数学与物理中的许多问题都与代数的Lie 结构有密切关系。申请人与合作者已对算子代数的线性Lie 结构进行了研究，并初步探讨了三角代数上的非线性Lie结构。本课题拟进一步以三角代数、tvon Neumann 代数的套子代数、CSL 代数、次对角代数为模型，应用"重叠分块，以块代点"和类似于刻画环的素度的方法，研究这几类非自伴算子代数的非线性Lie导子、Lie n-导子、线性或非线保持Lie乘积的映射，从而刻画非自伴算子代数的Lie结构。结合算子谱理论与函数方程，研究非自伴算子代数的各种多线性局部映射和局部保持问题，通过它们的研究探讨非自伴算子代数的结构与代数不变量，并应用于研究它们的上同调理论与相似问题。最后研究两类特殊的Kadison-singer代数的Lie 结构、导子、局部导子、局部同构。通过本项目的研究，以期对算子代数的研究产生积极影响。

Many problems og mathematics and physics are closely related with the Lie structures of algebras. The applicant and collaborators have investigated linear Lie structure of some operator algebras and studied preliminarily the nonlinear Lie structure of triangular algebras. In this project, firstly we will investigate nonlinear Lie derivation, Lie n-derivation, linear or nonlinear mappings of preserving Lie products on some non selfadjoint operator algebras such as triangular algebras, nest subalgebras of von Neumann, CSL algebras and subdiagonal algebras. The new methods of overlapping and separating operator block and using block replacing piont will be used. In addition, we will use the method similar to prime ring to study the above operator algebras. Secondly, we will discuss multilinear local mappings and local preserving problems of triangular algebras, nest subalgebras of von Neumann, CSL algebras and subdiagonal algebras. Operator spectrum theory and functional identities will be applicated. We will study the structure and algebraic invariant of non selfadjoint operator algebras through the study. Lastly, we will discuss Lie structure, derivation、local derivation and local isomorphism of two certain Kadison-Singer algebras. We hope that the research of this project will not only enrich theory of operator algebras but also have a positive impact to study operator algebras.