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The exceptional Lie algebra $\mathfrak{g}_2$ generated by three generators subject to quadruple relations

N. I. Stoilova, J. Van der Jeugt

Abstract

In this short communication we show how the Lie algebra $\mathfrak{g}_2$ can easily be described as a free Lie algebra on 3 generators, subject to some simple quadruple relations for these generators.

The exceptional Lie algebra $\mathfrak{g}_2$ generated by three generators subject to quadruple relations

Abstract

In this short communication we show how the Lie algebra can easily be described as a free Lie algebra on 3 generators, subject to some simple quadruple relations for these generators.
Paper Structure (1 section, 1 theorem, 5 equations, 1 table)

This paper contains 1 section, 1 theorem, 5 equations, 1 table.

Table of Contents

  1. Acknowledgments

Key Result

Theorem 1

The Lie algebra $\mathfrak{g}$ on three generators $x_1, x_2, x_3$ subject to the following list of quadruple relations is equal to $\mathfrak{g}_2$: Herein, $\epsilon_{ijk}$ is the common Levi-Civita symbol in three dimensions: $\epsilon_{ijk}$ is 1 if $(i,j,k)$ is an even permutation of $(1,2,3)$, $-1$ if it is an odd permutation of $(1,2,3)$, and 0 otherwise.

Theorems & Definitions (1)

  • Theorem 1