Split Malcev-Poisson-Jordan algebras

TL;DR

This paper introduces split Malcev-Poisson-Jordan algebras, extending split Malcev Poisson algebras, and demonstrates their decomposition into simple ideals with specific algebraic properties.

Contribution

It defines the class of split Malcev-Poisson-Jordan algebras and proves their decomposition into simple ideals under certain conditions.

Findings

Decomposition of split Malcev-Poisson-Jordan algebras into direct sums of ideals.

Conditions under which the algebra decomposes into simple ideals.

Extension of split Malcev Poisson algebra theory.

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 $P = \oplus_{j \in J}I_j$ with any $I_j$ a non-zero ideal of $P$ in such a way that satisfies $[I_{j_1},I_{j_2}] = I_{j_1} \circ I_{j_2} = 0$ for $j_1 \neq j_2.$ Under certain conditions, it is shown that the above decomposition of $P$ is by means of the family of its simple ideals.