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# Rational Homogeneous Algebras

Published:2011-05-13
Printed: Jun 2012
• J. A. MacDougall,
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
• L. G. Sweet,
Dept. of Mathemetics, University of Prince Edward Island, Charlottetown, PEI C1A 4P3
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## Abstract

An algebra $A$ is homogeneous if the automorphism group of $A$ acts transitively on the one-dimensional subspaces of $A$. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if $A$ is a rational homogeneous algebra with $\operatorname{dim} A>1$, then $A^{2}=0$.
 Keywords: non-associative algebra, homogeneous, automorphism
 MSC Classifications: 17D99 - None of the above, but in this section 17A36 - Automorphisms, derivations, other operators

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