Fractional Quantum Hall Excitations as RdTS Highest Weight State Representations

Using the Chern-Simons effective model of fractional quantum Hall (FQH) systems, we complete partial results obtained in the literature on FQHE concerning topological orders of FQH states. We show that there exists a class of effective FQH models having the same filling fraction $\nu$, interchanged under $ Gl(n,Z)$ transformations and extends results on Haldane hierarchy. We also show that Haldane states at any generic hierarhical level n may be realised in terms of n Laughlin states composites and rederive results for the n=2,3 levels respectively associated with $\nu= {{2}\over{5}}$ and $\nu= {{3}\over{7}}$ filling fractions. We study symmetries of the filling fractions series $\nu= {{p_2}\over{p_1 p_2 -1}}$ and $\nu= {{p_1 p_2 -1}\over{p_1 p_2 p_3 -p_1 -p_2}}$, with $p_1$ odd and $p_2$ and $p_3$ even integers, and show that, upon imposing the Gl(n,Z) invariance, we get remarkable informations on their stability. Then, we reconsider the Rausch de Traubenberg and Slupinsky (RdTS) algebra recently obtained in [1,2] and analyse its limit on the boundary $\partial({AdS_3})$ of the (1+2) dimensional manifold $AdS_3$. We show that generally one may distinguish bulk highest weight states (BHWS) living in $AdS_3$ and edge highest weight states (EHWS)living on the border $\partial({AdS_3})$. We explore these two kinds of RdTS representations carrying fractional values of the spin and propose them as condidates to describe the FQH states.