Relative Rota-Baxter operators and crossed homomorphisms on Lie 2-groups

TL;DR

This paper introduces relative Rota-Baxter operators on Lie 2-groups and crossed modules, establishing their properties, categorical solutions to the Yang-Baxter equation, and the relationship with crossed homomorphisms, extending existing theories.

Contribution

It defines relative Rota-Baxter operators on Lie 2-groups and crossed modules, and establishes their equivalence and connection with crossed homomorphisms, expanding the theoretical framework.

Findings

Induced a factorization theorem for Lie 2-groups

Provided a categorical solution to the Yang-Baxter equation

Extended the correspondence between Lie 2-groups and crossed modules

Abstract

A relative Rota-Baxter operator on Lie 2-groups is introduced as a pair of relative Rota-Baxter operators on the underlying Lie groups which is also a Lie groupoid morphism. Such an operator induces a factorization theorem for Lie 2-groups and gives rise to a categorical solution of the Yang-Baxter equation. We further define relative Rota-Baxter operators on Lie group crossed modules. The well-known one-to-one correspondence between Lie 2-groups and crossed modules is extended to an equivalence between the respective relative Rota-Baxter operators on these two structures. Finally, as the formal inverse of relative Rota-Baxter operators, crossed homomorphisms on Lie 2-groups are also studied.