Nonabelian Generalized Lax Pairs, The Classical Yang-Baxter Equation And PostLie Algebras

We generalize the classical study of (generalized) Lax pairs, the related (?)-operators and the (modified) classical Yang-Baxter equation by introducing the con-cepts of nonabelian generalized Lax pairs, extended (?)-operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized r-matrix ansatz and the related double Lie algebra structures. Relationship between extended (?)-operators and the extended classical Yang-Baxter equation is estab-lished, especially for self-dual Lie algebras. This relationship allows us to obtain explicit description of the Manin triples for a new class of Lie bialgebras. Further-more, we show that a natural structure of PostLie algebra is behind (?)-operators and fits in a setup of triple Lie algebra that produces self-dual nonabelian generalized Lax pairs.