The structure of the N= 4 supersymmetric linear W_∞[λ] algebra

For the vanishing deformation parameter λ , the full structure of the (anti)commutator relations in the N= 4 supersymmetric linear W_∞[λ= 0] algebra is obtained for arbitrary weights h₁ and h₂ of the currents appearing on the left hand sides in these (anti)commutators. The w_1+∞ algebra can be seen from this by taking the vanishing limit of other deformation parameter q with the proper contractions of the currents. For the nonzero λ , the complete structure of the N= 4 supersymmetric linear W_∞[λ] algebra is determined for the arbitrary weight h₁ together with the constraint h₁- 3 ≤ h₂≤ h₁+ 1 . The additional structures on the right hand sides in the (anti)commutators, compared to the above λ= 0 case, arise for the arbitrary weights h₁ and h₂ where the weight h₂ is outside of above region.