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.