Let L be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. The classification of the finite dimensional simple representations of L is one of the fundamental results in representation theory. Herzog's project is to extend the classification to the simple pseudo-finite dimensional representations of L -- a representation is called pseudo-finite dimensional if in the language of L-modules it satisfies the axioms for a finite dimensional representation. The project involves describing the topology induced by the Ziegler spectrum on the space of simple pseudo-finite dimensional representations of L. For the Lie algebra of two-by-two matrices of trace zero, it is a compact Hausdorf space the cardinality of the continuum, of which the finite dimensional simple representations form an open, dense and discrete subspace. A group is the mathematical concept used to study the structure of symmetries of an object that may occur in physics or the other natural sciences. In the mathematical theory of representations, the goal is to study 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 met with stunning success by associating to G a Lie algebra L whose representations mirror those of G, and then classifying the representations. The objective of L. Herzog's project is to study the finite dimensional representations of certain Lie algebras L, using the methods of mathematical logic. The project focuses on the infinite dimensional representations X of L that are called pseudo-finite dimensional. These representations are virtually finite dimensional, in the sense that the limited power of expression of the language being used cannot distinguish X from the finite dimensional representations. In previous work, Herzog has applied the Compactness Theorem of Goedel to show that there exist uncountably many ``simple'' pseudo-finite dimensional representations. However, a single such representation has yet to be discovered, and part of Herzog's project is to construct a concrete example. ***

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Christopher W. Stark
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Ohio State University
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