Some Properties And Lower Order Cohomology Theory Of Zinbiel Superalgebras

Zinbiel algebras first appeared in the study of Loday’s work in 1995,which have a Koszul dual relationship with Leibniz algebras,so they are called dual Leibniz algebras.Later,they are renamed Zinbiel algebras.Zinbiel algebras have a wide range of applications in mathematics and physics: Zinbiel algebras play an important role in the study of pre-Gerstenhaber algebras.They can also be used to study Tortkara algebras,rack homology,number theory,Cartesian differential category,and so on.In general,Zinbiel algebras are important in mathematics,which can be applied to various fields.As a generalization of Zinbiel algebras,Zinbiel superalgebras were introduced by Benayadi,Kaygorodov and Mhamdi in 2022.As a new class of algebraic objects,there are not many results on Zinbiel superalgebras.Therefore,the purpose of this thesis is to generalize some classical results on Zinbiel algebras to Zinbiel superalgebras.Specifically,this thesis studies some properties of Zinbiel superalgebras and low-order cohomology theory,which is divided into three chapters.The first chapter introduces the research background and structure of this paper,and briefly describes the latest results on Zinbiel superalgebras.In Chapter 2,we study derivations,centroids,invariant bilinear forms and dual representations of Zinbiel superalgebras.Specifically,the definitions of derivations and centroids on Zinbiel superalgebras are introduced,and some classical results on derivations and centroids on other algebras are extended to Zinbiel superalgebras.The projection and pullback of invariant bilinear forms on Zinbiel superalgebras are also introduced.We give the definitions of representations,adjoint representations and dual representations on Zinbiel superalgebras,which can be applied to contruct a larger class of Zinbiel superalgebras.In Chapter 3,we study the low-order cohomology theory and deformation theory of Zinbiel superalgebras.Firstly,we give the definitions of bimodules and cochain complexes of Zinbiel superalgebras,and obtain the cohomology groups of 1,2 and 3 orders.Secondly,we introduce the definition of one-parameter formal deformations on Zinbiel superalgebras,and prove that the infinitesimal of equivalent one-parameter formal deformations on Zinbiel superalgebras belong to the same equivalence class of second cohomology groups.When the second-order cohomology group is equal to 0,one-parameter formal deformations on Zinbiel superalgebras are analytically rigid.Finally,first,second and third cohomology groups of homomorphisms of Zinbiel superalgebras are given and used to study the deformation of homomorphisms of Zinbiel superalgebras.It is also proved that the infinitesimal of the equivalent deformations of homomorphisms of Zinbiel superalgebras belong to the same equivalence class of second cohomology groups,and when the second cohomology group is equal to 0,the deformation of homomorphisms of Zinbiel superalgebras is analytically rigid.