Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems (original) (raw)

Using Grozman's formalism of invariant differential operators we demonstrate the derivation of N = 2 Camassa-Holm equation from the action of Vect(S 1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N = 2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H 1-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N = 1 super peakon type equations, known as N = 1 super b-field equations, are derived from the action of Vect(S 1|1) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S 1|1) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N = 1 super b-field equations. Keywords Pseudo-differential symbols • Super KdV • Camassa-Holm equation • Geodesic flow • Super b-field equations • Moyal deformation • Noncommutative integrable systems Mathematics Subject Classification (2000) 17B68 • 37K10 • 58J40 1 Prelude to Noncommutative Integrable Systems Noncommutative geometry [5] extends the notions of classical differential geometry from differential manifold to discrete spaces, like finite sets and fractals, and noncommutative spaces which are given by noncommutative associative algebras. It was an idea of Descartes that we can study a space by means of functions on the space, in other words, the algebra