Cohomology And Crosed Modules Of Hom-Lie Antialgebras

In this paper,we introduce the notion of Hom-Lie antialgebras and define its representa-tions,cohomology theories and crossed modules.In particular,the second cohomology and third cohomology are mainly investigatedIn chapter 1,we introduce the origin,concepts of Lie antialgebras.Then,we show some important results obtained in this paper.In Chapter 2,after giving the definition of Hom-Lie antialgebras and its representations,we define the corresponding Chevalley-Eilenberg complex chain.We also study abelian extensions of Hom-Lie antialgebras.The notion of Nijienhuis operators of a Hom-Lie antialgebras is intro-duced to describe trivial deformations.It is proved that given an infinitesimal deformation of a Hom-Lie antialgebra(?,?,?),there is a 2-cocycle of a with coefficients in the representationIn Chapter 3,we introduced the concept of crossed module for Hom-Lie antialgebras and Cat1-Hom-Lie antialgebras.Then,we proof that the category of crossed modules for Hom-Lie antialgebras and the category of Cat1-Hom-Lie antialgebras are equivalent to each other.At last,we define crossed module extension of Hom-Lie antialgebras.The relationship between the crossed module extension of Hom-Lie antialgebras and the third cohomology group are found.