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In this dissertation, we establish the fundamental theory and methods to study systems of linear matrix equations over regular rings and polynomial rings over finite central algebras; and investigate some significant systems of linear matrix equations over the rings mentioned above.

本文在正则环和有限维中心代数上的多项式环上初步建立了研究矩阵方程组的基本理论和方法，并在上述环上研究了具体的具有重要意义的矩阵方程组。

Tilting theory is an important tool to study representation theory of algebras.

而倾斜理论是研究代数表示理论的重要工具之一。

At the side of the semigroup algebras, gives a similar result of inverse semigroup algebra.

在第2.1节中，基于[L9]中的结论，我们给出了一个正则Yang-Baxter方程的解，同时也给出了这个解的一个分解。

In this paper, we mainly study representations of strong semisimple n-Lie algebras, prove that a representation of a strong semisimple n-Lie algebra is a representation of the reductive Lie algebra LP, and the relative properties.

本文主要研究了强半单的n-李代数的表示，证明了强半单的n-李代数的表示可转化为一个约化李代数Lρ的表示，并证明了不变线性形等其它相关性质。

At last,we present a criterion for an Lie algebra with Heisenberg nilpotentradicals to be CS in terms of representation theory of semisimple Lie algebras.

在第二节中，我们推广了[42]中的主要定理3.3并由此构造了许多新的非极大秩可解完备李代数。1

The so-called AT-algebras are inductive limits of finite direct sums of matrices over the extension algeras of circle algebra by K, where K is the C~*— algebra of all compact operators on a separable infinite dimensional Hilbert space.

若V_*与V_*同构，且保持单位元等价类；T与T仿射同胚，且同构映射与同胚映射相容，则存在E与E′的同构导出上述同构和同胚，所谓AT-代数即为圆代数通过κ的本质酉扩张的矩阵代数的有限直和的归纳极限，这里κ为可分的无限维复Hilbert空间上的紧算子全体，不变量中的V*为三变元Abel半群，T为迹态空间，[1]为单位元所在的Murray-von Neumann等价类，r_E为连接映射。

The most intuitive development of boolean algebras arises from the concept of a set algebra.

布尔代数是集合代数的概念最自然的发展。

In chapter 1, similarity and perturbations of nest subalgebras of factor von Neumann algebras are studied. A similarity theorem is given (see Theorem 1. 2. 3), and it is shown that similar nests in a factor von Neumann algebra are close.

第一章研究因子von Neumann代数中套的相似性和套子代数的摄动，获到一个相似定理（定理1.2.3），其几何意义是：因子von Neumann代数中相似的套聚拢在一起并可向外扩展。

Classification of simple modules for Hecke algebra of type D_n is an open problem in representation theory of Hecke algebras.

D_n型黑克代数的单模的分类是黑克代数表示论中一个公开的问题。

Moreover, a main inconvenience of mixing assumptions is the difficulty of checking them because the calculation involves the complicated manipulation of taking the supremum over twoσ- algebras.

而且，混合假设主要的不便还在于很难验证，这是由于在两个σ-代数之间取上确界包含了复杂的计算。

I think people in this lifetime in a rebirth.

我觉得人在这一生里其实也在轮回。

He had to tie her hands together.

他不得不把她的双手绑在一起。