Using {\it weighted traces} which are linear functionals of the type A→trQ(A):=(tr(AQ−z)−z−1tr(AQ−z))z=0​ defined on the whole
algebra of (classical) pseudo-differential operators (P.D.O.s) and where Q is
some positive invertible elliptic operator, we investigate the geometry of loop
groups in the light of the cohomology of pseudo-differential operators. We set
up a geometric framework to study a class of infinite dimensional manifolds in
which we recover some results on the geometry of loop groups, using again
weighted traces. Along the way, we investigate properties of extensions of the
Radul and Schwinger cocycles defined with the help of weighted traces.Comment: 36 page