Some Study On Categorifications And Deformations Of 3-Lie-algebras

In this thesis,we mainly study 3-Lie 2-algebras,n-Lie algebras,(n-1)-order de-formations of an n-Lie algebra and Nijenhuis operators on an n-Lie algebra.We build the equivalent relations between 3-Lie 2-algebras and 2-term 3-Lie?-algebras.We prove that there is a one-to-one correspondence between(n-1)-order trivial deformations of an n-Lie algebra and Nijenhuis operators on an n-Lie algebra.We give various constructions of Nijenhuis operators.In Chapter 3,we define 3-Lie?-algebras which is a homotopification of a 3-Lie algebra.A 3-Lie?-algebra satisfy the fundamental identity up to all higher homotopies.We show that there is naturally a Leibniz?-algebra structure on the space of fundamental objects.In particular,we focus on 2-term 3-Lie?-algebras.First,we give explicit formulas for 2-term 3-Lie?-algebras and define homomorphisms between 2-term 3-Lie?-algebras and 2-homomorphisms between homomorphisms.We put 3-Lie algebra structures on 2-vector spaces and obtain 3-Lie 2-algebras which are the categorification of 3-Lie algebras.We define the homomorphisms between 3-Lie 2-algebras and 2-homomorphisms between homomorphisms.We prove that 2-categories 2Term 3-Lie? and 3Lie2Alg are 2-equivalent.We study two particular 3-Lie 2-algebras:skeletal 3-Lie 2-algebras and strict 3-Lie 2-algebras in detail.We classify skeletal 3-Lie 2-algebras via the third cohomology group.We introduce the notion of a crossed module of 3-Lie algebras and show that there is a one-to-one correspondence between crossed modules of 3-Lie algebras and strict 3-Lie 2-algebras.In the end,we construct a strict 3-Lie 2-algebra from a symplectic 3-Lie algebra.In Chapter 4,we study(n-1)-order deformations of an n-Lie algebra and Nijenhuis operators on an n-Lie algebra.In particular,we introduce the notion of a Nijenhuis operator on an n-Lie algebra by the(n-1)-order trivial deformation of an n-Lie algebra.We show that there is a one-to-one correspondence between(n-1)-order trivial deformations of an n-Lie algebra and Nijenhuis operators on an n-Lie algebra.We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator.We give a method to construct Nijenhuis operators on an(n + 1)-Lie algebra from Nijenhuis operators on an n-Lie algebra.We construct Nijenhuis operators on a 3-Lie algebra from Nijenhuis operators on a commutative associative algebra.We also construct Nijenhuis operators on a 3-Lie algebra from Rota-Baxter operators and derivations on a 3-Lie algebra.