), An explicit example of a noncrossed product division algebra, Math. Nachr.251 (2004) 51–68, A twisted Laurent series ring that is a noncrossed product, Israel. J. Math.150 (2005) 199–2003].

On the non-existence of cyclic splitting fields for division algebras

2018 ◽  
Vol 30 (2) ◽  
pp. 385-395
Author(s):  
Mehran Motiee
Keyword(s):  
Division Algebra ◽  
Commutator Subgroup ◽  
Cyclic Extension ◽  
Roots Of Unity ◽  
Division Algebras ◽  
Group Of Units ◽  
Positive Integers ◽  
Special Classes

AbstractLetDbe a division algebra over its centerFof degreen. Consider the group{\mu_{Z}(D)=\mu_{n}(F)/Z(D^{\prime})}, where{\mu_{n}(F)}is the group of all then-th roots of unity in{F^{*}}, and{Z(D^{\prime})}is the center of the commutator subgroup of the group of units{D^{*}}ofD. It is shown that if{\mu_{Z}(D\otimes_{F}L)\neq 1}for someLcontaining all the primitive{n^{k}}-th roots of unity for all positive integersk, thenDis not split by any cyclic extension ofF. This criterion is employed to prove that some special classes of division algebras are not cyclically split.

scholarly journals Nicely semiramified division algebras over Henselian fields

2005 ◽  
Vol 2005 (4) ◽  
pp. 571-577 ◽  
Author(s):  
Karim Mounirh
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Central Division ◽  
Field Extensions ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Radical Type ◽  
Maximal Subfield

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

Duplication methods for embeddings of real division algebras

2020 ◽  
pp. 2250063
Author(s):  
Marina Tvalavadze ◽  
Noureddine Motya ◽  
Abdellatif Rochdi
Keyword(s):  
Division Algebra ◽  
Infinite Family ◽  
Division Algebras ◽  
Unital Algebra

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every [Formula: see text]-dimensional (4D) real unital algebra [Formula: see text] into an 8D real unital algebra denoted by [Formula: see text] We also find the conditions on [Formula: see text] under which [Formula: see text] is a division algebra. This covers the most classes of known [Formula: see text]D real division algebras. The second process allows us to embed particular classes of [Formula: see text]D RDAs into [Formula: see text]D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic [Formula: see text]D real division algebras whose derivation algebras contain [Formula: see text]

scholarly journals HOMOLOGICAL TRANSCENDENCE DEGREE

2006 ◽  
Vol 93 (1) ◽  
pp. 105-137 ◽  
Author(s):  
AMNON YEKUTIELI ◽  
JAMES J. ZHANG
Keyword(s):  
Division Algebra ◽  
Main Tool ◽  
Transcendence Degree ◽  
Division Algebras ◽  
Base Field ◽  
Injective Dimension ◽  
Dualizing Complex

Let $D$ be a division algebra over a base field $k$. The homological transcendence degree of $D$, denoted by $\text{Htr}\; D$, is defined to be the injective dimension of the algebra $D \otimes_k D^{\circ}$. We show that $\text{Htr}$ has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute $\text{Htr}$ for several classes of division algebras. The main tool for the computation is Van den Bergh's rigid dualizing complex.

On Asano’s theorem

2019 ◽  
Vol 18 (10) ◽  
pp. 1950181
Author(s):  
Münevver Pınar Eroǧlu ◽  
Tsiu-Kwen Lee ◽  
Jheng-Huei Lin
Keyword(s):  
Vector Space ◽  
Division Algebra ◽  
Division Algebras ◽  
Infinite Field ◽  
Additive Subgroup ◽  
Normal Bases ◽  
Algebraic Elements ◽  
Inner Automorphisms

Let [Formula: see text] be a division algebra over an infinite field [Formula: see text] such that every element of [Formula: see text] is a sum of finitely many algebraic elements. As a generalization of Asano’s theorem, it is proved that every noncentral subspace of [Formula: see text] invariant under all inner automorphisms induced by algebraic elements contains [Formula: see text], the additive subgroup of [Formula: see text] generated by all additive commutators of [Formula: see text]. From the viewpoint we study the existence of normal bases of certain subspaces of division algebras. It is proved among other things that [Formula: see text] is generated by multiplicative commutators as a vector space over the center of [Formula: see text].

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