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A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$

Dimitar Grantcharov, Khoa Nguyen, Kaiming Zhao

TL;DR

This work classifies simple scalar-type rank-2 modules that are free over the Cartan subalgebra for $\mathfrak{sl}(2)$. By realizing the action through $K(h)\in\mathrm{GL}_2(\mathbb{C}[h])$ and employing twisted similarity $\sim_{\sigma^{-1}}$ together with Cohn’s standard form, the authors reduce simplicity and isomorphism to explicit matrix data. They construct a universal family $M(\alpha,\mathbf a,K(h))$ and identify sharp simplicity criteria, plus an isomorphism theorem governed by the group $G=\mathbb{C}^*\rtimes\mathbb{Z}$ acting on parameter spaces; this yields a complete classification of simple scalar-type rank-2 modules in terms of $G$-orbits. The results illuminate a rich non-weight facet of $\mathfrak{sl}(2)$-modules and connect the module classification to nonabelian cohomology interpretations and automorphisms of $\mathrm{GL}_2(\mathbb{C}[h])$.

Abstract

We study simple $\mathfrak{sl}(2)$-modules over $\mathbb C$ that are free of finite rank as $U(\mathfrak h)$-modules, where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak{sl}(2)$. Our main result is an explicit classification of the scalar-type simple modules of rank $2$. We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in $\mbox{GL}_2({\mathbb C}[h])$ and rely on Cohn's standard form.

A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$

TL;DR

This work classifies simple scalar-type rank-2 modules that are free over the Cartan subalgebra for . By realizing the action through and employing twisted similarity together with Cohn’s standard form, the authors reduce simplicity and isomorphism to explicit matrix data. They construct a universal family and identify sharp simplicity criteria, plus an isomorphism theorem governed by the group acting on parameter spaces; this yields a complete classification of simple scalar-type rank-2 modules in terms of -orbits. The results illuminate a rich non-weight facet of -modules and connect the module classification to nonabelian cohomology interpretations and automorphisms of .

Abstract

We study simple -modules over that are free of finite rank as -modules, where is a Cartan subalgebra of . Our main result is an explicit classification of the scalar-type simple modules of rank . We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in and rely on Cohn's standard form.
Paper Structure (12 sections, 29 theorems, 181 equations, 1 table)

This paper contains 12 sections, 29 theorems, 181 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and notation
  4. The category of $U(\mathfrak{h})$-free modules of finite rank
  5. $\sigma^{-1}$--similarity
  6. Standard Form for Matrices in $\mathop{\mathrm{GL}}\nolimits_2(\mathbb C[h])$
  7. Standard form via a Euclidean Algorithm on the first row
  8. The Set $\mathcal{S}$
  9. A family of $U(\mathfrak{h})$-free modules of rank 2
  10. Simplicity criteria
  11. Isomorphism theorem and the group $G$
  12. Interpretation of the group $G$ as a subgroup of $\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathrm{GL}}\nolimits_2(\mathbb C[h]))$

Key Result

Theorem 2.2

Let $M = \mathbb C[h]^{\oplus n}$ be a module in $\mathcal{M}$ with a central character $\gamma = (2\alpha-1)^2$. Then there exist $K(h)\in\mathop{\mathrm{GL}}\nolimits_n(\mathbb C[h])$ and ${\bf{a}} = (a_-, a_0, a_+)$, such that the $U(\mathfrak{sl}(2))$-action on $M$ is given by: where $g_i(h) \in \mathbb C[h]$ for all $i \in \{1,\ldots, n\}$. The matrix $P_{({\bf{a}},\alpha)}(h)$ is the Smith

Theorems & Definitions (73)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5: $\sigma$--similarity and $\sigma^{-1}$--similarity
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 63 more