Equations des algèbres Lie triple qui sont des algèbres train

Abstract

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank ≤4 with an idempotent are Jordan algebras.Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of e-stable power-associative algebras.We also prove that the algebra obtained by 2-gametization process of a Lie triple algebra is a Lie triple one.