This award supports research on the representation theory of Lie algebras. The principal investigator will continue work on the Lie cohomology of some vector fields algebras. He will also consider the modular representations of Kac-Moody algebras at the critical level. 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 spherical motion.