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 [Formula: see text] pair), which lead to the notions of the Nijenhuis operator on the Lie conformal algebra and the Nijenhuis structure on the [Formula: see text] pair, respectively. Then, by adding compatibility conditions between Nijenhuis structures and [Formula: see text]-operators, we introduce the notion of an [Formula: see text]-structure on an [Formula: see text] pair and show that an [Formula: see text]-structure gives rise to a hierarchy of pairwise compatible [Formula: see text]-operators. In particular, we show that compatible [Formula: see text]-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.