Enveloping algebras of the nilpotent Malcev algebra of dimension five

TL;DR

This paper investigates the structure of the universal enveloping algebra of a specific 5-dimensional nilpotent Malcev algebra, revealing its non-power-associativity and constructing its universal alternative enveloping algebra.

Contribution

It explicitly determines the structure constants of the enveloping algebra and constructs the universal alternative enveloping algebra for the nilpotent Malcev algebra of dimension five.

Findings

U(M) is not power-associative

Constructed the universal alternative enveloping algebra A(M)

Proved the Malcev algebra is special

Abstract

Perez-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 $\ne 2, 3$ 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.