Model Theory and Representation Theory

9972613Herzog设L是特征为零的代数闭域上的有限维简单李代数。L的有限维简单表示的分类是表示理论的基本结果之一。Herzog的计划是将分类扩展到L的简单伪有限维表示——在L模的语言中，如果一个表示满足有限维表示的公理，则称为伪有限维表示。该项目涉及描述由Ziegler谱在l的简单伪有限维表示空间上引起的拓扑。对于迹零的2乘2矩阵的Liealgebra，它是一个紧Hausdorf空间-连续体的基数，其中有限维简单表示形成一个开放、密集和离散的子空间。群是一个数学概念，用于研究物体的对称结构，可能出现在物理学或其他自然科学中。在表示的数学理论中，目标是通过对G是对称群的所有对象（G的表示）进行分类来研究群G。对于许多小组来说，这个程序已经取得了惊人的成功，它将G与一个李代数L联系起来，L的表示反映了G的表示，然后对表示进行分类。赫尔佐格项目的目标是用数学方法研究某些李代数L的有限维表示。该项目专注于l的无限维表示X，称为伪有限维。这些表示实际上是有限维的，从某种意义上说，所使用的语言的有限表达能力无法将X与有限维表示区分开来。在之前的工作中，Herzog已经应用了哥德尔的紧致性定理来证明存在无数个“简单”的伪有限维表示。然而，一个单一的这样的表现还没有被发现，赫尔佐格项目的一部分是构建一个具体的例子

9972613Herzog Let L be a finite dimensional simple Lie algebra over an algebraicallyclosed field of characteristic zero. The classification of the finitedimensional simple representations of L is one of the fundamental results inrepresentation theory. Herzog's project is to extend the classification tothe simple pseudo-finite dimensional representations of L -- a representationis called pseudo-finite dimensional if in the language of L-modules itsatisfies the axioms for a finite dimensional representation. The projectinvolves describing the topology induced by the Ziegler spectrum on the spaceof simple pseudo-finite dimensional representations of L. For the Liealgebra of two-by-two matrices of trace zero, it is a compact Hausdorf spacethe cardinality of the continuum, of which the finite dimensional simplerepresentations form an open, dense and discrete subspace. A group is the mathematical concept used to study the structure ofsymmetries of an object that may occur in physics or the other naturalsciences. In the mathematical theory of representations, the goal is tostudy a group G by classifying all the objects -- the representations of G--- of which G is a symmetry group. For many groups, this program has metwith stunning success by associating to G a Lie algebra L whoserepresentations mirror those of G, and then classifying the representations.The objective of L. Herzog's project is to study the finite dimensionalrepresentations of certain Lie algebras L, using the methods of mathematicallogic. The project focuses on the infinite dimensional representations X ofL that are called pseudo-finite dimensional. These representations arevirtually finite dimensional, in the sense that the limited power ofexpression of the language being used cannot distinguish X from the finitedimensional representations. In previous work, Herzog has applied theCompactness Theorem of Goedel to show that there exist uncountably many``simple'' pseudo-finite dimensional representations. However, a single suchrepresentation has yet to be discovered, and part of Herzog's project is toconstruct a concrete example.***