Kronecker factorization theorems for the exceptional Malcev algebra

TL;DR

This work analyzes Malcev (and Malcev super)algebras in Filippov’s variety $\mathcal{H}$ that contain the 7‑dimensional exceptional non‑Lie Malcev algebra $\mathbb{M}$. By exploiting representation theory, centroid arguments, and complete reducibility, the authors prove Kronecker‑type factorizations: any such algebra is isomorphic to $\mathbb{M}\otimes_F \mathcal{U}$ (or to $\mathbb{M}\otimes_F U$ in the super case) where $\mathcal{U}$ (resp. $U$) is a (super)commutative associative algebra. They also establish Morita‑type equivalences between categories of modules over $\mathcal{U}$ (or $U$) and modules over the Malcev algebra $\mathbb{M}$ (or $\mathbb{M}(U)$), and extend the framework to (super)algebras with (super)involutions under the J‑admissibility condition. Collectively, these results coordinatize the structure of prime non‑Lie Malcev algebras inside larger Malcev (super)algebras, offering a robust description of their representations and categorical relationships with base associative commutative algebras.

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.