Loop algebras and their relation to the conformal structure of integrable systems (Q582371)

In the first half, the author explains how to contruct the infinite set of conserved quantities of the integrable nonlinear evolution equation associated with a loop algebra \(\hat{\mathfrak g}\) on a simple Lie algebra \({\mathfrak g}\). The case \({\mathfrak g}=A_ 1\) is discussed in detail. In this case one has the conserved quantities of the KdV and MKdV equations. A relation with a Poisson bracket realization of the Virasoro algebra is emphasized. Then the author turns to the supersymmetric case, \({\mathfrak g}={\mathfrak b}(0,1)\), the (3\(| 2)\)-dimensional simple Lie superalgebra. In this case the Virasoro algebra is replaced by the Neveu-Schwarz algebra. Using the same argument as in the ordinary case, some supersymmetric integrable nonlinear evolution equations as well as their conserved quantities are obtained.