The structure of the \(\mathcal{N}=4\) supersymmetric linear \(W_{\infty }[\lambda ]\) algebra

For the vanishing deformation parameter \(\lambda \), the full structure of the (anti)commutator relations in the \(\mathcal{N}=4\) supersymmetric linear \(W_{\infty }[\lambda =0]\) algebra is obtained for arbitrary weights \(h_1\) and \(h_2\) of the currents appearing on the left hand sides in these (anti)commutators. The \(w_{1+\infty }\) 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 \(\lambda \), the complete structure of the \(\mathcal{N}=4\) supersymmetric linear \(W_{\infty }[\lambda ]\) algebra is determined for the arbitrary weight \(h_1\) together with the constraint \(h_1-3 \le h_2 \le h_1+1\). The additional structures on the right hand sides in the (anti)commutators, compared to the above \(\lambda =0\) case, arise for the arbitrary weights \(h_1\) and \(h_2\) where the weight \(h_2\) is outside of above region.