Finite subgroups of finite dimensional division algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

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Valuation rings in finite-dimensional division algebras

Journal of Algebra ◽

10.1016/0021-8693(89)90190-7 ◽

1989 ◽

Vol 120 (1) ◽

pp. 90-99 ◽

Cited By ~ 23

Author(s):

H.H Brungs ◽

Joachim Gräter

Keyword(s):

Division Algebras ◽

Finite Dimensional ◽

Valuation Rings

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LOCALLY FINITE-DIMENSIONAL DIVISION ALGEBRAS

Skew Linear Groups ◽

10.1017/cbo9780511600630.004 ◽

1987 ◽

pp. 80-121

Keyword(s):

Division Algebras ◽

Locally Finite ◽

Finite Dimensional

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Real Flexible Division Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-039-x ◽

1982 ◽

Vol 34 (3) ◽

pp. 550-588 ◽

Cited By ~ 19

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.

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K-theory of finite dimensional division algebras

Journal of Pure and Applied Algebra ◽

10.1016/0022-4049(78)90030-0 ◽

1978 ◽

Vol 12 (2) ◽

pp. 153-158 ◽

Cited By ~ 6

Author(s):

S. Green ◽

D. Handelman ◽

P. Roberts

Keyword(s):

Division Algebras ◽

Finite Dimensional ◽

K Theory

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RESTRICTIONS OF ALGEBRAIC GROUP REPRESENTATIONS TO FINITE SUBGROUPS

Bulletin of the London Mathematical Society ◽

10.1112/s0024609301008773 ◽

2002 ◽

Vol 34 (2) ◽

pp. 165-173

Author(s):

HARM DERKSEN

Keyword(s):

Algebraic Group ◽

Short Proof ◽

Linear Algebraic Group ◽

Finite Subgroup ◽

Group Representations ◽

Rational Representation ◽

Finite Dimensional ◽

Finite Subgroups

Suppose that H is a finite subgroup of a linear algebraic group, G. It was proved by Donkin that there exists a finite-dimensional rational representation of G whose restriction to H is free. This paper gives a short proof of this in characteristic 0. The author also studies more closely which representations of H can appear as a restriction of G.

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Nicely semiramified division algebras over Henselian fields

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.571 ◽

2005 ◽

Vol 2005 (4) ◽

pp. 571-577 ◽

Cited By ~ 4

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.

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