scholarly journals Levels of division algebras

1990 ◽  
Vol 32 (3) ◽  
pp. 365-370 ◽  
Author(s):  
David B. Leep
Keyword(s):  
Function Field ◽  
Algebraic Extension ◽  
Global Field ◽  
Real Closed Field ◽  
Closed Field ◽  
Division Algebras ◽  
Central Division ◽  
Product Level ◽  
Euclidean Field ◽  
Finite Dimensional

In [7] the level, sublevel, and product level of finite dimensional central division algebras D over a field F were calculated when F is a local or global field. In Theorem 1.4 of this paper we calculate the same quantities if all finite extensions K of F satisfy ū(K) ≤2, where ū is the Hasse number of a field as defined in [2]. This occurs, for example, if F is an algebraic extension of the function field R(x) where R is a real closed field or hereditarily Euclidean field (see [4]).

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

Going-Down Results for Ci-Fields

2006 ◽  
Vol 49 (1) ◽  
pp. 11-20
Author(s):  
Anthony J. Bevelacqua ◽  
Mark J. Motley
Keyword(s):  
Function Fields ◽  
Algebraic Extension ◽  
Real Closed Field ◽  
Closed Field ◽  
Difficult Case ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

AbstractWe search for theorems that, given a Ci-field K and a subfield k of K, allow us to conclude that k is a Cj -field for some j. We give appropriate theorems in the case K = k(t) and K = k((t)). We then consider the more difficult case where K/k is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field.

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.

scholarly journals Graded-division algebras over arbitrary fields

2020 ◽  
pp. 2140009
Author(s):  
Yuri Bahturin ◽  
Alberto Elduque ◽  
Mikhail Kochetov
Keyword(s):  
Finite Groups ◽  
Closed Field ◽  
Arbitrary Field ◽  
Division Algebras ◽  
Real Numbers ◽  
Arbitrary Group ◽  
Matrix Algebras ◽  
Finite Dimensional ◽  
Group Gradings

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field [Formula: see text] can be reduced to the following three classifications, for each finite Galois extension [Formula: see text] of [Formula: see text]: (1) finite-dimensional central division algebras over [Formula: see text], up to isomorphism; (2) twisted group algebras of finite groups over [Formula: see text], up to graded-isomorphism; (3) [Formula: see text]-forms of certain graded matrix algebras with coefficients in [Formula: see text] where [Formula: see text] is as in (1) and [Formula: see text] is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.

scholarly journals Représentations banales de

2013 ◽  
Vol 149 (4) ◽  
pp. 679-704 ◽  
Author(s):  
Alberto Mínguez ◽  
Vincent Sécherre
Keyword(s):  
Complex Numbers ◽  
Closed Field ◽  
Locally Compact ◽  
Central Division ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Compact Field ◽  
Residue Characteristic

AbstractLet${\rm F}$be a non-Archimedean locally compact field of residue characteristic$p$, let${\rm D}$be a finite-dimensional central division${\rm F}$-algebra and let${\rm R}$be an algebraically closed field of characteristic different from$p$. We definebanalirreducible${\rm R}$-representations of the group${\rm G}={\rm GL}_{m}({\rm D})$. This notion involves a condition on the cuspidal support of the representation depending on the characteristic of${\rm R}$. When this characteristic is banal with respect to${\rm G}$, in particular when${\rm R}$is the field of complex numbers, any irreducible${\rm R}$-representation of${\rm G}$is banal. In this article, we give a classification of all banal irreducible${\rm R}$-representations of${\rm G}$in terms of certain multisegments, called banal. When${\rm R}$is the field of complex numbers, our method provides a new proof, entirely local, of Tadić’s classification of irreducible complex smooth representations of${\rm G}$.

A note on Lie ideals of prime rings

2019 ◽  
Vol 18 (03) ◽  
pp. 1950045
Author(s):  
Tsiu-Kwen Lee ◽  
Muzibur Rahman Mozumder
Keyword(s):  
Division Algebras ◽  
Finitely Generated ◽  
Prime Rings ◽  
Central Division ◽  
Arbitrary Characteristic ◽  
Finite Dimensional

In this paper, we characterize Lie ideals, which are either finitely generated [Formula: see text]-modules or maximal, in (centrally closed) prime rings. As consequences, we extend the results proved in [1] for finite dimensional central division algebras of characteristic not [Formula: see text] to simple rings of arbitrary characteristic.

scholarly journals On the Cardinality of A Semi-Algebraic Set

1994 ◽  
Vol 1 (3) ◽  
pp. 277-286
Author(s):  
G. Khimshiashvili
Keyword(s):  
Quadratic Forms ◽  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraic Set

Abstract It is shown that the cardinality of a finite semi-algebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

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