Universal enveloping algebras of the four-dimensional Malcev algebra
We determine structure constants for the universal nonassociative enveloping algebra U(M) of the four-dimensional non-Lie Malcev algebra M by constructing a representation of U(M) by differential operators on the polynomial algebra P (M). These structure constants involve Stirling numbers of the second kind. This work is based on the recent theorem of Pérez-Izquierdo and Shestakov which generalizes the Poincaré-Birkhoff-Witt theorem from Lie algebras to Malcev algebras. We use our results for U(M) to determine structure constants for the universal alternative enveloping algebra A(M) = U(M)/I(M) where I(M) is the alternator ideal of U(M). The structure constants for A(M) were obtained earlier by Shestakov using different methods.
This article is published as Bremner, Murray R., Irvin R. Hentzel, Luiz A. Peresi, and Hamid Usefi. "Universal enveloping algebras of the four-dimensional Malcev algebra." Contemporary Mathematics 483 (2009): 73-89. Posted with permission.