African Journal Of Mathematical Physics Volume 6 (2008) 123-143 Integrable Systems: Bosonic and Supersymmetric Cases A. El Boukili, M.B. Sedra, A. Zemate Laboratoire de Physique de la Mati` ere et Rayonnement (LPMR), Facult´ e des Sciences, K´ enitra, Morocco. e-mail: Corresponding Author, sedra@ibntofail.ma Abstract The main task of this work concerns integrable models and supersymmetric extensions of the Gelfand-Dickey algebra of pseudo differential operators. The con- sistent and systematic study that we perform consists in describing in detail the relation existing between the algebra of super(pseudo-)differential operators on the ring of superfields u s 2 (z,θ),s ∈ Z and the higher and lower spin extensions of the conformal algebra. In relation to integrable systems, the supersymmetric GD bracket play a pioneering role as it gives in some sense a guarantee of integrability of the associated non linear supersymmetric systems. Keywords: SuperGD Bracket, Integrable systems Pacs: 11.10.Lm, 11.30.Pb, 02.30.Ik I. INTRODUCTION For several years, two dimensional integrable models 1 in connection with conformal field theories 2,3 and their underlying lower (s ≤ 2) 4–7 and higher (s ≥ 2) 8,9 spin symmetries, have occupied a central position in various areas of research. More particularly, a lot of interest has been paid to w-symmetries 8 , which are infinite dimensional algebras extending the conformal in- variance (Virasoro algebra) by adding to the energy momentum operator T (z) ≡ W 2 , a set of conserved currents w s (z), of conformal spin s> 2 with some composite operators necessary for the closure of the algebra. In the language of 2d conformal field theory, the above mentioned currents w s are taken in general as primary satisfying the OPE 2 T (z)W s (ω)= s (z − ω) 2 W s (ω)+ W ′ s (ω) (z − ω) , (I.1) or equivalently, W s = J s ˜ W s (I.2) under a general change of coordinate (diffeomorphism) x → ˜ x(x) with J = ∂ ˜ x ∂x is the associated Jacobian. These w-symmetries exhibit among others a non linear structure and are not Lie algebra 0 c  a GNPHE publication 2008, ajmp@fsr.ac.ma 123