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Generalized permutation matrices and non-weight modules over $\mathfrak{sl}(m|1)$

Ivan Dimitrov, Khoa Nguyen, Charles Paquette, David Wehlau

TL;DR

The paper addresses the problem of classifying and constructing $U(\mathfrak h)$-free $\mathfrak{sl}(m|1)$-modules of rank $(k|k)$, extending the rank-1 theory to higher ranks. It introduces a robust framework of generalized permutation matrices to realize modules $M(A_1,\dots,A_m)$, and shows how exponential modules $E\big(\sum a_i x_i^{k_i},\mathcal S\big)$ arise as explicit $M$-realizations with precise $h_i$-companion data. A central result is the indecomposability criterion: $E(\sum a_i x_i^{k_i},\mathcal S)$ is indecomposable iff $\gcd(k_1,\dots,k_m)=1$, together with explicit isomorphism criteria and endomorphism descriptions; when the gcd is $s>1$, the module splits into $s$ indecomposable blocks. The work also develops a duality theory for these modules and provides a moduli space description as a finite union of weighted projective spaces, offering a concrete, geometric classification of isomorphism classes. Overall, the results yield new, explicit constructions and a thorough understanding of non-weight $\mathfrak{sl}(m|1)$-modules of finite rank, enriching the representation theory of Lie superalgebras with concrete algebraic and geometric data.

Abstract

We study the category $\mathcal{M}_{\mathfrak{sl}(m|1)}(k|k)$ of $\mathcal U(\mathfrak h)\text{-free}$ $\mathcal U(\mathfrak{sl}(m|1))$-modules of rank $k$ in each parity (rank $(k|k)$), where $k\in\mathbb{Z}_{\geq1}$. We construct an explicit family of such modules, provide an isomorphism theorem, and establish an indecomposability criterion.

Generalized permutation matrices and non-weight modules over $\mathfrak{sl}(m|1)$

TL;DR

The paper addresses the problem of classifying and constructing -free -modules of rank , extending the rank-1 theory to higher ranks. It introduces a robust framework of generalized permutation matrices to realize modules , and shows how exponential modules arise as explicit -realizations with precise -companion data. A central result is the indecomposability criterion: is indecomposable iff , together with explicit isomorphism criteria and endomorphism descriptions; when the gcd is , the module splits into indecomposable blocks. The work also develops a duality theory for these modules and provides a moduli space description as a finite union of weighted projective spaces, offering a concrete, geometric classification of isomorphism classes. Overall, the results yield new, explicit constructions and a thorough understanding of non-weight -modules of finite rank, enriching the representation theory of Lie superalgebras with concrete algebraic and geometric data.

Abstract

We study the category of -modules of rank in each parity (rank ), where . We construct an explicit family of such modules, provide an isomorphism theorem, and establish an indecomposability criterion.
Paper Structure (17 sections, 31 theorems, 306 equations)

This paper contains 17 sections, 31 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Basis of $\mathfrak{sl} (m|1)$
  4. Automorphisms of ${\mathbb C} [{\bf h}]$
  5. Weyl superalgebras
  6. Weight modules over $\mathfrak{sl}(m|1)$ and $\mathcal{D}(m|1)$
  7. The category $\mathcal{M}_{\mathfrak{sl}(m|1)}\bigl(k|k\bigr)$ over $\mathcal{U}(\mathfrak{sl}(m|1))$
  8. The category $\mathcal{M}_{\mathfrak{sl}(m|1)}\bigl(1|1\bigr)$
  9. Modules over ${\mathfrak{sl}(1|1)}$
  10. Generalized permutation matrix and $\mathcal{U}(\mathfrak{h})$-free modules of finite rank over $\mathfrak{sl}(m|1)$
  11. Generalized permutation matrices and $h_i$-companion pairs
  12. The module $M(A_1,\dots,A_m)$
  13. $\mathcal{U}(\mathfrak{h})$-duality
  14. Exponential modules for $\mathcal{U}(\mathfrak{sl}(m|1))$
  15. The $M(A_1,\dots,A_m)$-realization of $E\!\left(\sum_{i=1}^m a_i x_i^{k_i},\,\mathcal{S}\right)$ and its properties
  16. ...and 2 more sections

Key Result

Lemma 2.7

For $M \in \mathcal{M}_{\mathfrak{sl}(m|1)}\bigl(k|k\bigr)$, fix an identification $M\;=\;\mathbb C[{\bf h}]^{\oplus k} \oplus \mathbb C[{\bf h}]^{\oplus k}$ and write $\mathbf f({\bf h}) := \bigl[f_1({\bf h})\;\;\dots\;\;f_{2k}({\bf h})\bigr]^{\mathsf T}$, where $f_r({\bf h}) \in \mathbb C[\mathbf Moreover, where $A_{IJ}({\bf h}), B_{IJ}({\bf h}) \in \mathop{\mathrm{Mat}}\nolimits_k\bigl(\mathb

Theorems & Definitions (82)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Remark 2.9
  • ...and 72 more