The Structure of the $\mathcal{N}=4$ Supersymmetric Linear $W_{\mathrm{\infty }}[\lambda ]$ Algebra

For the vanishing deformation parameter $\lambda$, the full structure of the (anti)commutator relations in the $\mathcal{N}=4$ supersymmetric linear $W_{\mathrm{\infty }}[\lambda =0]$ 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 $w_{1+\mathrm{\infty }}$ 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 $W_{\mathrm{\infty }}[\lambda ]$ 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.