Lie algebras with S4-action and structurable algebras

TL;DR

This paper explores the connection between certain algebraic structures with S4 symmetry and Lie algebras, showing how these structures classify Lie algebras with specific automorphism groups.

Contribution

It establishes a correspondence between normal symmetric triality algebras, Lie related triple algebras, and Lie algebras with S4 automorphism groups, linking these concepts.

Findings

Normal LRTA's are exactly structurable algebras.

Normal STA's and LRTA's classify Lie algebras with S4 automorphism groups.

Unital LRTA's coincide with known structurable algebras.

Abstract

The normal symmetric triality algebras (STA's) and the normal Lie related triple algebras (LRTA's) have been recently introduced by the second author, in connection with the principle of triality. It turns out that the unital normal LRTA's are precisely the structurable algebras extensively studied by Allison. It will be shown that the normal STA's (respectively LRTA's) are the algebras that coordinatize those Lie algebras whose automorphism group contains a copy of the alternating (resp. symmetric) group of degree 4.