On ${U}(\mathfrak{h})$-free modules over $\mathfrak{sl}(m|n)$

TL;DR

This paper classifies certain ${U}(rak{h})$-free modules over $rak{sl}(m|n)$, showing that for most cases these modules do not exist, and provides a complete classification for $rak{sl}(m|1)$.

Contribution

It offers a complete classification of ${U}(rak{h})$-free modules of rank 2 over $rak{sl}(m|1)$ and proves non-existence for higher ranks when $m,n geq 1$.

Findings

Complete classification for $rak{sl}(m|1)$ modules.

Non-existence of such modules for $m,n geq 2$.

Characterization of module categories $rak{M}_{rak{sl}(m|n)}(2)$ and $rak{M}_{rak{sl}(m|n)}(1|1)$.

Abstract

We study two categories of ${U}(\mathfrak h)$-free $\mathfrak{sl}(m|n)$-modules of total rank 2: $\mathcal{M}_{\mathfrak{sl}(m|n)}(2)$, whose objects are free of rank 2 over ${U}(\mathfrak h)$ which are not necessarily $\mathbb Z_2$-graded, and $\mathcal{M}_{\mathfrak{sl}(m|n)}(1|1)$, whose objects are supermodules with even and odd parts each isomorphic to ${U}(\mathfrak h)$. For $\mathfrak{sl}(m|1)$ we give a complete classification in both categories, and we prove that for $m,n\geq 2$ both categories are empty.