Abelian Extensions Of Hom-Pre-Lie Algebras

This article mainly gives the concept of abelian extensions of Hom-preLie algebras,and uses the cohomology group to give their classification and proves that the abelian extensions of the isomorphism corresponds to the fixed point set of the second-order homology group one-to-one.Later,we study the relationship between the abelian extensions of Hom-Lie algebra and the abelian extensions of Hom-pre-Lie algebra,and prove that the abelian extensions of a Hom-pre-Lie algebra can naturally give an abelian extensions of Hom-Lie algebra.The full text is divided into three chapters.In Chapter 1,we introduce the historical process of the research topic,the research content and progress of related directions,and then introduce the research purpose of this article and some main results.In Chapter 2,we review the basic definitions of Lie algebras,pre-Lie algebras,Hom-Lie algebras and Hom-pre-Lie algebras and their related results.In Chapter 3,we recall abelian extensions of Hom-Lie algebras.Secondly,the definition and isomorphism of abelian extensions of Hom-pre-Lie algebras are given.Thirdly,we use homology theory to study the abelian extensions of Hom-pre-Lie algebras,and get that the diagonal abelian extensions of Hom-pre-Lie algebras is classified by the set of second order fixed points.The diagonal abelian extensions of the isomorphism gives the same elements in the fixed point set of the second order homology group.In turn,the fixed point set of two-order closed chains of the second order homology group gives two isomorphic diagonally abelian extensions.Finally,it is proved that the abelian extensions of Hom-pre-Lie algebras can give an adjacent Hom-Lie algebra of abelian extensions.