Lie algebras with S3 or S4-action, and generalized Malcev algebras

TL;DR

This paper studies Lie algebras with S3 or S4 automorphism actions, decomposing them into irreducible modules and introducing generalized Malcev algebras that extend classical Malcev structures.

Contribution

It introduces generalized Malcev algebras with binary and ternary products, extending classical Malcev algebras and linking them to Lie algebras with symmetric group actions.

Findings

Decomposition of Lie algebras with S3/S4 symmetry into irreducible modules.

Introduction of generalized Malcev algebras with new algebraic structures.

Connection between these algebras and classical Malcev and Jordan systems.

Abstract

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In case of S3-symmetry, the Lie algebras are coordinatized by some nonassociative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary products, and include both the Malcev algebras and the Jordan triple systems.