Model Theory and Representation Theory

点击翻译按钮获取中文摘要

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.***