Infinitesmal deformations and central extensions of the $n$-th Schrödinger algebra

Abstract

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