Hom-pre-Lie Algebras In Dimension Three And Dipolarization Of Nilpotent Lie Algebras In Low Dimensions

G-Hom-associative algebra is structured by the deformation of the associative algebra along the algebra homomorphism.As a special case,we obtain Hom-pre-Lie algebra、Homassociative algebra、Hom-Lie algebra and Hom-Lie-admissible algebra as a deformation of pre-Lie algebra、associative algebra、Lie algebra and Lie admissiblie algebra.The algebra which is closely related to Hom-pre-Lie algebra is pre-Lie algebra(or Koszul-Vinberg algebras、left-symmetric algebra、quasi-associative algebras and so on).It is regarded as an underlying structure of those Lie algebras and should be given more attention.In this paper,we discuss some examples of Hom-pre-Lie algebra in dimension three and depolarization of nilpotent lie algebra in low dimensions.In Chapter 2,we give the definition of G-Hom-associative algebra and introduce the Hom-pre- Lie algebra as a special case of the G-Hom-associative algebra.According to the classification of 3-dimensional pre-Lie algebra over the complex field through the algebra homomorphism,I give some example of Hom-pre-Lie algebra in dimension three and give out results.In Chapter 3,we discuss the dipolarization of nilpotent Lie algebra in low dimensions. At first,we find the classification of nilpotent Lie algebra in low dimensions,then we find a class of dipolarization of the Lie algebra according to the definition of dipolarization of Lie algebra.We give the dipolarization of the Abel Lie algebra directly,and find a dipolarization of every non-Abel Lie algebra in low dimension.