Mathematical Sciences: Geometric Aspects of Representationsof Quantum Groups and Kac-Moody Lie Algebras

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This award supports research on two main topics: (1) Representations of quantum groups and Kac-Moody Lie algebras and perverse sheaves over configuration spaces; (2) Local structure of moduli spaces. The principal investigator will work on a construction relating certain tensor categories arising from q-deformations of enveloping algebras of simple Lie algebras and certain tensor categories of a geometrical nature. He will also work on describing the completion of the local ring at a point X in terms of the cohomology of certain sheaves of dg Lie algebras over X naturally associated to a deformation problem. This research is concerned with a mathematical object called a Lie algebra. Lie algebras arise from another object called a Lie group. An example of a Lie group is the rotations of a sphere where one rotation is followed by another. Lie groups and Lie algebras are important in areas involving analysis of sperical motion.