Split Malcev-Poisson-Jordan algebras

Abstract

We introduce the class of split Malcev-Poisson-Jordan algebras as the natural extension of the one of split Malcev Poisson algebras, and therefore split (non-commutative) Poisson algebras. We show that a split Malcev-Poisson-Jordan algebra P can be written as a direct sum P = ⊕ j∈J I j with any I j a non-zero ideal of P in such a way that satisfies [I j1 , I j2 ] = I j1 • I j2 = 0 for j 1 = j 2 . Under certain conditions, it is shown that the above decomposition of P is by means of the family of its simple ideals.