The BRST complex and the cohomology of compact lie algebras

Abstract

We construct the BRST and anti-BRST operator for a compact Lie algebra which is a direct sum of abelian and simple ideals. Two different inner products are defined on the ghost space and the hermiticity properties of the ghost and BRST operators with respect to these inner products are discussed. A decomposition theorem for ghost states is derived and the cohomology of the BRST complex is shown to reduce to the standard Lie-algebra cohomology. We show that the cohomology classes of the Lie algebra are given by all invariant anti-symmetric tensors and explain how these can be obtained as zero modes of an invariant operator in the representation space of the ghosts. Explicit examples are given.