On cohomology of the Lie algebra of pseudodifferential symbols on a circle (Q1356295)

The authors first review what is known in general about the computation of the cohomology of the Lie algebra \(\Psi\) of pseudodifferential operators on the circle and the adjoint cohomology of the Lie algebra \(C^\infty (T^2)\), and the Fuks principle that ``under deformation, cohomology can decrease in dimension but never increase'', the authors discuss the result of Khesin and Kravchenko that there exist two linearly independent cohomology classes in \(H^1 (\Psi,\Psi)\) whose images in \(H^2(\Psi,\mathbb{R})\) are also linearly independent and prove that the cup product of these two classes is nonzero in \(H^2(\Psi,\Psi)\). They also state the result that there exists a cohomology class in \(H^3(\Psi,\mathbb{R})\) whose image in the Gelfand-Fuks cohomology \(H^3({\mathcal X}(S^1),\mathbb{R})\) is the Godbillon-Vey class, and they give an expression for this class in terms of the two classes in the Khesin-Kravchenko theorem.