Compatible Lie Brackets and the Yang–Baxter Equation

We show that any pair of compatible Lie brackets with a common invariant form produces a nonconstant solution of the classical Yang–Baxter equation. We describe the corresponding Poisson brackets, Manin triples, and Lie bialgebras. It turns out that all bialgebras associated with the solutions found by Belavin and Drinfeld are isomorphic to some bialgebras generated by our solutions. For any compatible pair, we construct a double with a common invariant form and find the corresponding solution of the quantum Yang–Baxter equation for this double.