Using, the harmonic analysis of the 3- and 2-spheres, we study the compactification of the IKKT model on the F 0 Hirzebruch complex surface. Like for tori and orbifolds, we show that here there also exists a possibility of compactifications of matrix models of M-theory on non-commutative F 0 geometry. Other features, such as the extension of Connes et al 's projective module solutions to non-commutative F 0 are studied.

Using recent results on strings on AdS 3 × N d , where N is a d-dimensional compact manifold, we re-examine the derivation of the non trivial extension of the (1+2)dimensional-Poincaré algebra obtained by Rausch de Traubenberg and Slupinsky, refs[1]and [29] . We show by explicit computation that this new extension is a special kind of fractional supersymmetric algebra which may be derived from the deformation of the conformal structure living on the boundary of AdS 3 . The two so(1, 2) Lorentz modules of spin ± 1 k used in building of the generalisation of the (1+2) Poincaré algebra are re-interpreted in our analysis as highest weight representations of the left and right Virasoro symmetries on the boundary of AdS 3 . We also complete known results on 2d-fractional supersymmetry by using spectral flow of affine Kac-Moody and superconformal symmetries. Finally we make preliminary comments on the trick of introducing Fth-roots of g-modules to generalise the so(1, 2) result to higher rank Lie algebras g.

Using parafermionic field theoretical methods, the fundamentals of 2d fractional supersymmetry 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 ( 1 3 , 0) and ( ( 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 , (

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