OSP(1-VERTICAL-BAR-2) AND SL(2) REDUCTIONS IN GENERALIZED SUPER-TODA MODELS AND FACTORIZATION OF SPIN-1 2 FIELDS/

I show that the classical Toda models built on superalgebras, and obta ined from a reduction with respect to an Sl(2) algebra, are ''linearly supersymmetrizable'' (by adding spin 1/2 fields) if and only if the S l(2) is the bosonic part of an OSp(1\2) algebra. In that case, the mod el is equivalent to the one constructed from a reduction with respect to the OSp(1\2) algebra, up to spin 1/2 fields. The corresponding W al gebras are related through a factorization of spin 1/2 fields (bosons and fermions). I illustrate this factorization on an example: the supe rconformal algebra built on Sl(n + 1\2).