Lie theory and cohomology of relative Rota-Baxter operators

TL;DR

This paper develops a local Lie theory for relative Rota-Baxter operators of weight 1, establishing cohomology theories, deformation analysis, and integration methods between Lie algebras and Lie groups, with applications to factorization and matched pairs.

Contribution

It introduces a cohomology framework for relative Rota-Baxter operators on Lie algebras and groups, and constructs an integration/differentiation correspondence extending classical Lie theory.

Findings

Cohomology theory for relative Rota-Baxter operators on Lie algebras and groups

Van Est theorem linking the two cohomologies

Explicit formulas for factorization and matched pair integration

Abstract

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.