V ect(S¹) Action on Pseudodifferential Symbols on S¹ and (Noncommutative) Hydrodynamic Type Systems

Abstract

The standard embedding of the Lie algebra V ect(S ¹) of smooth vector fields on the circle V ect(S ¹) into the Lie algebra ψD(S ¹) of pseudodifferential symbols on S ¹ identifies vector field and its dual as π(u(x)dx ²) = u(x)ξ ⁻². The space of symbols can be viewed as the space of functions on T ^× S ¹. The natural lift of the action of Diff(S ¹) yields Diff(S ¹)-module. In this paper we demonstate this construction to yield several examples of dispersionless integrable systems. Using Ovsienko and Roger method for nontrivial deformation of the standard embedding of V ect(S ¹) into ψD(S ¹) we obtain the celebrated Hunter-Saxton equation. Finally, we study the Moyal quantization of all such systems to construct noncommutative systems.

Dedicated to Professor Dieter Mayer on his 60th birthday