On ${U}(\mathfrak{h})$-free modules over $\mathfrak{sl}(m|n)$
Ivan Dimitrov, Khoa Nguyen
TL;DR
The work provides a detailed classification of ${U}(\mathfrak h)$-free modules of total rank 2 over the Lie superalgebras $\mathfrak{sl}(m|n)$. It gives a complete description for $\mathfrak{sl}(1|1)$, showing only two isomorphism classes with infinite length in $\mathcal{M}_{\mathfrak{sl}(1|1)}(2)$, linked to a string-algebra picture. For $\mathfrak{sl}(m|1)$ with $m\ge 2$, objects in $\mathcal{M}_{\mathfrak{sl}(m|1)}(2)$ and $\mathcal{M}_{\mathfrak{sl}(m|1)}(1|1)$ are parametrized by $\mathbf a\in(\mathbb{C}^\times)^m$ and $\mathcal S\subset{\bf m}$ with explicit action data, and isomorphism classes are governed by the subset $\mathcal S$ and a common scalar multiple of $\mathbf a$; parity considerations yield corresponding graded descriptions. Crucially, the paper proves that for $m,n\ge 2$, the rank-2 categories vanish, marking a sharp boundary in the landscape of $U(\mathfrak h)$-free modules for basic Lie superalgebras.
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.