Matched pairs of Lie algebras and Rota-Baxter Lie algebras

2025/10/01 by Wang, Shukun
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2510.00834

Abstract

In this paper, we investigate the relationship between matched pairs of Lie algebras and Rota-Baxter Lie algebras. First, we show that every Rota-Baxter Lie algebra (\mathfrakg,B) of weight -1 gives rise to a matched pair of Lie algebras (\mathfrakg₊,\mathfrakg_-,\rhd,\bhd), and we prove that the bicrossed product Lie algebra decomposes as \mathfrakg₊\bowtie\mathfrakg_-=\mathfrakg₁⊕\mathfrakg₂. Moreover, we establish a Rota-Baxter Lie algebra structure on \mathfrakg₁ which is isomorphic to (\mathfrakg,B) as a Rota-Baxter Lie algebra, and we endow \mathfrakg₂ with a Rota-Baxter Lie algebra structure. Then we study the connection between quadratic Rota-Baxter Lie algebras and Manin triples. We prove that every quadratic Rota-Baxter Lie algebra of weight -1 gives rise to a Manin triple, and we obtain a decomposition theorem for this Manin triple. Finally, we show that every Rota-Baxter group induces a matched pair of groups and investigate the internal structure of the induced matched pair of groups.

Citations

• Factorizable Lie Bialgebras, Quadratic Rota–Baxter Lie Algebras and Rota–Baxter Lie Bialgebras
• Integrable Systems and Factorization Problems
• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem

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