Enveloping Algebras of the Nilpotent Malcev Algebra of Dimension Five

Abstract

Pérez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra M over a field of characteristic ≠ 2, 3 there is a representation of the universal nonassociative enveloping algebra U(M) by linear operators on the polynomial algebra P(M). For the nilpotent non-Lie Malcev algebra $\mathbb{M}$ of dimension 5, we use this representation to determine explicit structure constants for $U(\mathbb{M})$; from this it follows that $U(\mathbb{M})$ is not power-associative. We obtain a finite set of generators for the alternator ideal $I(\mathbb{M}) \subset U(\mathbb{M})$ and derive structure constants for the universal alternative enveloping algebra $A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})$, a new infinite dimensional alternative algebra. We verify that the map $\iota\colon \mathbb{M} \to A(\mathbb{M})$ is injective, and so $\mathbb{M}$ is special.