代数群

Suppose G is a finite group, then QG is a semisimple Q-algera, obviously ZG is a Z-order of QG,we denote the maximal Z-order by ?.If G is not abelian, it is not an easy thing to determine г; if G is an abelian group, then QG is isomorphic to the direct sum of a finite number of number fields, and r is the direct sum of these rings of algebraic integers of those number fields, but which elements of QG belong to r is not clear.

设G是一个有限群，那么QG是一个半单代数，ZG是QG的一个Z-序，设Γ是QG的一个极大Z-序，当G是一个非交换群时，Γ的求解是困难的问题；当G是一个交换群时，QG同构于有限多个数域的直和，Γ相应的就是各数域代数整数环的直和，但Γ具体是QG中那些元素不清楚。

The course contains four sections as follows: mathematical logic (including basic concepts of propositional logic and predicate logic, propositional calculuses and inference theories), set theory (including set algebras, relations, functions and cardinal numbers), algebraic structure (including algebraic systems, semigroups and groups, rings and fields, lattices and Boolean algebras), graph theory (including basic concepts of graph, Euler graphs and Hamiltonian graphs, trees, planar graphs and coloring graphs, some special vertex subsets and edge subsets).

本课程包含四部分内容：数理逻辑（包含命题逻辑与一阶逻辑的基本概念、等值演算以及推理理论），集合论（包含集合代数、二元关系、函数和基数），代数结构（包含代数系统、半群与群、环与域、格与布尔代数），图论（包含图的基本概念、欧拉图与哈密顿图、树、平面图及图的着色、图的某些特殊的顶点子集与边子集）。

The main purpose of this thesis is to study the application of the representation theory in Hochschild homology and cohomology groups of algebras, Hopf algebras and quantum groups, which are very active branches at present.

本学位论文主要研究代数表示论在代数的Hochschild同调群和上同调群，Hopf代数和量子群这几个当今很活跃的数学分支中的应用。

In particular, we compute Hochschild homology and cohomology groups of infinitedimensional path algebras and some of their quotient algebras, and we prove that for a general monomial algebra (not necessary finite-dimensional), all Hochschild cohomology groups of positive degrees vanish if and only if its Gabriel quiver is a finite tree.

特别地，我们计算了无限维路代数以及某些商代数的Hochschild同调群和上同调群，而且给出了一般单项代数的各正次Hochschild上同调群为零的充分必要条件，即它的Gabriel箭图是有限树。

Libermann.The early researcheson this kind of manifolds were closely related to Physics and Mechanics.But since1991,S.Kaneyuki published his result on the algebraic condition for the existence ofinvariant〓structures on a coset space,Lie theory has played the most impor-tant role in the study of this kind of manifolds.In particular,dipolarizations in a Liealgebra are closely related to the homogeneous〓manifolds.Dipolarizationsin semisimple Lie algebras and the homogeneous〓manifolds associated withthese dipolarizations have been studied by S.Kaneyuki,Z.X.Hou and S.Q.Deng.Inthe partⅡ of this thesis we study the dipolarizations in some quadratic Lie algebrasand the homogeneous parakahler manifolds associated with these dipolarizations.

Libermann给出的，早期的有关类流形的研究与物理和力学密切相关，自从1991年金行壮二发表了陪集空间上存在不变仿凯勒结构的代数化结果后，李群及李代数理论在这类流形的研究中起着主要作用，特别地，李代数的双极化与这类流形密切相关，半单李代数的双极化的相关几何，金行壮二，候自新和邓少强等人已作了研究，二次李代数是比半单李代数更广且带有非退化不变双线性型的李代数，本文主要研究了二次代数的双极化及相关几何。

Firstly, we prove the existence and the uniqueness of the hyperfocal subalgebra when the coefficient field is split for a defect pointed group; then, considering the structural pattern of source algebras over arbitrary fields, by means of G-acted groups, we reduce the existence and the uniqueness of the hyperfocal subalgebra over an arbitrary field to the case that the coefficient field is split for a defect pointed group and the inertia group stablizes a hyperfocal subalgebra.

首先我们证明了如果系数域对亏点群是分裂的，则源代数的超聚焦子代数存在且共轭唯一；然后通过考虑任意域上源代数的结构模式，以G-作用群为手段，将超聚焦子代数的存在性和唯一性归结为系数域对亏点群是分裂的情况下惯性群稳定的超聚焦子代数的存在性和唯一性。

In recent years, the theory of infinite dimensional simple Lie algebra has become an important branch in Lie theory.

近年来无限维单李代数的结构理论以及表示理论已经成为李代数研究中的重要分支，无限维单李代数的构造，自同构群的确定，同构分类，二上同调群的计。。。

The quantum deformation of a Lie algebra is obtained by adding one parameter q,which is reduced to the original Lie algebra when taking the limit q→1;some properties of the original Lie algebra remain.

在Hopf代数或量子群理论中，构造李双代数的量子化是产生新的量子群的一个十分重要方法，研究李双代数的重要目的之一就是对其量子化。

Its main goal is to explore information in the K-theory groups of the index C*-algebras, the Roe algebras C*, by using the large-scale geometrical structure of proper metric spaces, including noncompact complete Riemannian manifolds, finitely generated groups, etc., so as to establish connections among geometry, topology and analysis of the geometric spaces, and furthermore, to solve other relating problems, say, the Novikov conjecture, the Gromov-Lawson-Rosenberg conjecture on positive scalar curvature, the idempotent problem in the theory of C*-algebras.

粗几何上的指标理论是"非交换几何"领域九十年代以来发展起来的重要研究方向，它孕育于非紧流形上的指标理论，其主要目标是通过几何空间（如非紧完备黎曼流形、有限生成群等）的大尺度几何结构探索指标代数，即 Roe代数，的K-理论群的信息，从而建立几何空间的几何、拓扑与分析之间的联系，并应用于解决其他重要问题，如Novikov猜测、Gromov-Lawson-Rosenberg正标量曲率猜测、群C*-代数幂等元问题等。

The semigroup algebra theory,which has a huge development under interior and exterior double Mathematical conditions form 1950"s to 1960"s,is a new branch of algebra theory.In 1990,R.Biswas gave definition of anti-fuzzy subgroup.In 1995,Shen gave definitions of anti-fuzzy subgroup and normal anti-fiizzy subgroup of group.

半群的代数理论是在数学内部和外部双重条件下，从20世纪50年代到60年代发展起来的一个崭新的代数分支。1990年，Biswas R提出了反Fuzzy子群的定义；1995年沈正维提出了一个群的反Fuzzy子群和正规反Fuzzy子群的定义。

I'm strongly against the death penalty — it's an eye for an eye.

我不赞成死刑——这是以牙还牙的报复行为。

And to get you the support you need, we're enlisting all elements of our national power: our diplomacy and development, our economic might and our moral suasion, so that you and the rest of our military do not bear the burden of our security alone.

并给你们所须的支援，我们正徵召国家所有各种的力量：我们的外交及发展，我们的经济力量与道德劝说，所以你们与其他军人不须要孤独地负起国家安全的责任。