scholarly journals Positive Cones and Gauges on Algebras With Involution

2020 ◽  
Author(s):  
Vincent Astier ◽  
Thomas Unger
Keyword(s):  
Finite Dimensional ◽  
Simple Algebras

Abstract We extend the classical links between valuations and orderings on fields to Tignol–Wadsworth gauges and positive cones on finite-dimensional simple algebras with involution. We also study the compatibility of gauges and positive cones and prove lifting results in the style of the Baer–Krull theorem for fields.

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.

UNIT GROUPS OF CENTRAL SIMPLE ALGEBRAS AND THEIR FRATTINI SUBGROUPS

2010 ◽  
Vol 09 (06) ◽  
pp. 921-932 ◽  
Author(s):  
R. FALLAH-MOGHADDAM ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Global Field ◽  
Central Simple Algebras ◽  
Frattini Subgroup ◽  
Division Rings ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Central Simple

Given a finite dimensional F-central simple algebra A = Mn(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that [Formula: see text] where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D.

Polynomial Identities of Finite Dimensional Simple Algebras

2011 ◽  
Vol 39 (3) ◽  
pp. 929-932 ◽  
Author(s):  
Ivan Shestakov ◽  
Mikhail Zaicev
Keyword(s):  
Polynomial Identities ◽  
Finite Dimensional ◽  
Simple Algebras

scholarly journals Admissible groups, symmetric factor sets, and simple algebras

1984 ◽  
Vol 7 (4) ◽  
pp. 707-711
Author(s):  
R. A. Mollin
Keyword(s):  
Division Algebra ◽  
Multiplicative Group ◽  
Sufficient Conditions ◽  
Simple Algebra ◽  
Roots Of Unity ◽  
Finite Dimensional ◽  
Odd Order ◽  
Simple Algebras ◽  
Necessary And Sufficient ◽  
Maximal Subfield

LetKbe a field of characteristic zero and suppose thatDis aK-division algebra; i.e. a finite dimensional division algebra overKwith centerK. In Mollin [1] we proved that ifKcontains no non-trivial odd order roots of unity, then every finite odd order subgroup ofD*the multiplicative group ofD, is cyclic. The first main result of this paper is to generalize (and simplify the proof of) the above. Next we generalize and investigate the concept of admissible groups. Finally we provide necessary and sufficient conditions for a simple algebra, with an abelian maximal subfield, to be isomorphic to a tensor product of cyclic algebras. The latter is achieved via symmetric factor sets.

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