Normed Right Alternative Algebras Over the Reals

1972 ◽  
Vol 24 (6) ◽  
pp. 1183-1186 ◽  
Author(s):  
José I. Nieto
Keyword(s):  
Division Algebra ◽  
Real Field ◽  
Division Algebras ◽  
Classical Theorem ◽  
Complex Field ◽  
The Real ◽  
Alternative Algebras ◽  
Finite Dimensional

One of the most interesting results on real normed division algebras says that every real normed associative division algebra is finite dimensional [6, Theorem 1.7.6], and hence by a classical theorem of Frobenius either isomorphic to the real field, the complex field, or the algebra of quaternions. Thus the dimension of the algebra can only be either 1, 2 or 4.

Classes of Functions on Algebras

1966 ◽  
Vol 18 ◽  
pp. 139-146 ◽  
Author(s):  
C. G. Cullen ◽  
C. A. Hall
Keyword(s):  
Associative Algebra ◽  
Real Field ◽  
Complex Field ◽  
The Real ◽  
Finite Dimensional

Let be a finite-dimensional linear associative algebra over the real field R or the complex field C and let F be a function with domain and range in .Several classes of functions on have been discussed in the literature, and it is the purpose of this paper to discuss the relationships between these classes and to present some interesting examples. First we shall list the definitions of the classes we wish to consider here.

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.

Real Flexible Division Algebras

1982 ◽  
Vol 34 (3) ◽  
pp. 550-588 ◽  
Author(s):  
Georgia M. Benkart ◽  
Daniel J. Britten ◽  
J. Marshall Osborn
Keyword(s):  
Jordan Algebra ◽  
Nonassociative Algebra ◽  
Division Algebras ◽  
Hermitian Matrices ◽  
Real Numbers ◽  
The Real ◽  
Precise Statement ◽  
Finite Dimensional ◽  
Dimension One

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.

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.

Intrinsic Functions on Semi-Simple Algebras

1967 ◽  
Vol 19 ◽  
pp. 590-598 ◽  
Author(s):  
C. A. Hall
Keyword(s):  
Associative Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Ground Field ◽  
The Real ◽  
Simple Algebras

Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on a linear associative algebra , with identity, over the real field R or the complex field C. In this paper we shall consider a semi-simple algebra = ⊕ … ⊕ over R or C with simple components . Let G be the group of all automorphisms or anti-automorphisms of which leave the ground field elementwise invariant, and let H be the subgroup of G such that Ω = (i = 1, 2, … , t) for each Ω in H.

On Involutions of Quasi-Division Algebras

1975 ◽  
Vol 17 (5) ◽  
pp. 723-725 ◽  
Author(s):  
Lowell Sweet
Keyword(s):  
Division Algebra ◽  
Short Paper ◽  
Division Algebras ◽  
Finite Dimensional

All algebras are assumed to be finite dimensional and not necessarily associative. An involution of an algebra is an algebra automorphism of order two. A quasi-division algebra is any algebra in which the non-zero elements form a quasi-group under multiplication. The purpose of this short paper is to determine the structure of all involutions of quasi-division algebras and to give an application of this result.

On a Theorem of Hermite and Joubert

1999 ◽  
Vol 51 (1) ◽  
pp. 69-95 ◽  
Author(s):  
Zinovy Reichstein
Keyword(s):  
Division Algebra ◽  
Minimal Polynomial ◽  
Field Extension ◽  
Division Algebras ◽  
Classical Theorem

AbstractA classical theorem of Hermite and Joubert asserts that any field extension of degree n = 5 or 6 is generated by an element whose minimal polynomial is of the form λn + c1λn−1 + ··· + cn−1λ + cn with c1 = c3 = 0. We show that this theorem fails for n = 3m or 3m + 3l (and more generally, for n = pm or pm + pl, if 3 is replaced by another prime p), where m > l ≥ 0. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra UD(n).We also prove a similar result for division algebras and use it to study the structure of the universal division algebra UD(n).

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.

Order Characterization of the Complex Field

1978 ◽  
Vol 21 (3) ◽  
pp. 313-318
Author(s):  
Lino Gutierrez Novoa
Keyword(s):  
Upper Bound ◽  
Ordered Set ◽  
Real Field ◽  
Complex Field ◽  
The Real ◽  
Ordered Field ◽  
The One ◽  
Definition Of ◽  
Real Number Field

It is well known that the real number field can be characterized as an ordered field satisfied the “least upper bound” property.Using the idea of n -ordered set, introduced in [3], and generalizing the notion of l.u.b. in a suitable way, it is possible to give a similar categorical definition of the complex field.With these extended meanings, the main theorem of this paper (Theorem 7 in the text) is stated almost identically to the one for the real field. Any directly two-ordered field, in which the "supremum property" holds, is isomorphic to the complex field.

scholarly journals The characterization of generalized Jordan centralizers on algebras

2017 ◽  
Vol 35 (3) ◽  
pp. 225 ◽  
Author(s):  
Quanyuan Chen ◽  
Xiaochun Fang ◽  
Changjing Li
Keyword(s):  
Continuous Mapping ◽  
Additive Mapping ◽  
Real Field ◽  
Complex Field ◽  
Von Neumann ◽  
The Real

In this paper, it is shown that if $\mathcal {A}$ is a CSL subalgebra of a von Neumann algebr and $\phi$ is a continuous mapping on $\mathcal {A}$ such that $(m+n+k+l)\phi(A^{2})-(m\phi(A)A+nA\phi(A)+k\phi(I)A^2+l A^2 \phi(I))\in \mathbb{F}I $ for any $A\in \mathcal {A}$, where $\mathbb{F}$ is the real field or the complex field, then $\phi$ is a centralizer. It is also shown that if $\phi$ is an additive mapping on $\mathcal {A}$ such that $(m+n+k+l)\phi(A^{2})=m\phi(A)A+nA\phi(A)+k\phi(I)A^2+l A^2 \phi(I) $for any $A\in\mathcal{A}$, then $\phi$ is a centralizer.

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