Strong Ad-nilpotent Elements Of Lie Algebra

In the study of Lie algebras,the conjugate theorem of Cartan subalgebra is particularly impor-tant.The strongly ad-nilpotent element plays a decisive role in the proof of the Cartan conjugate theorem of solvable Lie algebras.In order to further study the strongly ad-nilpotent elements of solvable Lie algebras,we choose a special class of solvable Lie algebras,that is,Lie algebras t(n,F)composed of upper triangular matrices over the algebraic closed field IF of characteristics 0.In this paper,we study the set of t(n,IF)strongly ad-nilpotent elements by using the derivations of Lie algebras and the eigenvalues of matrices.In the first chapter,the development of Lie algebras is introduced,and the research status of solvable Lie algebras of upper triangular matrices and strongly ad-nilpotent elements are also introduced.In the second chapter,we introduce some definitions and related lemmas about the article.In the third chapter,we study the strongly ad-nilpotent elements of t(3,F)and its orbits under the action of automorphism groups.Using the definition of strongly ad-nilpotent elements,we calculate all the strongly ad-nilpotent elements of t(3,F)by algebraic method.In the fourth chapter,the strongly ad-nilpotent elements of t(4,F)are studied.By using the definition of strongly ad-nilpotent elements,all the strongly ad-nilpotent elements of t(4,F)are calculated by algebraic methods.In the fifth chapter,we study the strongly ad-nilpotent element of t(n,F).In this paper,the strongly ad-nilpotent elements of t(n,F)is studied,which creates a new research content of solvable Lie algebras of upper triangular matrices.The results provide a certain theoretical basis for the research of solvable Lie algebras of upper triangular matrices.