Model Theory and Representation Theory

小行星9972613 设L是特征为零的代数闭域上的有限维单李代数。 L的有限维单表示的分类是表示论的基本结果之一。 Herzog的计划是将分类扩展到L的简单伪有限维表示--如果在L-模的语言中它满足有限维表示的公理，则表示称为伪有限维。 该项目涉及描述L的简单伪有限维表示空间上的齐格勒谱所诱导的拓扑。 对于迹为零的2 × 2矩阵的李代数，它是一个紧致的Hausdorf空间，它的有限维简单表示构成一个开的、稠密的、离散的子空间。 群是一个数学概念，用于研究物理学或其他自然科学中可能出现的物体的对称性结构。 在表示的数学理论中，目标是通过对所有对象--G的表示--进行分类来研究一个群G，其中G是一个对称群。 对于许多群，这个程序取得了惊人的成功，它把一个李代数L与G联系起来，L的表示与G的表示是镜像的，然后对这些表示进行分类。赫尔佐格的项目是研究有限维表示的某些李代数L，使用的方法，代数逻辑。 该项目的重点是无限维表示X ofL被称为伪有限维。 这些表示实际上是有限维的，在某种意义上，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。 在以前的工作中，赫尔佐格应用哥德尔的紧性定理证明了存在无数个“简单”的伪有限维表示。 然而，一个单一的这样的表示还没有被发现，赫尔佐格的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.***