Matched pairs of Lie algebras and Rota-Baxter Lie algebras

In this paper, we investigate the relationship between Rota-Baxter Lie algebras of weight Formula: see text and matched pairs of Lie algebras, providing an operator-theoretic interpretation of Semenov-Tian-Shansky’s Infinitesimal Factorization Theorem. For a Rota-Baxter Lie algebra Formula: see text of weight Formula: see text, we prove that Formula: see text gives rise to a matched pair of Lie algebras Formula: see text. Conversely, we show that the bicrossed product of Formula: see text and Formula: see text admits a decomposition Formula: see text. Next, we establish Rota-Baxter Lie algebra structures on Formula: see text and Formula: see text respectively. Finally, we globalize these results to the level of Rota-Baxter groups. We prove that every Rota-Baxter group of weight Formula: see text Formula: see text induces a matched pair of groups Formula: see text, and investigate the internal decomposition of Formula: see text through group projections.