The well known prolongation technique of Wahlquist and Estabrook for nonlinear evolution equations is generalized for supersymmetric equations and applied to the supersymmetric extension of the K d V equation of Manin-Hadul. Using the theory of Kac-Moody Lie superalgebra, the explicit form of the resulting Lie superalgebra is determined. It is shown to be isomorphic to R M R x Covs(C(2),u), where R M R is an eight dimensional radical. An auto-Backlund transformation is derived from the prolongation structure and the relationship with known solution methods of the S K d V equation is analysed. In addition it is indicated how a superposition principle for the S K d V equation can be obtained.
We prove that the second cohomology group with coefficients in the adjoint module of both the infinite-dimensional Lie superalgebras k (1) and k + ( 1), as well as their unique central extensions, which are known as the Neveu-Schwarz and the Ramond superalgebras, respectively, is equal to zero. This particularly implies that all these Lie superalgebras are rigid, i.e. they can only be deformed in a trivial manner.
We discuss a method to construct a De Rham complex (differential algebra) of Poincare-Birkhoff-Witt type on the universal enveloping algebra of a Lie algebra g. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of U(g). The construction of such differential structures is interpreted in terms of color Lie superalgebras.
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