Infinite Dimensional Lie Algebras, Quantum Groups, & their Applications

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9401163 Reshetikhin This project is concerned with the study of quantized universal enveloping algebras and their representations together with their applications to low-dimensional topology and to 1+1-dimension models of quantum field theory. The principal investigator will investigate the relations between the structure of the category of modules over quantized universal enveloping algebras and corresponding affine Lie algebras. He will further study invariants of 3-manifolds to better understand the relation between the combinatorial approach to topological field theory and the differential-geometric one. He will continue studying the quantized universal enveloping algebras of infinite-dimensional Lie algebras and their representation theory and, in particular, when the parameter of deformations is a root of 1. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal quantum field theory.