Conformal R-matrix-Nijenhuis Structures,Symplectic-Nijenhuis Structures And ON-structures

In this thesis,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 Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the LCMod pair,respectively.Then by adding compatibility conditions between Nijenhuis structures and Ooperators,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 conformal r-matrixNijenhuis structure and symplectic-Nijenhuis structure on Lie conformal algebras and study their relations.This thesis is divided into five chapters.The main contents are as follows:In Chapter 1,we introduce the backgrounds of the Lie conformal algebras.In Chapter 2,we recall the notions of Lie conformal algebras,the module and cohomology theory of Lie conformal algebras.In Chapter 3,we study the infinitesimal deformations of Lie conformal algebras and LCMod pairs.Then we introduce the notions of Nijenhuis operator on a Lie conformal algebra and Nijenhuis structure on an LCMod pair.In Chapter 4,we add compatibility conditions between an O-operator and a Nijenhuis structure to define the notion of ON-structure on an LCMod pair.Then we study compatible O-operators on LCMod pairs,and show that a Nijenhuis operator connects two compatible Ooperators.We also show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators.In Chapter 5,we first introduce the notion of conformal r-matrix-Nijenhuis structure on a Lie conformal algebra A.We show that a conformal r-matrix-Nijenhuis structure gives rise to a hierarchy of pairwise compatible r-matrices.Furthermore,we introduce the notion of symplectic-Nijenhuis structure on a Lie conformal algebra A.The relations between ONstructures on LCMod pairs,conformal r-matrix-Nijenhuis structures and symplectic-Nijenhuis structures on Lie conformal algebras are given.