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

TL;DR

The paper addresses the explicit construction and classification of non-weight $U(\mathfrak{osp}(1|2))$-modules that are $U(\mathfrak{h})$-free of finite rank, via exponential modules arising from $D(1)$ and oscillator embeddings. It develops a detailed framework using two oscillator homomorphisms $\Phi_{\pm}$ to pull back exponential $D(1)$-modules $E(g)$ to modules $E_{\pm}(g)$, proving their simplicity and providing complete isomorphism criteria. A central contribution is the $F(X_{+},X_{-})$ matrix realization of these modules, along with explicit bases, rank-$n$ decompositions, and rank-one classification with concrete isomorphisms $\Psi_{\pm}$, which are further described through generating functions and ODEs. The work delivers explicit, computable descriptions of the exponential $\mathfrak{osp}(1|2)$-modules, clarifies the structure of the FR(n) category for this algebra, and opens pathways for applications in representation theory and mathematical physics where non-weight modules with controlled central characters are relevant.

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 $E_-(g)$.

Theorems & Definitions (31)

• Lemma 3.1
• Remark 3.2
• Proposition 4.1
• proof
• Definition 4.2
• Remark 4.3
• Definition 5.1
• Proposition 5.2
• proof
• Proposition 5.3