Classification of Hom- preLie Algebras in Dimension Two

In this paper,we mainly discuss the basic properties and the classification of Hom-Novikov algebras and Hom- preLie algebras in dimension two in the complex field. At first,we give the definition of the Hom- Novikov algebras,Hom- preLie algebras and some related definitions. Then we discuss regular Hom- preLie algebras and give the necessary conditions for an Hom- preLie algebras to be pre- Lie type. We also give the direct sum of Hom- preLie algebras and get the necessary and sufficient condition for the existence of homomorphism between two Hom- preLie algebras. At last,with these definitions and their basic properties,we obtain the classification of the Hom- Novikov algebras and Hom- preLie algebras in two dimensions.