The Structure Constants of the Exceptional Lie Algebra \mathfrak g₂ in the Cartan-Weyl Basis

2016/03/16 by H. Fakhri, Fakhri, H., M. Sayyah-Fard +3
Chemistry · Mathematics · #17B22 #22E60 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1603.05606

openalex publication_date 2016/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to answer the question whether it is possible to realize simultaneously the relations N_α,β=-N_-α,-β, N_α,β=N_β,-α-β=N_-α-β,α and N_α,βN_-α,-β=-(1)/(2)q(p+1)⟨α,H_α⟩ by the structure constants of the Lie algebra \mathfrak g₂. 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 ⟨β,H_α⟩. 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.

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