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A Generalization of Degree Two Simple Finite-Dimensional Noncommutative Jordan Algebras

Published online by Cambridge University Press:  20 November 2018

Mary Ellen Conlon
Mary Ellen Conlon*
Affiliation:
Illinois Institute of Technology, Chicago, Illinois
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Let be an algebra over a field . For x, y, z in , write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached algebra is the same vector space as , but the product of x and y is x · y. We aim to prove the following result.

THEOREM 1. Let be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in , suppose

1

Then is noncommutative Jordan.

The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace in the Peirce decomposition . In §4 we construct an ideal of which we then use to show that the nilpotent elements of form a subalgebra of for i = 0, 1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 480 - 493
Copyright
Copyright © Canadian Mathematical Society 1980

References

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