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

The standard embedding of the Lie algebra V ect(S1) of smooth vector fields on the circle V ect(S1) into the Lie algebra ΨD(S1) of pseudodifferential symbols on S1 identifies vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)\,\frac{\partial }{{\partial x}}\, \in \,Vect\,({S^1})$$\end{document} and its dual as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi \,(f(x)\frac{\partial }{{\partial x}})\, = \,f\,(x)\,\xi \,\pi (u(x)d{x^2})\, = \,u\,(x){\xi ^2}.$$\end{document} The space of symbols can be viewed as the space of functions on T*S1. The natural lift of the action of Diff(S1) yields Diff(S1)-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(S1) into ΨD(S1) we obtain the celebrated Hunter-Saxton equation. Finally, we study the Moyal quantization of all such systems to construct noncommutative systems.