Matched pairs of Lie algebras and Rota-Baxter Lie algebras

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 $(\mathfrak{g},B)$ of weight $-1$ gives rise to a matched pair of Lie algebras $({\mathfrak{g}}_{+},{\mathfrak{g}}_{-},⊳,\text{bhd})$, and we prove that the bicrossed product Lie algebra decomposes as ${\mathfrak{g}}_{+}\bowtie {\mathfrak{g}}_{-}={\mathfrak{g}}_1\oplus {\mathfrak{g}}_2$. Moreover, we establish a Rota-Baxter Lie algebra structure on ${\mathfrak{g}}_1$ which is isomorphic to $(\mathfrak{g},B)$ as a Rota-Baxter Lie algebra, and we endow ${\mathfrak{g}}_2$ 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.