2024/11/25 by Jun Jiang, Jiang, Jun
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2411.16040
openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we construct a categorical solution (\huaC, R) of the Yang-Baxter equation, i.e. \huaC is a small category and R: \huaC×\huaC\lon\huaC×\huaC is an invertible functor satisfying (R×\Id_\huaC)(\Id_\huaC× R)(R×\Id_\huaC)=(\Id_\huaC× R)(R×\Id_\huaC)(\Id_\huaC× R), where \huaC×\huaC 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.