The Structure Constants of the Exceptional Lie Algebra ${\mathfrak g}_2$ in the Cartan-Weyl Basis

TL;DR

This paper investigates the structure constants of the exceptional Lie algebra g2 in the Cartan-Weyl basis, analyzing the compatibility of certain relations and establishing an isomorphism between different basis representations.

Contribution

It demonstrates the incompatibility of specific structure constant relations and constructs an explicit isomorphism between two basis approaches for g2.

Findings

Certain structure constant relations cannot be simultaneously satisfied.

An explicit isomorphism between different Cartan-Weyl basis representations is established.

The paper clarifies the derivation of structure constants using Cartan matrix elements.

Abstract

The purpose of this paper is to answer the question whether it is possible to realize simultaneously the relations $N_{\alpha,\beta}=-N_{-\alpha,-\beta}$, $N_{\alpha,\beta}=N_{\beta,-\alpha-\beta}=N_{-\alpha-\beta,\alpha}$ and $N_{\alpha,\beta}N_{-\alpha,-\beta}=-\frac{1}{2}q(p+1)\langle\alpha,H_{\alpha}\rangle$ by the structure constants of the Lie algebra ${\mathfrak g}_2$. We show that if the structure constants obey the first relation, the three last ones are violated, and vice versa. Contrary to the second case, the first one uses the Cartan matrix elements to derive the structure constants in the form of $\langle\beta,H_{\alpha}\rangle$. The commutation relations corresponding to the first case are exactly documented in the prior literature. However, as expected, a Lie algebra isomorphism is established between the Cartan-Weyl bases obtained in both approaches.