Two Mal'cev-type theorems in universal algebra

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 general...

Universal Algebra in Type Theory

We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. types endowed with an arbitrary equivalence relation, as carriers for algebras. In this way it is possible to define the quotient of an algebra by a congruence. Standard constructions over algebra...

Enveloping algebras of the nilpotent Malcev algebra of dimension five

Pérez-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 6= 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 rep...

The Simple Non-lie Malcev Algebra as a Lie-yamaguti Algebra

The simple 7-dimensional Malcev algebra M is isomorphic to the irreducible sl(2,C)-module V (6) with binary product [x, y] = α(x ∧ y) defined by the sl(2,C)-module morphism α : Λ2V (6)→ V (6). Combining this with the ternary product (x, y, z) = β(x∧y) ·z defined by the sl(2,C)-module morphism β : Λ2V (6)→ V (2) ≈ sl(2,C) gives M the structure of a generalized Lie triple system, or Lie-Yamaguti ...

Universal Algebra