Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation

Abstract

In this paper, we construct a categorical solution $(C,R)$ of the Yang-Baxter equation, i.e. $C$ is a small category and $R:C\times C\to C\times C$ is an invertible functor satisfying$(R\times {\mathrm{Id}}_C)({\mathrm{Id}}_C\times R)(R\times {\mathrm{Id}}_C)=({\mathrm{Id}}_C\times R)(R\times {\mathrm{Id}}_C)({\mathrm{Id}}_C\times R),$ where $C\times C$ is the product category. First, the notion of Rota-Baxter operators on crossed modules of Lie groups is defined and its various properties are established. Then, we use Rota-Baxter operators on crossed modules of Lie groups to construct categorical solutions of the Yang-Baxter equation. We also study the Rota-Baxter operators on crossed modules of Lie algebras which are infinitesimals of Rota-Baxter operators on crossed modules of Lie groups, they can give connections on manifolds. Finally, we study the integration of Rota-Baxter operators on crossed modules of Lie algebras and the differentials of Rota-Baxter operators on crossed modules of Lie groups.