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
J. A. MacDougall
  Department of Mathematics
  University of Newcastle
  Callaghan NSW 2308

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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