Enveloping algebras of the nilpotent Malcev algebra of dimension five

Details Enveloping algebras of the nilpotent Malcev algebra of dimension five Enveloping algebras of the nilpotent Malcev algebra of dimension five

2010-08-16

arxiv.org/abs/1008.2728v1

Mathematics

Rings and Algebras

15 pages

Scientific paper

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.

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