Non-Splittable Extensions of the Virasoro and Superconformal Algebras via Pseudodifferential Operators

Applying the Weyl symbols techniques to the algebra of pseudodifferential operators on a circle and its extensions, a number of non-splittable Abelian, as well as infinite tower of non-Abelian, kernel extensions of the Virasoro and superconformal algebras have been explicitly constructed. The corresponding non-trivial cocycles appear as quantum corrections to the Poisson brackets when developing the Weyl multiplication formula. As an application, this construction may yield interesting generalizations of the KdV equation.