Kronecker factorization theorems for the exceptional Malcev algebra

TL;DR

This paper establishes that certain Malcev algebras and superalgebras containing the 7-dimensional simple Malcev algebra are structurally isomorphic to tensor products of this algebra with associative (super)algebras, revealing a Kronecker factorization.

Contribution

It proves Kronecker factorization theorems for Malcev algebras and superalgebras containing the 7-dimensional simple Malcev algebra, extending structural understanding.

Findings

Malcev algebra containing the 7-dimensional simple Malcev algebra is isomorphic to a tensor product with a commutative algebra.

Malcev superalgebra with the simple Malcev algebra in its even part is isomorphic to a tensor product with a supercommutative superalgebra.

Structural decomposition results for Malcev algebras and superalgebras containing the simple Malcev algebra.

Abstract

We prove that a Malcev algebra $\mathcal{M}$ containing the $7$-dimensional simple non-Lie Malcev algebra $\mathbb{M}$ such that $m\mathbb{M}\neq 0$ for any $m\neq 0$ from $\mathcal{M}$, is isomorphic to $\mathbb{M}\otimes_\textup{F} \mathcal{U}$, where $\mathcal{U}$ is a certain commutative associative algebra. Also, we prove that a Malcev superalgebra $\mathcal{M}=\mathcal{M}_0\oplus \mathcal{M}_1$ whose even part $\mathcal{M}_0$ contains $\mathbb{M}$ with $m\mathbb{M}\neq 0$ for any homogeneous element $0\neq m\in \mathcal{M}_0\cup \mathcal{M}_1$, is isomorphic to $\mathbb{M}\otimes_\textup{F}U$, where $U$ is a certain supercommutative associative superalgebra.