On a problem by Nathan Jacobson for Malcev algebras

TL;DR

This paper addresses a structural problem for a class of Malcev algebras, providing a coordinatization theorem and describing their structure when containing the simple Lie algebra sl_2.

Contribution

It extends Jacobson's problem to Malcev algebras, offering a new coordinatization theorem and structural description for specific classes of Malcev algebras.

Findings

Proves a coordinatization theorem for Malcev algebras containing sl_2.

Describes the structure of Malcev algebras containing sl_2 without the non-zero product condition.

Provides a classification of a certain class of Malcev algebras.

Abstract

In this paper we solve a problem for a certain class of Malcev algebras, which is an analogous of an old problem posed by Nathan Jacobson for alternative algebras. Specifically we prove a coordinatization theorem for a class of Malcev algebras containing the 3-dimensional simple Lie algebra $\mathfrak{s l}_{2}(\mathbb{F})$ such that $m\,\mathfrak{s l}_{2}(\mathbb{F})\neq 0$ for any $0\neq m\in\mathcal{M}.$ We drop the last condition and we describe the structure of the same class of Malcev algebras $\mathcal{M}$ that contains $\mathfrak{s l}_{2}(\mathbb{F})$.