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.PT: J; CR: DJOKOVIC DZ, 1973, P AM MATH SOC, V41, P457 DJOKOVIC DZ, 1999, P AM MATH SOC, V127, P3169 GROSS F, 1971, P AM MATH SOC, V31, P10 IVANOV DN, 1982, VESTNIK MOSKOV U MAT, V37, P69 KOSTRIKIN AI, 1965, IZVESTIYA AKAD NAUK, V29, P471 MACDOUGALL JA, 1978, PAC J MATH, V74, P153 SHULT EE, 1969, ILLINOIS J MATH, V13, P654 SWEET LG, 1975, P AM MATH SOC, V48, P321 SWEET LG, 1975, PAC J MATH, V59, P385 SWEET LG, 1987, PAC J MATH, V129, P375; NR: 10; TC: 0; J9: J AUST MATH SOC; PN: Part 1; PG: 10; GA: 590RVSource type: Electronic(1