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

TL;DR

This paper investigates the second cohomology space of the $n$-th Schrödinger algebra, revealing it vanishes for $n eq 2$ and identifying its dimension for $n=2$, thus classifying infinitesimal deformations and central extensions.

Contribution

It provides a complete computation of the second cohomology for the $n$-th Schrödinger algebra, highlighting the unique case of $n=2$ with a two-dimensional cohomology space.

Findings

Second cohomology vanishes for $n eq 2$.

Dimension of $H^2$ for $n=2$ is 2.

Cohomology of the factor algebra by its center is determined.

Abstract

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.