A Result on Derivations with Algebraic Values

1986 ◽  
Vol 29 (4) ◽  
pp. 432-437 ◽  
Author(s):  
Onofrio M. Di Vincenzo
Keyword(s):  
Division Algebra ◽  
Bounded Degree ◽  
Prime Algebra ◽  
Central Division ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Algebra Over A Field ◽  
Primitive Ring

AbstractLet R be a prime algebra over a field F and let d be a non-zero derivation in R such that for every x ∊ R, d(x) is algebraic over F of bounded degree. Then R is a primitive ring with a minimal right ideal eR, where e2 = e and eRe is a finite dimensional central division algebra.

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

2014 ◽  
Vol 21 (03) ◽  
pp. 483-496 ◽  
Author(s):  
H. R. Dorbidi ◽  
R. Fallah-Moghaddam ◽  
M. Mahdavi-Hezavehi
Keyword(s):  
Maximal Subgroup ◽  
Division Algebra ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Central Division ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Free Subgroups ◽  
Maximal Subfield

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.

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.

A Double-Centralizer Theorem for Simple Associative Algebras

1969 ◽  
Vol 21 ◽  
pp. 477-478 ◽  
Author(s):  
W. L. Werner
Keyword(s):  
Division Algebra ◽  
Classical Result ◽  
Classical Case ◽  
Associative Algebras ◽  
Double Centralizer ◽  
Central Division ◽  
Central Division Algebra ◽  
General Proposition ◽  
Real Quaternions ◽  
Algebra Over A Field

Consider the following result.PROPOSITION. Let D be a finite-dimensional central division algebra over a field F, and let Dn be the algebra (over F) of all n × n matrices with entries in D. Let A and B be in Dn, and suppose that BX = XB for every X in Dn such that XA = AX. Then B is a polynomial in A with coefficients in F.The case D = F is a well-known classical result. Recently, the particular case where D is the algebra of real quaternions was established by Cullen and Carlson (2). In this note, the general proposition is proved by reduction to the classical case by way of tensor products.

Radicability condition on GLn(D)

2018 ◽  
Vol 17 (11) ◽  
pp. 1850203
Author(s):  
R. Fallah-Moghaddam ◽  
H. Moshtagh
Keyword(s):  
Division Algebra ◽  
Characteristic Zero ◽  
Field Extension ◽  
Central Division ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Quaternion Division Algebra

Given an indivisible field [Formula: see text], let [Formula: see text] be a finite dimensional noncommutative [Formula: see text]-central division algebra. It is shown that if [Formula: see text] is radicable, then [Formula: see text] is the ordinary quaternion division algebra and [Formula: see text] is divisible. Also, it is shown that when [Formula: see text] is a field of characteristic zero and [Formula: see text], then [Formula: see text] is radicable if and only if for any field extension [Formula: see text] with [Formula: see text], [Formula: see text] is divisible.

SLn, Orthogonality Relations and Transfer

2007 ◽  
Vol 59 (3) ◽  
pp. 449-464 ◽  
Author(s):  
Alexandru Ioan Badulescu
Keyword(s):  
Division Algebra ◽  
Local Field ◽  
Finite Dimension ◽  
Complex Conjugate ◽  
Orbital Integrals ◽  
Central Division ◽  
Zero Characteristic ◽  
Central Division Algebra ◽  
Orthogonality Relations ◽  
Square Integrable Representation

AbstractLet π be a square integrable representation of G′ = SLn(D), with D a central division algebra of finite dimension over a local field F of non-zero characteristic. We prove that, on the elliptic set, the character of π equals the complex conjugate of the orbital integral of one of the pseudocoefficients of π. We prove also the orthogonality relations for characters of square integrable representations of G′. We prove the stable transfer of orbital integrals between SLn(F) and its inner forms.

On some arithmetic properties of automorphic forms ofGLmover a division algebra

2014 ◽  
Vol 10 (04) ◽  
pp. 963-1013 ◽  
Author(s):  
Harald Grobner ◽  
A. Raghuram
Keyword(s):  
Division Algebra ◽  
Automorphic Forms ◽  
Local Version ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
Central Division ◽  
Arithmetic Properties ◽  
Central Division Algebra ◽  
Archimedean Place ◽  
Split Form

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GLm/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e. D = F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke–Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G′ to statements on automorphic forms of its split form using the global Jacquet–Langlands correspondence developed by Badulescu and Badulescu–Renard. Beside that we prove that the local version of the Jacquet–Langlands transfer at an archimedean place preserves the property of being cohomological.

Intrinsic Functions on Matrices of Real Quaternions

1963 ◽  
Vol 15 ◽  
pp. 456-466 ◽  
Author(s):  
C. G. Cullen
Keyword(s):  
Division Algebra ◽  
Matrix Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Real Quaternions ◽  
Algebra Over A Field

It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.

scholarly journals Commuting rings of simple A(k)-modules

1981 ◽  
Vol 31 (2) ◽  
pp. 142-145 ◽  
Author(s):  
Daniel R. Farkas ◽  
Robert L. Snider
Keyword(s):  
Division Algebra ◽  
Division Ring ◽  
Characteristic Zero ◽  
Weyl Algebra ◽  
Division Algebras ◽  
Finite Dimensional ◽  
Algebra Over A Field

AbstractFor the Weyl algebra A(k) and each finite dimensional division ring D over k, there exists a simple A(k)-module whose commuting ring is D.It has been known for some time that if A(k) denotes the Weyl algebra over a field k of characteristic zero, the commuting ring of a simple A(k)-module is a division algebra finite dimensional over k (see the introduction of [1]). Which division algebras actually appear? Quebbemann [1] showed that if D is a finite dimensional division algebra whose center is k, then it occurs as a commuting ring. We complete this circle of ideas by showing that any D appears: a division algebra over k appears as the commuting ring of a simple A(k)-module if and only if it is finite dimensional over k.

On Crossed Product Algebras over Henselian Valued Fields

2020 ◽  
Vol 27 (03) ◽  
pp. 389-404
Author(s):  
Driss Bennis ◽  
Karim Mounirh
Keyword(s):  
Positive Integer ◽  
Division Algebra ◽  
Residue Field ◽  
Crossed Product ◽  
Valued Fields ◽  
Central Division ◽  
Valued Field ◽  
Central Division Algebra ◽  
Henselian Valued Fields ◽  
Maximal Subfield

Let D be a tame central division algebra over a Henselian valued field E, [Formula: see text] be the residue division algebra of D, [Formula: see text] be the residue field of E, and n be a positive integer. We prove that Mn([Formula: see text]) has a strictly maximal subfield which is Galois (resp., abelian) over [Formula: see text] if and only if Mn(D) has a strictly maximal subfield K which is Galois (resp., abelian) and tame over E with ΓK ⊆ ΓD, where ΓK and ΓD are the value groups of K and D, respectively. This partially generalizes the result proved by Hanke et al. in 2016 for the case n = 1.

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