J. Austral. Math. Soc.  72 (2002), 47-56
A decomposition theorem for homogeneous algebras

L. G. Sweet
  Department of Mathematics
  and Computer Science
  University of Prince Edward Island
  Charlottetown PEI C1A 4P3
  Canada
  sweet@upei.ca
and
J. A. MacDougall
  Department of Mathematics
  University of Newcastle
  Callaghan NSW 2308
  Australia
  mmjam@cc.newcastle.edu.au


Abstract
An algebra $A$ is homogeneous if the automorphism group of $A$ acts transitively on the one dimensional subspaces of $A$. Suppose $A$ is a homogeneous algebra over an infinite field ${\bf k}$. Let $L_a$ denote left multiplication by any nonzero element $a \in A$. Several results are proved concerning the structure of $A$ in terms of $L_a$. In particular, it is shown that $A$ decomposes as the direct sum $A = \ker L_a \oplus \operatorname{Im} L_a$. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.
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