Introduction to the 1-Cohomology of Lie Groups

Abstract

Let G be a real Lie group, ℋ a locally convex barreled complete topological vector space (C.T.V.S.) on ℂ. Let (U, ℋ) be a continuous representation of G in ℋ. A 1-cocycle on G with values in the G-module ℋ is a continuous mapping a: →ℋ such that $${a_{gg'}} - {a_g} = U{\,_g}{a_{g'}}$$ (1.1) for every g, g’ ∈ G. We denote by Z c 1 (G, ℋ) the space of the 1-cocycles on G with values in ℋ and by B 1(G, ℋ) the space of the coboundaries (i.e. the space of the 1-cocycles of the form a g = U g φ − φ for a given φ ∈ ℋ).