Rota-Baxter型算子的形变理论、上同调理论及其应用

Rota-Baxter型算子在微扰量子场论的重整化理论和经典Yang-Baxter方程等领域中起到核心作用。形变理论和上同调理论给出数学结构的不变量，并且是理解数学结构的重要方法。目前，Rota-Baxter型算子是数学物理中的热点课题，受到越来越多的重视。本项目主要研究：（1）相对Rota-Baxter李代数的形变理论、上同调理论和同伦理论。（2）李代数上权为1的相对Rota-Baxter算子的形变理论、上同调理论和李代数上的post-Lie代数结构。（3）莱布尼茨代数上的相对Rota-Baxter算子、twisting理论和双代数理论。

Rota-Baxter type operators play a central role in the renormalization in perturbative quantum field theory and classical Yang-Baxter equations. Deformation theories and cohomology theories can give invariants of mathematical structures. They are also important methods to understand the mathematical structures. Recently, Rota-Baxter type operators are hot topics in mathematical physics. They have attracted more and more attentions. In this project, we mainly study: (1)Deformation theories, cohomology theories and homotopy theories of relative Rota-Baxter Lie algebras. (2)Deformation theories, cohomology theories of relative Rota-Baxter operators of weight 1 on Lie algebras and post-Lie algebra structures on Lie algebras. (3)Relative Rota-Baxter operators, twisting theories and bialgebraic theories on Leibniz algebras.