On a problem by Nathan Jacobson for Malcev algebras

TL;DR

The work determines the structure of Malcev algebras that contain the simple Lie algebra sl2 over a field F, situating the problem in Filippov's subvariety H and extending Jacobson's classical coordinatization ideas. It proves a coordinatization theorem showing that, under a nondegeneracy condition, such algebras are built from a unital commutative associative algebra B and a B-bimodule V with a skew form ⟨.,.⟩: V×V→B satisfying Plucker relations, yielding M ≅ sl2(B) ⊕ V2 with a specific product. The converse holds, and dropping the nondegeneracy yields a general decomposition Ann_M L ⊕ (sl2(U) ⊕ V2) that encompasses exceptional and Cayley–Dickson type examples, linking to the 7-dimensional exceptional Malcev algebra M7 and Cayley–Dickson extensions. The results provide a Malcev analogue of Kronecker factorization for algebras containing sl2 and reveal how these algebras realize either M7-like or Cayley–Dickson–type constructions, with implications for non-Lie modules and graded structures in Malcev theory.

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 $\mathfrak{s l}_{2}(\mathbb{F})$.