Split Lie algebras of order 3

摘要

We introduce the class of split. Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form L = u (Sigma(j) I-j) with u a linear subspace of H and any I-j a well-described (split) ideal of L satisfying [I-j, ((0) over bar), I-k] = {I-j,I- ((i) over bar), I-k,((i) over bar), L-(i) over bar} = 0, with (i) over bar is an element of {(1) over bar, (2) over bar}, if j not equal k. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that L is the direct sum of the family of its (split) simple ideals) is stated.