On Non-abelian Extensions Of 3-Leibniz Algebras

In this paper we focus on the theory of non-abelian extensions of 3-Leibniz algebras.First we define non-abelian cohomology of 3-Leibniz algebras,and show that non-abelian extensions of 3-Leibniz algebras are classified by non-abelian cohomology.Then we construct a differential graded Lie algebra and prove that there is a one-to-one correspondence between the isomorphism classes of non-abelian extensions of 3-Leibniz algebras and the equivalence classes of Maurer-Cartan elements of this differential graded Lie algebra.Finally we show that a non-abelian extension of a 3-Leibniz algebra induces a non-abelian extension of a Leibniz algebra.The full text is divided into five chapters.In Chapter 1,we introduce the background and its recent development and analyze the motivations and the main results of this thesis.In Chapter 2,We review the relevant results of non-abelian extensions of 3-Lie algebras and the definition and properties of 3-Leibniz algebras.In Chapter 3,We define the non-abelian extensions of 3-Leibniz algebras and their isomorphism classes,we find an approach to constructe a new 3-Leibniz algebra on the direct sum of two 3-Leibniz algebras.We define the non-abelian cohomology of 3-Leibniz algebras,and obtain the equivalence relationship between the isomorphism classes of non-abelian extensions of3-Leibniz algebras and the non-abelian second cohomology group.In Chapter 4,we prove that there is a one-to-one correspondence between the isomorphism classes of non-abelian extensions of 3-Leibniz algebras and the equivalence classes of Maurer-Cartan elements on a certain differential graded Lie algebra.In Chapter 5,we analyze the Leibniz algebra structure on the space of fundamental objects of 3-Leibniz algebras,and show that a non-abelian extension of 3-Leibniz algebras naturally induces a non-abelian extension of Leibniz algebras.