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

TL;DR

This paper develops a comprehensive cohomology and extension theory for Nijenhuis Lie conformal algebras, linking their algebraic, homotopical, and automorphism structures with new cohomological invariants.

Contribution

It introduces cohomology, $ ext{L}_ ext{infty}$-structures, and classification methods for non-abelian extensions of Nijenhuis Lie conformal algebras, advancing their algebraic and homotopical understanding.

Findings

Defined cohomology for Nijenhuis Lie conformal algebras.

Established correspondence between 2-term Nijenhuis $ ext{L}_ ext{infty}$-conformal algebras and 3-cocycles.

Developed classification of non-abelian extensions via second cohomology group.

Abstract

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.