标题： 同调，同伦，扩张和Nijenhuis李共形代数的自同构 显示英文标题

标题： Cohomology, Homotopy, Extensions, and Automorphisms of Nijenhuis Lie Conformal Algebras

摘要： 本文探讨了Nijenhuis李共形代数的各种代数和同伦方面，包括它们的上同调理论、$\mathcal{L}_\infty$-结构、非交换扩张以及自同构群。我们定义了Nijenhuis李共形代数的上同调，并将其与这类结构的形变理论联系起来。我们还引入了$2$-项Nijenhuis$\mathcal{L}_\infty$-共形代数，并建立了它们与Nijenhuis李共形代数上同调中的交叉模和$3$-上循环之间的对应关系。此外，我们通过Nijenhuis李共形代数的第二非交换上同调群发展了一种分类理论来研究其非交换扩张。最后，我们研究了在这种扩张下自同构的可诱导性问题，引入了一种Wells型映射并推导出相应的正合序列。 显示英文摘要

摘要： This paper explores various algebraic and homotopical aspects of Nijenhuis Lie conformal algebras, including their cohomology theory, $\mathcal{L}_\infty$-structures, non-abelian extensions, and automorphism groups. We define the cohomology of a Nijenhuis Lie conformal algebra and relate it to the deformation theory of such structures. We also introduce $2$-term Nijenhuis $\mathcal{L}_\infty$-conformal algebras and establish their correspondence with crossed modules and $3$-cocycles in the cohomology of Nijenhuis Lie conformal algebras. Furthermore, we develop a classification theory for non-abelian extensions of Nijenhuis Lie conformal algebras via the second non-abelian cohomology group. Finally, we study the inducibility problem for automorphisms under such extensions, introducing a Wells-type map and deriving an associated exact sequence.