Enveloping algebras of Malcev algebras

TL;DR

This paper explores the structure and properties of universal enveloping algebras of Malcev algebras, including explicit calculations and isomorphisms with known algebraic structures like octonions.

Contribution

It provides explicit structure constants for certain Malcev algebras and proves isomorphisms with well-known algebraic systems, expanding understanding of Malcev algebra enveloping algebras.

Findings

Explicit structure constants for 4D solvable Malcev algebra

Explicit structure constants for 5D nilpotent Malcev algebra

Isomorphism between the 7D simple Malcev algebra's enveloping algebra and octonions

Abstract

We first discuss the construction by Perez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some brief remarks on tangent algebras of analytic Bol loops and monoassociative loops. This is a revised and expanded version of the survey talk given by the first author at the Second Mile High Conference on Nonassociative Mathematics at the University of Denver (Denver, Colorado, USA, June 22 to 26, 2009).