Three Kinds Of Cohomologies On Leibniz Triple Systems And Their Applications

Leibniz triple systems are a kind of new algebraic structures,which are obtained by the Kolesnikov-Pozhideav algorithm to Lie triple systems.They are the generalization of Leibniz algebras and Lie triple systems,and are closely related to both Jordan algebras and Malcev algebras.This dissertation is divided into five chapters.In Chapter 1,we introduce backgrounds and main results.In Chapter 2,we study cohomologies of Leibniz triple systems.We define the first and third cohomology groups on Leibniz triple systems,and show that there is a one-to-one corre-spondence between equivalent classes of central extensions of Leibniz triple systems and the third cohomology group.We consider the T~*-extension of a Leibniz triple system and prove that every even-dimensional quadratic Leibniz triple system is isomorphic to a T~*-extension of another Leibniz triple system under a suitable condition.Meanwhile,we give a necessary and sufficient condition for a quadratic Leibniz triple system to admit a symplectic form.Moreover,we develop the 1-parameter formal deformation theory of Leibniz triple systems and prove that it is governed by the cohomology groups.At last,we consider the relationship between the cohomology of Leibniz algebras and the associated Leibniz triple system.In Chapter 3,we study product structures and complex structures on Leibniz triple sys-temes.We first introduce the notion of a Nijenhuis operator on Leibniz triple systems,which can generate a trivial deformation.Then we use Nijenhuis operators to define product structures on a Leibniz triple system,and prove that there exists a product structure on a Leibniz triple system if and only if it is the direct sum of eigenpaces with eigebvalue of 1 and-1.There are four kinds of special product structures,each of which corresponds to a special decomposition of a Leibniz triple system.Similarly,we study complex structures on Leibniz triple systems,then we define four kinds of special complex structures,and the necessary and sufficient con-ditions for the existence of these complex structures are given.Finally,we add a compatibility condition between a product structure and a complex structure in order to introduce the notion of a complex product structure on a Leibniz triple system and give the sufficient and necessary condition for the existence of this structure.In Chapter 4,we study the cohomology of Leibniz triple systems with a derivation and extension problems of a pair of derivations.We introduce the notion of the Leib TSDer pair,i.e.,a Leibniz triple system with a derivation.We define a representation and lower order co-homology groups of the Leib TSDer pair,and prove that a Leib TSDer pair is rigid if the third cohomology group is zero,the necessary and sufficient conditions for extensibility from order n deformations to order n+1 deformations are also given,and the central extensions of it can be classified by the third cohomology group.At last,we define a concrete representation on an abelian extension of a Leibniz triple system by another Leibniz triple system.We con-sider the abelian extension of derivations of these two Leibniz triple systems,and prove that their splitting property can be characterized by this representation and the third cohomology class corresponding to their derivations.We also prove that the set of compatible derivation pairs can define a Lie algebra,whose representation can also characterize the extensibility of a compatible pair derivations.In Chapter 5,we study deformations of relative Rota-Baxter operators on Leibniz triple systems.We define the first and third cohomology groups of relative Rota-Baxter operators on Leibniz triple systems and use them to study formal deformations of the relative Rota-Baxter operators and the extensibility of deformations from order n to order n+1.Finally,we estab-lish the relationship between cohomology of relative Rota-Baxter operators on Leibniz algebras and associated Leibniz triple systems.