Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O} N$-structures

TL;DR

This paper explores advanced algebraic structures related to Lie conformal algebras, introducing Nijenhuis operators, $ ext{O}N$-structures, and their relations to $r$-matrix and symplectic structures, expanding the theoretical framework of conformal algebra deformations.

Contribution

It introduces new concepts like $ ext{O}N$-structures and studies their compatibility with existing structures, providing a hierarchy of compatible $ ext{O}$-operators and linking Nijenhuis operators to conformal algebra deformations.

Findings

Defined Nijenhuis operators on Lie conformal algebras.

Established the hierarchy of compatible $ ext{O}$-operators.

Connected conformal $r$-matrix-Nijenhuis and symplectic-Nijenhuis structures.

Abstract

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an $\mathsf{LCMod}$ pair), which lead to the notions of Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the $\mathsf{LCMod}$ pair, respectively. Then by adding compatibility conditions between Nijenhuis structures and $\mathcal{O}$-operators, we introduce the notion of an $\mathcal{O} N$-structure on an $\mathsf{LCMod}$ pair and show that an $\mathcal{O} N$-structure gives rise to a hierarchy of pairwise compatible $\mathcal{O}$-operators. In particular, we show that compatible $\mathcal{O}$-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of conformal $r$-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on the Lie conformal algebra and study their relations.