标题： 等变Nijenhuis李代数：对经典李理论结构的扩展 显示英文标题

标题： Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures

摘要： 我们发展了等变Nijenhuis李代数（ENL代数）的结构理论，即配备满足关于伴随表示的等变条件的Nijenhuis算子的李代数。这种刚性使得经典的李双代数构造可以系统地扩展到带有算子的设置中。在这个框架内，我们定义了ENL双代数，并建立了相关的匹配对、马宁三元组和Drinfel'd双代数的概念。我们证明了余边界ENL双代数由满足等变经典Yang-Baxter方程的EN$r$-矩阵所表征。我们进一步引入了EN相对Rota-Baxter算子，并证明它们为这样的$r$-矩阵提供了算子理论实现，从而得到后代ENL代数以及半直积ENL代数上的经典Yang-Baxter方程的解。在二次情况下，此构造退化为零权的Rota-Baxter算子。最后，我们将EN框架扩展到预李代数，并表明预-ENL代数自然地诱导出相关的ENL结构。 显示英文摘要

摘要： We develop a structural theory of equivariant Nijenhuis Lie algebras (ENL algebras), namely, Lie algebras equipped with Nijenhuis operators satisfying an equivariance condition with respect to the adjoint representation. This rigidity allows classical Lie bialgebra constructions to extend systematically to the operator-equipped setting. Within this framework, we define ENL bialgebras and establish the associated notions of matched pairs, Manin triples, and Drinfel'd doubles. We show that coboundary ENL bialgebras are characterized by EN $r$-matrices satisfying an equivariant classical Yang-Baxter equation. We further introduce EN-relative Rota-Baxter operators and prove that they provide an operator-theoretic realization of such $r$-matrices, leading to descendent ENL algebras and to solutions of the classical Yang--Baxter equation on semidirect ENL algebras. In the quadratic case, this construction reduces to Rota-Baxter operators of weight zero. Finally, we extend the EN framework to pre-Lie algebras and show that pre-ENL algebras naturally induce associated ENL structures.