Every Cayley algebra of dimension <mml:math alttext="$2^{r}, r \geq 2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn><mml:mi>r</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>&#8805;</mml:mo><mml:mn>2</mml:mn></mml:math>, contains an embedded invertible loop of order <mml:math alttext="$2^{r + 1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> generated by its basis. Such a loop belongs to a class of non-abelian invertible
loops that are flexible and power-associative.

Loops embedded in generalized Cayley algebras of dimension <mml:math alttext="$2^{r}, r \geq 2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn><mml:mi>r</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math>

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International Journal of Mathematics and Mathematical Sciences

Loops embedded in generalized Cayley algebras of dimension 2r,r≥2

Abstract

Copyright