# A DECOMPOSITION THEOREM FOR HOMOGENEOUS ALGEBRAS

MacDougall, James A. and Sweet, L. G. (2002) A DECOMPOSITION THEOREM FOR HOMOGENEOUS ALGEBRAS. Journal of the Australian Mathematical Society Series A, 72 . pp. 47-56.

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An algebra $A$ is {\em 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 $k$. 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 $A = ker L_a \oplus Im L_a$. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.