标题： 拟双李代数的变形映射 显示英文标题

标题： Deformation maps of quasi-twilled Lie algebras

摘要： 在本文中，我们提供了一种统一的方法来研究李代数范畴中各种类型算子的上同调理论和形变理论，包括修正的$r$矩阵、交叉同态、导子、同态、相对 Rota-Baxter 算子、扭曲 Rota-Baxter 算子、Reynolds 算子以及李代数匹配对的形变映射。主要元素是准扭结李代数。我们引入了准扭结李代数的两种类型的形变映射。类型 I 的形变映射统一了修正的$r$矩阵、交叉同态、导子和李代数之间的同态，而类型 II 的形变映射统一了相对 Rota-Baxter 算子、扭曲 Rota-Baxter 算子、Reynolds 算子以及李代数匹配对的形变映射。我们进一步给出了这两种类型形变映射的控制代数和上同调，这不仅恢复了现有关于交叉同态、导子、同态、相对 Rota-Baxter 算子、扭曲 Rota-Baxter 算子和 Reynolds 算子的结果，还导致了一些以前无法解决的新结果，例如修正的$r$矩阵以及李代数匹配对的形变映射的控制代数和上同调。 显示英文摘要

摘要： In this paper, we provide a unified approach to study the cohomology theories and deformation theories of various types of operators in the category of Lie algebras, including modified $r$-matrices, crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. The main ingredients are quasi-twilled Lie algebras. We introduce two types of deformation maps of a quasi-twilled Lie algebra. Deformation maps of type I unify modified $r$-matrices, crossed homomorphisms, derivations and homomorphisms between Lie algebras, while deformation maps of type II unify relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. We further give the controlling algebras and cohomologies of these two types of deformation maps, which not only recover the existing results for crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators, but also leads to some new results which are unable to solve before, e.g. the controlling algebras and cohomologies of modified $r$-matrices and deformation maps of matched pairs of Lie algebras.