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On the full set of unitarizable supermodules over $\mathfrak{sl}(m\vert n)$

Steffen Schmidt

Abstract

We present a novel classification of unitarizable supermodules over special linear Lie superalgebras using an algebraic quadratic Dirac operator introduced by Huang and Pandžić and a corresponding Dirac inequality.

On the full set of unitarizable supermodules over $\mathfrak{sl}(m\vert n)$

Abstract

We present a novel classification of unitarizable supermodules over special linear Lie superalgebras using an algebraic quadratic Dirac operator introduced by Huang and Pandžić and a corresponding Dirac inequality.
Paper Structure (28 sections, 32 theorems, 147 equations)

This paper contains 28 sections, 32 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. The Classification Method
  3. The Finite-Dimensional Case
  4. Infinite-Dimensional Case
  5. Comparison to the Literature
  6. Notation
  7. Leitfaden
  8. Special Linear Lie Superalgebras sl(m|n)
  9. Structure Theory
  10. Real Forms su(p,q|r,s)
  11. Unitarizable Supermodules and Dirac Operators
  12. Fundamentals
  13. Intermezzo: Highest Weight Supermodules
  14. Unitarizable Highest Weight -Modules
  15. Highest Weight Supermodules
  16. ...and 13 more sections

Key Result

Theorem 1

$\Lambda \in {\mathfrak h}^{\ast}$ is the highest weight of a unitarizable highest weight ${\mathfrak{g}}$-supermodule if and only if the following conditions hold:

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Lemma 3: jakobsen1994full, SchmidtDirac
  • Definition 4: jakobsen1994full
  • Theorem 5: furutsu1991classificationneeb2011lie
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8: NeebHW
  • Theorem 9
  • ...and 37 more