Hom-lie Algebra And Its Center Expansion And Enveloping Algebra,

This article consists of two parts. The first part is hom-Lie algebra and its central extenxion and enveloping algebra. The sencond part contains two papers in appendix. One is some properties of two dimensional noncommutative Lie algebra and its complete form. The other is the inference method and its application on three dimensional simple Lie algebra.In the first part, we first get hom-Lie algebra by deforming the definition of Lie algebra in terms of Jacobi identity in its definition.For the aim, we give the definition of UFD, and study an important property of UFD(Theorem 2.3). We construct a hom-Lie algebra in method of Jonast. Hartwig. Then we give a concrete example of hom-Lie algebra and study the related properties. We study the homomorphism of hom-Lie algebra by analogue of methods of studying Lie algebra. Then we study central extension, which is an extension of hom-Lie algebra . To do this , we first generalize the definition of 2-cocycle for Lie algebra and define endomorphism s of hom-Lie algebra. Then we construct Lie bracket [.,.]l,, which allows us to get the central extension. Finally, we give the definition of enveloping algebra of hom-Lie algebra and study some related properties(Theorem 3.7, Corolary 3.8, Corolary 3.9).In the first paper of the second part , it studies two dimensional noncommutative Lie algebra and its solvability, completeness and nonsemisimplicity and so on .In the second paper of the second part, we gives a inference method of three dimensional simple Lie algebra and display the internal relation among three dimensional simple Lie algebra.