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

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 $\alpha :{\Lambda }^{2}V(6)\to V(6)$. 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 $β\colon Λ^2 V(6) \to V(2) \approx \s$ 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.

arXiv comment20 pages