Representation Theory of Infinite-Dimensional Lie Algebras and Applications

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9622870 Kac This grant supports the research of Professor V. Kac to work on various problems in representation theory and mathematical physics. He will study the representation theory of the Lie algebra of differential operators on the circle as well vertex operators, quantum groups and conformal quantum field theory. He will also study soliton equations, integral representations of affine superalgebras and the relationship between reciprocity laws in number theory and the representation theory of infinite dimensional groups. This is research that starts out in the field of algebra but moves beyond that to touch both number theory and theoretical physics. The reasons for these connections are that algebra can be thought of as the study of symmetry in the abstract. As such algebra has direct applications to areas of physics and chemistry. In particular the modern theory of gauge fields in physics uses this area of algebra extensively and this project will study some of these connections at length. There are also connections to coding theory and the transmission of data across communication lines.