Algebraic structure of Galilean superconformal symmetries

The semisimple part of $d$-dimensional Galilean conformal algebra $g^{(d)}$ is given by $h^{(d)}=O(2,1)\oplus O(d)$, which after adding via a semidirect sum the $3d$-dimensional Abelian algebra $t^{(d)}$ of translations, Galilean boosts, and constant accelerations completes the construction. We obtain Galilean superconformal algebra $G^{(d)}$ by first defining the semisimple superalgebra $H^{(d)}$ which supersymmetrizes $h^{(d)}$, and further by considering the expansion of $H^{(d)}$ by tensorial and spinorial graded Abelian charges in order to supersymmetrize the Abelian generators of $t^{(d)}$. For $d=3$ the supersymmetrization of $h^{(3)}$ is linked with a specific model of $\mathcal{N}=4$ extended superconformal mechanics, which is described by the superalgebra $D(2,1;\alpha )$ if $\alpha =1$. We shall also present the alternative derivations of extended Galilean superconformal algebras for $1\le d\le 5$ by employing the Inönü-Wigner contraction method.