The Lie Algebra Cohomology of Jets

摘要

Let g be a finite-dimensional complex semi-simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra zg[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible representation of g at most once.