Enveloping Algebras of the Nilpotent Malcev Algebra of Dimension Five

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 not equal 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 M of dimension 5, we use this representation to determine explicit structure constants for U (M); from this it follows that U (M) is not power-associative. We obtain a finite set of generators for the alternator ideal I (M) subset of U(M) and derive structure constants for the universal alternative enveloping algebra A (M) = U(M)/I(M), a new infinite dimensional alternative algebra. We verify that the map M -> A (M) is injective, and so M is special.