Weighted Traces on Algebras of Pseudo-Differential Operators and Geometry of Loop Groups

摘要

Using {\it weighted traces} which are linear functionals of the type $$A\totr^Q(A):=(tr(A Q^{-z})-z^{-1} tr(A Q^{-z}))_{z=0}$$ defined on the wholealgebra of (classical) pseudo-differential operators (P.D.O.s) and where $Q$ issome positive invertible elliptic operator, we investigate the geometry of loopgroups in the light of the cohomology of pseudo-differential operators. We setup a geometric framework to study a class of infinite dimensional manifolds inwhich we recover some results on the geometry of loop groups, using againweighted traces. Along the way, we investigate properties of extensions of theRadul and Schwinger cocycles defined with the help of weighted traces.