Integration and geometrization of Rota-Baxter Lie algebras

This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on Lie group so that its differentiation gives corresponding algebra. Direct products groups, including decompositions Iwasawa and Langlands, carry natural operators. Formal inverse is precisely crossed homomorphism group, whose tangent map differential $1$ A factorization theorem groups proved, deriving directly level, well-known global theorems Semenov-Tian-Shansky in his study integrable systems. As geometrization, notions algebroids groupoids are introduced, with former latter. Further, algebroid naturally rise to post-Lie algebroid, generalizing fact for algebras algebras. It shown geometrization algebra or can be realized by action manifold. Examples applications provided these new notions.