罗塔 - 巴克斯特李\(H -\)伪代数的高阶结构
Higher Structures of Rota--Baxter Lie $H$-Pseudoalgebras
浏览论文内容
中文总结 AI 辅助
研究罗塔 - 巴克斯特李\(H -\)伪代数，通过Maurer - Cartan方法构建微分分次李代数，利用二阶上同调分类非阿贝尔扩张，引入\(2 -\)项结构探索同伦理论，建立严格\(2 -\)项结构与交叉模的对应，为其高阶范畴理论发展奠定基础。
中文摘要 AI 辅助
本文研究罗塔 - 巴克斯特李\(H -\)伪代数。通过Maurer - Cartan方法为\(\lambda -\)加权相对罗塔 - 巴克斯特算子发展了上同调理论，构建了基础的微分分次李代数。利用二阶上同调对非阿贝尔扩张进行分类并推导Wells正合序列以解决自同构的可诱导性。通过引入\(2 -\)项骨架和严格罗塔 - 巴克斯特\(L_\infty -\)\(H -\)伪代数探索其同伦理论，建立了严格\(2 -\)项结构与罗塔 - 巴克斯特李\(H -\)伪代数交叉模之间的一一对应。这些结果为带代数算子的伪代数高阶范畴理论的未来发展建立了基础框架。
英文摘要
This paper investigates Rota--Baxter Lie $H$-pseudoalgebras. We develop a cohomology theory for $λ$-weighted relative Rota--Baxter operators via a Maurer--Cartan approach, constructing the underlying differential graded Lie algebra. We classify non-abelian extensions using second cohomology and derive the Wells exact sequence to address the inducibility of automorphisms. Furthermore, we explore the homotopy theory of these structures by introducing $2$-term skeletal and strict Rota--Baxter $L_\infty$-$H$-pseudoalgebras. In particular, we establish a one-to-one correspondence between strict $2$-term structures and crossed modules of Rota--Baxter Lie $H$-pseudoalgebras. These results establish a foundational framework for future advancements in the higher categorical theory of pseudoalgebras with algebraic operators. Ultimately, this work provides a robust foundation for the higher categorical study of pseudoalgebras equipped with algebraic operators.
