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I am studying Rota-Baxter Lie algebras. I do not know whether there exists the notion of free product, semi-direct product and derivation map for these types of algebras. May you introduce some papers or give me some explanation regarding this concepts.

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For a starter, this article is quite nice; it has a reference to the authors resources (in particular to his book on Rota-Baxter (Lie) algebras). We have an article on Rota-Baxter operators and Lie algebras, which might be helpful, including the references. Rota-Baxter Lie algebras are closely related to pre-Lie algebras and post-Lie algebras.

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