In this paper, we establish a local Lie theory for relative Rota–Baxter operators of weight 1. First we recall the category of relative Rota–Baxter operators of weight 1 on Lie algebras and construct a cohomology theory for them. We use the second cohomology group to study infinitesimal deformations of relative Rota–Baxter operators and modified $r$-matrices. Then we introduce a cohomology theory of relative Rota–Baxter operators on a Lie group. We construct the differentiation functor from the category of relative Rota–Baxter operators on Lie groups to that on Lie algebras, and extend it to the cohomology level by proving the Van Est theorem between the two cohomology theories. We integrate a relative Rota–Baxter operator of weight 1 on a Lie algebra to a local relative Rota–Baxter operator on the corresponding Lie group, and show that the local integration and differentiation are adjoint to each other. Finally, we give two applications of our integration of Rota–Baxter operators: one is to give an explicit formula for the factorization problem, and the other is to provide an integration for matched pairs.

中文翻译：

---

相关 Rota-Baxter 算子的李理论和上同调

在本文中，我们建立了权重为1的相对Rota-Baxter算子的局部李理论。首先我们回顾了李代数上权重为1的相对Rota-Baxter算子的范畴，并为它们构造了一个上同调理论。我们使用第二上同调群来研究相对 Rota-Baxter 算子的无穷小变形，并修改$r$-矩阵。然后我们介绍了李群上相关 Rota-Baxter 算子的上同调理论。我们构造了从李群上的相对Rota-Baxter算子范畴到李代数上的微分函子，并通过证明两种上同调理论之间的Van Est定理将其扩展到上同调水平。我们将李代数上权重为 1 的相对 Rota-Baxter 算子积分到相应李群上的局部相对 Rota-Baxter 算子，并证明局部积分和微分是相互伴随的。最后，我们给出了 Rota-Baxter 算子积分的两个应用：一个是给出分解问题的显式公式，另一个是提供匹配对的积分。