Non-trivial extension of the (1+2)-Poincaré algebra and conformal invariance on the boundary of ${\mathrm{AdS}}_3$

Using recent results on strings on AdS$_3\times N^d$, 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. 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 \(\pm 1/ k\) used in building of the generalization 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 generalize the so(1,2) result to higher rank Lie algebras g.