Structure of non-solvable cyclic metric Lie algebras

2025/09/18 by Huihui An, Ju Tan, Huihui, An +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2509.14695

openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper presents a systematic study of the structure of non-solvable cyclic metric Lie algebras. A cyclic metric is a symmetric bilinear form satisfying a cyclic cocycle condition, which arises naturally in the contexts of non-associative algebras and homogeneous pseudo-Riemannian manifolds. Firstly, we drive some sufficient conditions for a cyclic metric Lie algebra to be an orthogonal direct product of its semisimple and solvable parts. Then we introduce the notion of cyclic quadruples to analyze the interaction between semisimple and radical components. Finally, we use the double extension method to provide a complete characterization of non-degenerate cyclic metric Lie algebras that are neither semisimple nor solvable, over the fields of complex and real numbers.

Citations

• Cyclic Riemannian Lie groups: description and curvatures
• Lie algebras with ad-invariant metrics. A survey
• Cyclic Lorentzian Lie Groups
• Cyclic homogeneous Riemannian manifolds
• Cyclic metric Lie groups

Related

• Cyclic metric Lie groups
• On the cohomology of Lie algebras with an invariant inner product
• Cyclic Lorentzian Lie Groups
• Metric Lie algebras and quadratic extensions
• Cyclic Riemannian Lie groups: description and curvatures