The ${\cal N}=2,4$ Supersymmetric Linear $W_{\infty}[\lambda]$ Algebras for Generic $\lambda$ Parameter

The four different kinds of currents are given by the multiple $(\beta,\gamma)$ and $(b,c)$ ghost systems with a multiple product of derivatives. We determine their complete algebra where the structure constants depend on the deformation parameter $\lambda$ appearing in the conformal weights of above fields nontrivially and depend on the generic spins $h_1$ and $h_2$ appearing on the left hand sides in the (anti)commutators. By taking the linear combinations of these currents, the ${\cal N}=4$ supersymmetric linear $W_{\infty}[\lambda]$ algebra (and its ${\cal N}=4$ superspace description) for generic $\lambda$ is obtained explicitly. Moreover, we determine the ${\cal N}=2$ supersymmetric linear $W_{\infty}[\lambda]$ algebra for arbitrary $\lambda$. As a by product, the $\lambda$ deformed bosonic $W_{1+\infty}[\lambda] \times W_{1+\infty}[\lambda+\frac{1}{2}]$ subalgebra (a generalization of Pope, Romans and Shen's work in $1990$) is obtained. The first factor is realized by $(b,c)$ fermionic fields while the second factor is realized by $(\beta,\gamma)$ bosonic fields. The degrees of the polynomials in $\lambda$ for the structure constants are given by $(h_1+h_2-2)$. Each $w_{1+\infty}$ algebra from the celestial holography is reproduced by taking the vanishing limit of other deformation prameter $q$ at $\lambda=0$ with the contractions of the currents.