Formal Deformation Of Leibniz Pairs

The deformation theory plays an important role in algebra,which is close-ly related to algebraic geometry,representation theory of algebras,homological algebras,noncommutative geometry and algebraic topology.In order to study the deformation theory of Poisson algebras,M.Flato,M.Gerstenhaber and A.A.Vovonov introduced the notion of Leibniz Pair in 1995,which is widely investigated in recent years.This thesis aims to study some kinds of formal deformation of Leibniz Pairs.In Chapter 1,we mainly recall the formal deformation theory of associa-tive algebras and Lie algebras,such as their deformation equations,Hochschild cohomology and Chevalley-Eilenberg cohomology.In Chapter 2,we firstly recall the definitions of Leibniz Pair and its Leibniz Pair cohomolgy.Secondly,we discuss the global deformation of Leibniz Pair,in which the associative algebra and the Lie algebra are deformed simultaneously.Finally,we show that this deformation is exactly controlled by Leibniz Pair cohomology.It mentioned that this fact has been given by M.Flato,et.al.without a proof.Chapter 3 deals with two kinds of one-sided deformations of Leinbiz pairs,say the associative(Lie,resp.)algebra is deformed but the Lie(associative,resp.)algebra is not.Considering the spectral sequences of Leibniz Pair bi-complex,we introduce two new cohomology groups for a Leibniz Pair,which control these two kinds of one-sided deformations,respectively.