On the triviality of homogeneous algebras over an algebraically closed field

Sweet, Lowell

doi:10.1090/S0002-9939-1975-0364382-7

On the triviality of homogeneous algebras over an algebraically closed field
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Proc. Amer. Math. Soc. 48 (1975), 321-324 Request permission

Abstract:

Let $A$ be a finite-dimensional algebra (not necessarily associative) over a field $K$. Then $A$ is said to be homogeneous if $\operatorname {Aut} (A)$ acts transitively on the one-dimensional subspaces of $A$. If $A$ is homogeneous and $K$ is algebraically closed, then it is shown that either ${A^2} = 0$ or $\dim A = 1$.

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