Cohomology theory of Nijenhuis Lie algebras and (generic) Nijenhuis Lie bialgebras
Apurba Das
TL;DR
The paper develops a cohesive cohomology framework for Nijenhuis Lie algebras by coupling the Chevalley–Eilenberg theory with the Frölicher–Nijenhuis formalism, enabling simultaneous deformation analyses of Lie brackets and Nijenhuis operators. It extends to representations and builds a mapping-cone cohomology $H^ullet_{NLie}( rak g,N)$ that interplays with $H^ullet_{CE}( rak g^N)$ via long exact sequences, thereby grounding deformation theory in concrete cohomological invariants. The work further introduces homotopy Nijenhuis operators on 2-term $L_0_fty$-algebras, revealing skeletal–3-cocycle and strict–crossed-module correspondences, and connects these ideas to NS-Lie algebras. On the bialgebra side, it generalizes NLie bialgebras to generic Nijenhuis Lie bialgebras, proves equivalence with matched pairs and Manin triples, and relates coboundary and admissible CYBE constructions to antisymmetric $r$-matrices and relative Rota-Baxter operators. Overall, the article provides a robust toolkit for cohomology, deformation, and bialgebra theories in the presence of Nijenhuis operators, with potential applications in integrable systems and higher-algebraic structures.
Abstract
The aim of this paper is twofold. In the first part, we define the cohomology of a Nijenhuis Lie algebra with coefficients in a suitable representation. Our cohomology of a Nijenhuis Lie algebra governs the simultaneous deformations of the underlying Lie algebra and the Nijenhuis operator. Subsequently, we define homotopy Nijenhuis operators on $2$-term $L_\infty$-algebras and show that in some cases they are related to third cocycles of Nijenhuis Lie algebras. In another part of this paper, we extend our study to (generic) Nijenhuis Lie bialgebras where the Nijenhuis operators on the underlying Lie algebras and Lie coalgebras need not be the same. In due course, we introduce matched pairs and Manin triples of Nijenhuis Lie algebras and show that they are equivalent to Nijenhuis Lie bialgebras. Finally, we consider the admissible classical Yang-Baxter equation whose antisymmetric solutions yield Nijenhuis Lie bialgebras.