中文摘要：

研究李代数的同构条件以实现其完全分类。基于复数域上对称矩阵的正交相似分类，利用正交矩阵、伴随矩阵的性质和矩阵计算的技巧，给出Hom—Lie代数同构的条件，得到Lie代数gl2（C）上Hom—Lie代数的同构条件及其同构意义下的一个完全分类。

英文摘要：

Isomorphism conditions of Lie algebra are studied to achieve the complete classification. Based on the orthogonal similar classification of complex symmetric matrices, an isomorphism condition of Hom-Lie algebra is given by employing some properties of adjoint and orthogonal matrices and using skills of matrix calculation. An isomorphism condition of Horn-Lie algebra on the Lie algebra gl2 （C） is also obtained, from which a complete classification of Horn-Lie algebras on the Lie algebra gl2 （C） is presented.