Infinite Dimensional Lie algebras, Quantum Groups and their Applications

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The concept behind proposed research can be characterized as the study of mathematical aspects of symmetry properties in quantum theory and statistical mechanics. The first part of the project is focused on the objects which appear in the analysis of models of quantum field theory in two-dimensional space time with special conditions on the boundary. In the second part the PI will study the reasons why some mechanical systems can be solved exactly analytically and the symmetry arguments that go with this. The third and the fifth parts are related to questions involving statistical aspects of symmetry. The fourth part of the project is to connect quantum mechanics and a special two dimensional quantum field theory. This part is expected to bring new insight into now relatively the old subject of quantum mechanics.In the first part of the project the PI proposes to investigate Harish-Chandra series in a broad setting of finite dimensional simple Lie groups, affine Lie groups and their quantum counterparts. The second part can be viewed as a classical counterpart of the first part. It appears that a classical Hamiltonian counterpart of Harish-Chandra theory is a superintegrable system. The third and the fifth parts outline projects at the interface of representation theory and statistical mechanics. One of the specific directions here is the study of of limit shapes in integrable models of statistical mechanics. The fourth part of the project is built of prior results of the PI and the main goal is to give a description of a full semiclassical asymptotic of eigenfunctions of integrable systems in terms of Feynman diagrams for the corresponding Poisson sigma model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.