A decomposition theorem for homogeneous algebras

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 k. Let L-a denote left mulfiplication by any nonzero element a is an element of 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 circle plus Im L-a. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.