This award supports research on certain generalizations of affine Lie algebras. The first of these is the quantized analogue of the enveloping algebra of an affine Kac-Moody algebra, the quantum affine algebra. The finite dimensional representations of this have very subtle behaviour, similar to the spherical representations of a p-adic group, and the proposed approach is to use techniques developed by Kazhdan and Lusztig in their study of affine Hecke algebras. The principal investigator intends to construct the representations of these quantum affine algebras and determine which admit perfect crystals, study their tensor category structure, and investigate the relations with the highest weight category. The second theme of this project is to define and study elliptic quantum groups. The principal investigator will study their more arithmetic behaviour, arising from the moduli of elliptic curves. It is expected that these algebras will have serious implications for the study of the XYZ Hamiltonian of statistical mechanics. 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.
