Irreducible weight modules over the Schr{ö}dinger Lie algebra in $(n+1)$ dimensional space-time

Abstract

In this paper, we study weight representations over the Schr{ö}dinger Lie algebra ${\mathfrak{s}}_n$ for any positive integer $n$. It turns out that the algebra ${\mathfrak{s}}_n$ can be realized by polynomial differential operators. Using this realization, we give a complete classification of irreducible weight ${\mathfrak{s}}_n$-modules with finite dimensional weight spaces for any $n$. All such modules can be clearly characterized by the tensor product of ${\mathfrak{so}}_n$-modules, ${\mathfrak{sl}}_2$-modules and modules over the Weyl algebra.