On $\mathcal{O}$-operators on modules over Lie algebras

The notion of $\mathcal{O}$-operators on modules over Lie algebras generalize Rota-Baxter operators. They also generalize Poisson structures on Lie algebras in the presence of modules. Motivated from Poisson structures, we define gauge transformations and reductions of $\mathcal{O}$-operators. Next we consider compatible $\mathcal{O}$-operators on modules over Lie algebras. We define $\mathcal{O}\mathcal{N}$-structures which give rise to hierarchy of compatible $\mathcal{O}$-operators. We show that a solution of the strong Maurer-Cartan equation on a twilled Lie algebra associated to an $\mathcal{O}$-operator gives rise to an $\mathcal{O}\mathcal{N}$-structure, hence, a hierarchy of compatible $\mathcal{O}$-operators. Finally, we also introduce generalized complex structures and holomorphic $\mathcal{O}$-operators on modules over Lie algebras and show how they incorporate $\mathcal{O}$-operators.