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Simple $\mathfrak{sl}_2$-modules that are torsion free $U(\mathfrak{h})$-modules of rank $1$

Dimitar Grantcharov, Libor Krizka, Volodymyr Mazorchuk

Abstract

We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{osp}(1|2)$.

Simple $\mathfrak{sl}_2$-modules that are torsion free $U(\mathfrak{h})$-modules of rank $1$

Abstract

We provide an explicit classification of all simple -modules that are torsion free of rank over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra .
Paper Structure (24 sections, 17 theorems, 41 equations)

This paper contains 24 sections, 17 theorems, 41 equations.

Table of Contents

  1. Introduction and description of the results
  2. Historical overview
  3. Description of the results
  4. Methods
  5. Structure of the paper
  6. Modules over skew Laurent polynomial algebras
  7. Skew Laurent polynomial algebras over fields
  8. The $\sigma$-subgroup of $\Bbbk^{\times}$
  9. Another example
  10. $\Bbbk[x,x^{-1},\sigma]$-modules of dimension $1$
  11. Application to $\mathfrak{sl}_2$
  12. General setup
  13. The Lie algebra $\mathfrak{sl}_2$
  14. $U_\vartheta$ and skew Laurent polynomial algebra
  15. Simple $R$-modules and the corresponding simple $U_\vartheta$-modules
  16. ...and 9 more sections

Key Result

Lemma 1

The subset $G_\sigma$ is, in fact, a subgroup of $\Bbbk^{\times}$.

Theorems & Definitions (37)

  • Lemma 1
  • proof
  • Example 2
  • Example 3
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • ...and 27 more