Group Rings and Related Algebras

9820271唐纳德·S.帕斯曼计划在接下来的几年里沿着研究三个主要问题，以及他通常研究的一系列杂项问题。 主要问题涉及（1）群生成群的群环的半连续性，（2）群代数的双边理想格的结构，（3）广义Cartan型李代数的单性。 提议者回到研究组环在1990年和在过去的几年中，在一个长期的一系列文件，他能够描述的雅各布森根群环的局部有限群。 从某种意义上说，这解决了半连续性问题的一半，他现在建议研究另一半，即半连续生成群的情况。 第二个问题是由亚历克斯Zalesskii访问麦迪逊的1999年春季学期的动机。 他和提议者计划研究群代数何时只有平凡理想的问题的变化。 这将是长期合作的开始。 最后，提出者最近采取了环理论的方法来研究广义Cartan型李代数的单性。 他已经解决了这个问题的维特型代数，目前正在与杰夫卑尔根的特殊代数，并希望处理的汉密尔顿和接触类型在以后的某个时候。历史上，团体首先出现的一套所有对称的几何对象。 由于两个对称的乘积（合成）又是一个对称，我们看到一个群是一个具有“好”乘法的元素的集合。 相反，如果我们从一个抽象群开始，那么我们可以通过允许它作为某些代数对象上的对称来更好地理解它。 这里要考虑的适当对象是由向量构建的，称为向量空间。 该动作通过形成群代数来实现，群代数是由群确定的代数对象，并且具有加法和乘法。 感兴趣的向量空间就是所谓的群代数的不可约模，不可约是因为我们希望它们尽可能小。 事实证明，这个过程有一个根本的障碍，即群代数的雅各布森根。 实际上，这个过程有效当且仅当这个根式是平凡的，在这种情况下我们说群代数是半本原的。 因此，判定群代数何时是半本原的是一个自然而重要的问题，这也是上述建议（1）的目标。建议（2）涉及到更仔细地研究群代数，但在更特殊的情况下。

9820271Donald S. Passman plans to work on three main problems during the next few years along with his usual slew of miscellaneous ones. The main problems concern (1) the semiprimitivity of groups rings of finitely generated groups, (2) the structure of the lattice of two-sided ideals of a group algebra, and (3) the simplicity of Lie algebras of generalized Cartan type. The proposer returned to the study of group rings in 1990 and during the past few years, in a long series of papers, he was able to describe the Jacobson radicals of group rings of locally finite groups. In some sense, this solves half of the semiprimitivity problem, and he now proposes to work on the other half, namely the case of finitely generated groups. The second problem is motivated by the visit of Alex Zalesskii to Madison for the Spring semester of 1999. He and the proposer plan to study variations of the question of when group algebras have only trivial ideals. This will hopefully be the start of a long term collaboration. Finally, the proposer has recently taken a ring theoretic approach to study the simplicity of Lie algebras of generalized Cartan type. He has already settled the problem for Witt type algebras, is presently working with Jeff Bergen on the special algebras, and hopes to deal with the Hamiltonian and contact types at some later time.Historically, groups first arose as the set of all symmetries of a geometric object. Since the product (composition) of two symmetries is again a symmetry, we see that a group is a collection of elements with a ``nice'' multiplication. Conversely, if we start with an abstract group, then we can better understand it by allowing it to ``act'' as symmetries on certain algebraic objects. The appropriate objects to consider here are built out of vectors and are called vector spaces. The action is achieved by forming the group algebra, which is an algebraic object determined by the group and having both an addition and a multiplication. The vector spaces of interest are then what are known as the irreducible modules for the group algebra, irreducible because we would like them to be as small as possible. As it turns out, there is a fundamental obstacle to this procedure, namely the Jacobson radical of the group algebra. Indeed, the procedure works if and only if this radical is trivial, in which case we say that the group algebra is semiprimitive. Thus, it is a natural and important problem to decide when group algebras are semiprimitive, and this is the goal of proposal (1), described above. Proposal (2) is concerned with taking an even closer look at the group algebra, but under more special circumstances.