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

N. I. Stoilova, J. Van der Jeugt

TL;DR

This work constructs and analyzes a ${f Z}_2^3$-grading of the exceptional Lie algebra ${G}_2$, realizing it via a 14-dimensional basis $A_\alpha^\zeta$ aligned with the oriented Fano plane. It derives a uniform closed-form description of all commutators among these generators and uses this grading to build three distinct ${Z}_2^3$-graded color Lie algebras of type ${G}_2$, each with two basis elements per nonzero degree and explicit bracket structures. It also identifies a ${Z}_2^2$-graded color algebra compatible with a Cartan-Weyl basis, providing a toral grading where the Cartan subalgebra is diagonal and root vectors are organized accordingly. The results yield explicit matrix realizations and bracket tables that facilitate applications in (super)conformal quantum mechanics and motivate extensions to higher-arity gradings and exceptional types such as ${G}_3$.

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 $G_2$.

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

TL;DR

This work constructs and analyzes a -grading of the exceptional Lie algebra , realizing it via a 14-dimensional basis aligned with the oriented Fano plane. It derives a uniform closed-form description of all commutators among these generators and uses this grading to build three distinct -graded color Lie algebras of type , each with two basis elements per nonzero degree and explicit bracket structures. It also identifies a -graded color algebra compatible with a Cartan-Weyl basis, providing a toral grading where the Cartan subalgebra is diagonal and root vectors are organized accordingly. The results yield explicit matrix realizations and bracket tables that facilitate applications in (super)conformal quantum mechanics and motivate extensions to higher-arity gradings and exceptional types such as .

Abstract

We present a special and attractive basis for the exceptional Lie algebra , which turns into a -graded Lie algebra. There are two basis elements for each degree of , thus yielding 14 basis elements. We give a general and simple closed form expression for commutators between these basis elements. Next, we use this -grading in order to examine graded color algebras. Our analysis yields three different -graded color algebras of type . Since the -grading is not compatible with a Cartan-Weyl basis of , we also study another grading of . This is a -grading, compatible with a Cartan-Weyl basis, and for which we can also construct a -graded color algebra of type .
Paper Structure (10 sections, 2 theorems, 55 equations, 1 figure)

This paper contains 10 sections, 2 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. ${\mathbb Z}_2^3$-grading of the Lie algebras $\mathfrak{so}(7)$ and $G_2$
  3. The grading group $\Gamma$ and the oriented Fano plane
  4. Basis of $\mathfrak{so}(7)$ and $G_2$
  5. ${\mathbb Z}_2^3$-graded color Lie algebras of type $G_2$
  6. Case 1
  7. Case 2
  8. Case 3
  9. ${\mathbb Z}_2^2$-graded color Lie algebra of type $G_2$ in the Cartan-Weyl basis
  10. Concluding remarks

Key Result

Proposition 1

The 21 matrices $A_\alpha^\lambda$, with $\alpha\in\Gamma^*$ and $\lambda\in\alpha^\perp$ satisfy: otherwise (under the assumption that $\sigma(\alpha,\beta,\alpha+\beta)=+1$)

Figures (1)

  • Figure 1: The oriented Fano plane, with the 7 points labeled by encircled elements of $\Gamma^*$, and the 7 lines labeled in italic by the elements of $\Gamma^*$.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2