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

Sania Asif

TL;DR

The paper develops a comprehensive framework for Nijenhuis Lie conformal algebras by establishing a cohomology theory tied to deformation and extension problems. It introduces a two-tiered homotopy perspective via $2$-term ${\mathcal{L}}_\infty$-conformal algebras and shows skeletal cases correspond to $3$-cocycles while strict cases align with crossed modules. Non-abelian and abelian extension theories are classified through second cohomology groups, with a Wells-type map governing automorphism inducibility and yielding an exact sequence that reveals the interplay between automorphisms and cohomology. The results provide a rigorous toolkit for deformation, higher-homotopy structures, and extension problems in the conformal algebra setting, with potential implications for vertex algebras and conformal field theory.

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.

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

TL;DR

The paper develops a comprehensive framework for Nijenhuis Lie conformal algebras by establishing a cohomology theory tied to deformation and extension problems. It introduces a two-tiered homotopy perspective via -term -conformal algebras and shows skeletal cases correspond to -cocycles while strict cases align with crossed modules. Non-abelian and abelian extension theories are classified through second cohomology groups, with a Wells-type map governing automorphism inducibility and yielding an exact sequence that reveals the interplay between automorphisms and cohomology. The results provide a rigorous toolkit for deformation, higher-homotopy structures, and extension problems in the conformal algebra setting, with potential implications for vertex algebras and conformal field theory.

Abstract

This paper explores various algebraic and homotopical aspects of Nijenhuis Lie conformal algebras, including their cohomology theory, -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 -term Nijenhuis -conformal algebras and establish their correspondence with crossed modules and -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.

Paper Structure

This paper contains 16 sections, 25 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of Nijenhuis operators on Lie conformal algebra
  3. Cohomology theory of Nijenhuis operators and Nijenhuis Lie conformal algebras
  4. Cohomology of the Lie conformal algebra
  5. Cohomology of Nijenhuis operators on a Lie conformal algebra
  6. Formal deformations of Nijenhuis operators on Lie conformal algebra
  7. Extensions of the deformation of finite order
  8. Representation and Cohomology of Nijenhuis Lie conformal algebra
  9. The cohomology of Nijenhuis Lie conformal algebra with the coefficients in the adjoint Nijenhuis representation
  10. The cohomology of the Nijenhuis Lie conformal algebra, with the coefficients in the Nijenhuis representation
  11. Skeletal and Strict $2$-terms Homotopy Nijenhuis Lie conformal algebra
  12. Skeletal $2$-terms Nijenhuis ${\mathcal{L}}_\infty$-conformal algebras
  13. Strict $2$-terms Nijenhuis ${\mathcal{L}}_\infty$-conformal algebras
  14. Non-Abelian Extension of Nijenhuis Lie conformal algebra
  15. Special Case: Abelian Extension
  16. ...and 1 more sections

Key Result

Proposition 2.5

Let ${\mathcal{L}}$ be a Lie conformal algebra, and ${\mathcal{N}}$ be a Nijenhuis operator. Then $({\mathcal{L}},[\cdot_\lambda\cdot], {\mathcal{N}})$ yield another Nijenhuis Lie conformal algebra with the new Lie bracket given by $[p_\lambda q]_{\mathcal{N}}= [{\mathcal{N}}(p)_{\lambda}q]+ [p_{\la

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • ...and 55 more