Fractional Supersymmetry As a Matrix Model

Using parafermionic field theoretical methods, the fundamentals of 2d fractional supersymmetry ${\mathbf{Q}}^K=P$ are set up. Known difficulties induced by methods based on the $U_q(sl(2))$ quantum group representations and non commutative geometry are overpassed in the parafermionic approach. Moreover we find that fractional supersymmetric algebras are naturally realized as matrix models. The K=3 case is studied in details. Links between 2d $(\frac{1}{3},0)$ and $(({\frac{1}{3}}^2),0)$ fractional supersymmetries and N=2 U(1) and N=4 su(2) standard supersymmetries respectively are exhibited. Field theoretical models describing the self couplings of the matter multiplets $(0^2,(\frac{1}{3}{)}^2,(\frac{2}{3}{)}^2)$ and $(0^{4},({1\over 3})^{4},({2\over 3})^{4})$ are given.