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Real Flexible Division Algebras

Published online by Cambridge University Press:  20 November 2018

Georgia M. Benkart ,
Daniel J. Britten  and
J. Marshall Osborn
Georgia M. Benkart
Affiliation:
University of Wisconsin, Madison, Wisconsin
Daniel J. Britten
Affiliation:
University of Windsor, Windsor, Ontario
J. Marshall Osborn
Affiliation:
University of Windsor, Windsor, Ontario
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 550 - 588
Copyright
Copyright © Canadian Mathematical Society 1982

References

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