Post-Lie algebra structures on real oscillator algebras under a diagonal ansatz and applications

We study a class of post-Lie algebra structures on the finite-dimensional real oscillator algebras [Formula: see text]. More precisely, we impose a diagonal condition with respect to the standard basis [Formula: see text] and classify all post-Lie products satisfying this condition. The resulting solutions are organized into three explicitly parametrized families. As applications, we determine the diagonal Rota–Baxter operators of weight [Formula: see text] compatible with this ansatz and obtain the corresponding solutions of the classical Yang–Baxter equation on the finite-dimensional cotangent Lie algebra [Formula: see text].