The Characteristic Polynomial Of The Subalgebra Of Gl(3,C)

Characteristic polynomials have been widely used in many branches of mathematics,such as operator theory,matrix theory,group theory and Lie algebra theory.Characteristic polynomials are helpful to the classification of Lie algebras.Classification is a fundamental problem in the study of Lie algebra.This paper classifies the subalgebras of gl(3,C)with the help of character-istic polynomials.The article mainly studies the classification of no more than three-dimensional subalgebras of gl(3,C)under the action of the inner automorphism group,and then we calculate characteristic polynomials of the corresponding subalgebras.In the first chapter,the current status of characteristic polynomials and classification about Lie algebras are introduced.Besides,the main work of the article is briefly introduced.In the second chapter,we introduce some definitions and related lemmas about the article.In the third chapter,the characteristic polynomials of one-dimensional subalgebras are studied.Any 3 x 3 non-zero complex matrix can generate one-dimensional subalgebra of gl(3,C).Every matrix is equivalent under the action of inner automorphism group to Jordan standard form.So the classification of one-dimensional subalgebras can be obtained through Jordan standard form in the sense of inner automorphism,and then we calculate the characteristic polynomials of the corresponding subalgebras.In the fourth chapter,the characteristic polynomials of two-dimensional subalgebras of gl(3,C)are discussed.We know that two-dimensional abstract Lie algebra includes abelian Lie algebra and non-abelian algebra.To begin with,let a matrix in the standard basis be Jordan standard form.We can get another matrix through the Lie bracket rules and normalize the entries of another matrix with the help of linear independence of basis.And then we discuss whether are isomorphism between two generated Lie algebras.Finally,we can calculate the characteristic polynomials of corresponding subalgebras.In the fifth chapter,the characteristic polynomials of three-dimensional subalgebras of gl(3,C)are investigated.The process of classification discussion can be simplified by using matrix theory.We are able to normalize the entries of matrix in the standard basis with the help of linear indepen-dence of basis.And then we discuss whether are isomorphism between two generated Lie algebras through the action of similar matrices.Finally,we can calculate the characteristic polynomials of corresponding subalgebras.In the sixth chapter,the characteristic polynomials of the structure constant matrices which two and three-dimensional Lie algebras are calculated.We can obtain the structure constant matrix of Lie algebra according to the operation of Lie brackets of the standard basis.And then we calculate the characteristic polynomials of the structure constant matrix.The conclusions obtained in this paper can distinguish whether the subalgebras of linear Lie algebras are isomorphic in a sense and provide ideas and methods for the study of subalgebras of gI(n,C).It is helpful to the study of complex Lie algebras.