Exponential modules of $ \mathfrak{osp}(1|2)$

TL;DR

This paper investigates new families of non-weight modules over the Lie superalgebra rak{osp}(1|2), establishing their simplicity and isomorphism properties, and linking them to oscillator homomorphisms and Weyl algebra modules.

Contribution

It introduces and analyzes two parametric families of modules over rak{osp}(1|2), providing foundational results on their structure and classification.

Findings

Proved simplicity of the modules

Established isomorphism criteria for the modules

Connected modules to oscillator homomorphisms and Weyl algebra

Abstract

We study properties of two families, $E_{+}(g)$ and $E_-(g)$, of non-weight modules over the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ that are parameterized by a nonconstant polynomial $g(x) \in \mathbb C [x]$. These families appear naturally from the two oscillator homomorphisms and the exponential modules over the first Weyl algebra $\mathcal D(1)$. We prove simplicity and isomorphism theorems for $E_{+}(g)$ and $E_-(g)$.