$\mathbb{Z}_2^3$-grading of the Lie algebra $G_2$ and related color algebras

Abstract

We present a special and attractive basis for the exceptional Lie algebra $G_2$, which turns $G_2$ into a $\mathbb{Z}_2^3$-graded Lie algebra. There are two basis elements for each degree of $\mathbb{Z}_2^3\setminus\{(0,0,0)\}$, thus yielding 14 basis elements. We give a general and simple closed form expression for commutators between these basis elements. Next, we use this $\mathbb{Z}_2^3$-grading in order to examine graded color algebras. Our analysis yields three different $\mathbb{Z}_2^3$-graded color algebras of type $G_2$. Since the $\mathbb{Z}_2^3$-grading is not compatible with a Cartan-Weyl basis of $G_2$, we also study another grading of $G_2$. This is a $\mathbb{Z}_2^2$-grading, compatible with a Cartan-Weyl basis, and for which we can also construct a $\mathbb{Z}_2^2$-graded color algebra of type $G_2$.