The simple non-Lie Malcev algebra as a Lie-Yamaguti algebra

TL;DR

This paper shows that the 7-dimensional simple Malcev algebra can be viewed as a Lie-Yamaguti algebra by defining compatible binary and ternary products, and uses computer algebra to analyze its identities.

Contribution

It demonstrates that the simple Malcev algebra admits a Lie-Yamaguti algebra structure and characterizes its polynomial identities using computer algebra.

Findings

The Malcev algebra can be structured as a Lie-Yamaguti algebra.

Polynomial identities of low degree are determined.

The structure links Malcev algebras with Lie-Yamaguti algebras.

Abstract

The simple 7-dimensional Malcev algebra $M$ is isomorphic to the irreducible $\mathfrak{sl}(2,\mathbb{C})$-module V(6) with binary product $[x,y] = \alpha(x \wedge y)$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism $\alpha\colon \Lambda^2 V(6) \to V(6)$. Combining this with the ternary product $(x,y,z) = \beta(x \wedge y) \cdot z$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism $\beta\colon \Lambda^2 V(6) \to V(2) \approx \s$ gives $M$ the structure of a generalized Lie triple system, or Lie-Yamaguti algebra. We use computer algebra to determine the polynomial identities of low degree satisfied by this binary-ternary structure.