On the integration of relative Rota-Baxter Lie algebras

TL;DR

This paper establishes criteria for integrating relative Rota-Baxter Lie algebras using Lie group structures and explores their implications for Poisson-Lie groups and Lie algebra cases, including a counterexample.

Contribution

It provides necessary and sufficient conditions for the integrability of relative Rota-Baxter Lie algebras and constructs a non-integrable matched pair of Lie algebras.

Findings

Criteria for integrability via double Lie groups, matched pairs, and diffeomorphism factorization

Characterization of Poisson-Lie groups related to Rota-Baxter operators

Construction of a non-integrable matched pair of Lie algebras

Abstract

In this paper, we give the necessary and sufficient conditions of the integrability of relative Rota-Baxter Lie algebras via double Lie groups, matched pairs of Lie groups and factorization of diffeomorphisms respectively. We use the integrability of Rota-Baxter operators to characterize whether the Poisson-Lie group integrating a factorizable Lie bialgebra is again factorizable. We thoroughly study the integrability of Rota-Baxter operators on the unique nontrivial 2-dimensional Lie algebra. As a byproduct, we construct a matched pair of Lie algebras that can not be integrated to a matched pair of Lie groups.