Infinitesmal deformations and central extensions of the $n$-th Schr\"odinger algebra

In this paper we study the second cohomology space for the $n$-th Schr\"{o}dinger algebra $\mathfrak{sch}_n$. We prove that the second cohomology space is vanishing for $n \geq 3$ and show that $\dim(H^2(\mathfrak{sch}_2, \mathfrak{sch}_2)) =2.$ Moreover, we investigate the factor algebra of the Schr\"{o}dinger algebra by its center and determine the second cohomology group of this factor algebra.