BiHom-型代数及罗巴算子的若干研究

本项目主要研究BiHom-型代数的一些结构、性质及其上的罗巴算子。研究分为三部分：首先，定义BiHom-双代数上的Yang-Baxter方程并揭示该方程与罗巴算子之间的关系；通过拟三角BiHom-双代数的有限维表示范畴和BiHom-双代数上的Yetter-Drinfeld模范畴来刻画该方程的解。其次，研究BiHom-Lie代数上的罗巴算子的性质，并给出BiHom-型下的dendriform、quadri、pre-Lie、Leibniz和PostLie代数的定义及性质，构造自由罗巴BiHom-结合代数并利用罗巴算子来考虑这些代数上的范畴、函子间的关系。最后，引入广义罗巴算子和广义导子的定义并且发现广义罗巴算子来源于结合(Bi)Hom-Yang-Baxter的解，同时利用广义罗巴算子得到由Hom-Infinitesimal代数构造Hom-pre-Lie代数的方法。

In this project, we will study the constructions and properties of the BiHom-type algebras and the Rota-Baxter operators on these algebras. Firstly, we will define the Yang-Baxter equation on BiHom-type algebras and explore the relation between this new equation and Rota-Baxter operators. The finite-dimensional module category of the quasitriangular BiHom-bialgebra and the category of Yetter-Drinfeld modules on BiHom-bialgebra can character the solution to the Yang-Baxter equation. Secondly, we investigate some properties of Rota-Baxter operators on BiHom-Lie algebras. Along the way, we introduce and discuss the properties of the notions of BiHom analogues of dendriform, quadri, pre-Lie, Leibniz and PostLie algebras. We construct the free Rota-Baxter BiHom-associative algebra and present some observations about categories and functor related to Rota-Baxter structures. Finally, we introduce the concepts of generalized Rota-Baxter operator and generalized derivation. We show how to obtain a certain generalized Rota-Baxter operator from a solution of the associative (Bi)Hom-Yang-Baxter equation, and in a compatible way, a Hom-pre-Lie algebra from a Hom-Infinitesimal algebra.