Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deformations of Nijenhuis Lie algebras and Nijenhuis Lie algebroids

Chao Song, Kai Wang, Yuanyuan Zhang, Guodong Zhou

TL;DR

The paper develops a comprehensive operadic and homotopical framework for deformations of Nijenhuis structures, both algebraic (Nijenhuis Lie algebras) and geometric (Nijenhuis Lie algebroids). It constructs the minimal model ${\mathfrak{NjL}_{\infty}}$ via a Koszul dual cooperad ${\mathfrak{NjL}^{\!'}}$ and uses this to define an $L_\infty$-deformation complex that encodes simultaneous deformations of Lie brackets and Nijenhuis operators. A major outcome is the establishment of a Poicare-type vanishing result for a class of Nijenhuis operators (the Poincaré Lemma for diagonal operators), confirming a Bolsinov–Konyaev conjecture. The work also extends these algebraic structures to Lie algebroids and dg-manifolds, yielding a corresponding $L_\infty$-algebra controlling deformations of geometric Nijenhuis structures and bridging algebraic and geometric cohomology theories. Overall, the paper significantly deepens the operadic understanding of Nijenhuis geometry and its deformation theory, with concrete computational applications in cohomology vanishing results.

Abstract

This paper is the second in a series dedicated to the operadic study of Nijenhuis structures, focusing on Nijenhuis Lie algebras and Nijenhuis geometry. We introduce the concept of homotopy Nijenhuis Lie algebras and establish that the differential graded (=dg) operad $\mathfrak{NjL}_{\infty}$ governing these structures serves as the minimal model of the operad $\mathfrak{NjL}$ for Nijenhuis Lie algebras. We construct an $L_\infty$-algebra that encodes the simultaneous deformations of Lie brackets and Nijenhuis operators, leading to the deformation cochain complex and an associated cohomology theory for Nijenhuis Lie algebras. Extending these ideas to geometry, we investigate the deformations of geometric Nijenhuis structures. We introduce the notion of a Nijenhuis Lie algebroid-a Lie algebroid equipped with a Nijenhuis structure, which generalizes the classical Nijenhuis structure on vector fields of manifolds. Using the framework of dg manifolds, we construct an $L_\infty$-algebra that governs the simultaneous deformations of Lie algebroid structures and Nijenhuis operators. As a computational application, we prove that a certain class of Nijenhuis operators satisfies the Poincaré Lemma, meaning its cohomology vanishes, which confirms a conjecture by Bolsinov and Konyaev.

Deformations of Nijenhuis Lie algebras and Nijenhuis Lie algebroids

TL;DR

The paper develops a comprehensive operadic and homotopical framework for deformations of Nijenhuis structures, both algebraic (Nijenhuis Lie algebras) and geometric (Nijenhuis Lie algebroids). It constructs the minimal model via a Koszul dual cooperad and uses this to define an -deformation complex that encodes simultaneous deformations of Lie brackets and Nijenhuis operators. A major outcome is the establishment of a Poicare-type vanishing result for a class of Nijenhuis operators (the Poincaré Lemma for diagonal operators), confirming a Bolsinov–Konyaev conjecture. The work also extends these algebraic structures to Lie algebroids and dg-manifolds, yielding a corresponding -algebra controlling deformations of geometric Nijenhuis structures and bridging algebraic and geometric cohomology theories. Overall, the paper significantly deepens the operadic understanding of Nijenhuis geometry and its deformation theory, with concrete computational applications in cohomology vanishing results.

Abstract

This paper is the second in a series dedicated to the operadic study of Nijenhuis structures, focusing on Nijenhuis Lie algebras and Nijenhuis geometry. We introduce the concept of homotopy Nijenhuis Lie algebras and establish that the differential graded (=dg) operad governing these structures serves as the minimal model of the operad for Nijenhuis Lie algebras. We construct an -algebra that encodes the simultaneous deformations of Lie brackets and Nijenhuis operators, leading to the deformation cochain complex and an associated cohomology theory for Nijenhuis Lie algebras. Extending these ideas to geometry, we investigate the deformations of geometric Nijenhuis structures. We introduce the notion of a Nijenhuis Lie algebroid-a Lie algebroid equipped with a Nijenhuis structure, which generalizes the classical Nijenhuis structure on vector fields of manifolds. Using the framework of dg manifolds, we construct an -algebra that governs the simultaneous deformations of Lie algebroid structures and Nijenhuis operators. As a computational application, we prove that a certain class of Nijenhuis operators satisfies the Poincaré Lemma, meaning its cohomology vanishes, which confirms a conjecture by Bolsinov and Konyaev.

Paper Structure

This paper contains 28 sections, 49 theorems, 210 equations.

Table of Contents

  1. Preliminaries
  2. Notations
  3. $L_\infty$-algebras and Maurer-Cartan elements
  4. Symmetric homotopy (co)operads
  5. Cohomology and homotopy theory of Lie algebras
  6. Nijenhuis Lie algebras
  7. Nijenhuis Lie algebras and Nijenhuis representations
  8. Cohomology theory of Nijenhuis Lie algebras
  9. Cohomology of Nijenhuis operators
  10. Cohomology of Nijenhuis Lie algebras
  11. The Koszul dual homotopy cooperad of the operad for Nijenhuis Lie algebras
  12. The Minimal model of the operad for Nijenhuis Lie algebras
  13. The symmetric dg operad for homotopy Nijenhuis Lie algebras
  14. The minimal model
  15. Homotopy Nijenhuis Lie algebras
  16. ...and 13 more sections

Key Result

Proposition 2.5

Get09CGWZ24 Given a Maurer-Cartan element $\alpha$ in $L_\infty$-algebra $(L, \{l_n\}_{n\geqslant1})$, the twisting procedure gives a new $L_\infty$-structure $\{l_n^\alpha\}_{n\geqslant 1}$ on graded space $L$, by defining $l_n^{\alpha}: L^{\otimes n}\rightarrow L$ as: for $n \geqslant 1$ and homogeneous elements $x_1, \dots, x_n\in L$, whenever these infinite sums exist. The new $L_\infty$-alge

Theorems & Definitions (126)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5: Twisting procedure
  • Remark 2.6
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.5
  • ...and 116 more