Homotopy relative Rota-Baxter Lie algebras, triangular $L_\infty$-bialgebras and higher derived brackets

We describe $L_\infty$-algebras governing homotopy relative Rota-Baxter Lie algebras and triangular $L_\infty$-bialgebras, and establish a map between them. Our formulas are based on a functorial approach to Voronov's higher derived brackets construction which is of independent interest.

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