Supersymmetrization of deformed BMS algebras

$W(a,b)$ and $W(a,b;\bar{a},\bar{b})$ algebras are deformations of ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$ and ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra respectively. We present an $\mathcal{N}=2$ supersymmetric extension of $W(a,b)$ and $W(a,b;\bar{a},\bar{b})$ algebra in presence of $R-$symmetry generators that rotate the two supercharges. For $W(a,b)$ our construction includes most generic central extensions of the algebra. In particular we find that $\mathcal{N}=2$ ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$ algebra admits a new central extension that has so far not been reported in the literature. For $W(a,b;\bar{a},\bar{b})$, we find that an infinite $U(1{)}_V\times U(1{)}_A$ extension of the algebra is not possible with linear and quadratic structure constants for generic values of the deformation parameters. This implies a similar constraint for $U(1{)}_V\times U(1{)}_A$ extension of $\mathcal{N}=2$ ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra.