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Lie theory and cohomology of relative Rota-Baxter operators

Jun Jiang, Yunhe Sheng, Chenchang Zhu

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.

Lie theory and cohomology of relative Rota-Baxter operators

Abstract

In this paper, we establish a local Lie theory for relative Rota-Baxter operators of weight . First we recall the category of relative Rota-Baxter operators of weight 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 -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.

Paper Structure

This paper contains 18 sections, 33 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Relative Rota-Baxter operators on Lie algebras and their cohomology theory
  3. The category $\mathsf{RB_\mathfrak g^\mathfrak h}$ of relative Rota-Baxter operators on Lie algebras
  4. A cohomology theory in $\mathsf{RB_\mathfrak g^\mathfrak h}$
  5. Cohomology theory and deformations in $\mathsf{RB_\mathfrak g^\mathfrak h}$ and modified $r$-matrices
  6. Relative Rota-Baxter operators on Lie groups and their cohomology theory
  7. The category $\mathsf{RB_G^H}$ of relative Rota-Baxter operators on Lie groups
  8. A cohomology theory in $\mathsf{RB_G^H}$
  9. Differentiation and Van Est theory
  10. The differentiation functor for relative Rota-Baxter operators
  11. The Van Est homomorphism for the cohomologies
  12. An explicit example
  13. Local integration of relative Rota-Baxter operators on Lie algebras
  14. Local integration functor and adjointness
  15. Geometric applications
  16. ...and 3 more sections

Key Result

Proposition 2.5

Let $\phi: \mathfrak g\rightarrow\mathrm{Der}(\mathfrak h)$ be an action of a Lie algebra $(\mathfrak g, [\cdot,\cdot]_{\mathfrak g})$ on a Lie algebra $(\mathfrak h, [\cdot,\cdot]_{\mathfrak h})$. Then a linear map $B: \mathfrak h\rightarrow\mathfrak g$ is a relative Rota-Baxter operator if and onl

Theorems & Definitions (84)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • ...and 74 more