More non semigroup Lie gradings

TL;DR

This paper constructs simple examples of graded Lie algebras in dimensions 4 and 6 where the grading is not derived from a semigroup, demonstrating that 4 is the minimal such dimension.

Contribution

It provides the first explicit examples of non-semigroup Lie gradings in low dimensions, including the minimal dimension of 4.

Findings

Constructed 4- and 6-dimensional graded Lie algebras with non-semigroup gradings

Proved that 4 is the minimal dimension for such examples

Included a semisimple algebra among the examples

Abstract

This note is devoted to the construction of two very easy examples, of respective dimensions 4 and 6, of graded Lie algebras whose grading is not given by a semigroup, the latter one being a semisimple algebra. It is shown that 4 is the minimal possible dimension.