Conformal r-matrix-Nijenhuis structures, symplectic-Nijenhuis structures, and ON-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 LCMod pair), which lead to the notions of the Nijenhuis operator on the Lie conformal algebra and the Nijenhuis structure on the LCMod pair, respectively. Then, by adding compatibility conditions between Nijenhuis structures and O-operators, we introduce the notion of an ON-structure on an LCMod pair and show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. In particular, we show that compatible O-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of the conformal r-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on Lie conformal algebras and study their relations.