ON D=2(1 3,1 3) SUPERSYMMETRIC THEORIES .1.

Using the D = 2 (1/k, 1/k) superalgebra (i.e. the two-dimensional gene ralized supersymmetry generated by spin s = 1/k and -1/k; k = 2,3,4,.. ., charge operators Q and (Q) over bar satisfying among other conditio ns Q(k) = P and <(Q)over bar (k)> = (P) over bar where P and (P) over bar are the light components of the energy-momentum operator), we buil d a superspace representation of this exotic symmetry. This realizatio n generalizing the usual D = 2 (1/2, 1/2) supersymmetric one is based on the use of parafermionic variables theta and <(theta)over bar> of s pin s = -1/k and 1/k obeying theta(k) = <(theta)over bar (k)> = 0 as w ell as generalized commutation relation rules. Two-dimensional (1/3, 0 ) and (1/3, 1/3) invariant scalar superfield models extending the well -known D = 2 (1/2,0) and (1/2, 1/2) supersymmetric models are given. T he link between this exotic symmetry and the periodic representation o f U-q(sl(2)) with q(k) = 1 is worked out. Other features are also disc ussed.