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Dirac Cohomology and Unitarizable Supermodules over Lie Superalgebras of Type $A(m\vert n)$

Steffen Schmidt

TL;DR

This work extends the Dirac-operator framework to basic classical Lie superalgebras of type $A(m\vert n)$ to illuminate unitarizable supermodules. It establishes a Dirac-inequality criterion for unitarity, computes Dirac cohomology for unitarizable simple modules, and proves that Dirac cohomology determines infinitesimal characters via a super-Vogan-like conjecture. The paper develops Dirac induction, connects Dirac cohomology with Kostant cohomology, and introduces a Dirac index that matches the Euler characteristic in the unitarizable setting; it also provides two complementary formulas for ${\mathfrak k}^{\mathbb C}$-characters via Kostant cohomology and Dirac index. Overall, Dirac theory yields a structural, computational, and character-theoretic toolkit for understanding unitarizable ${\mathfrak g}$-supermodules and their ${\mathfrak g}_{\bar0}$-constituents, with potential applications to representation theory and harmonic analysis on Lie supergroups.

Abstract

Dirac operators and Dirac cohomology for Lie superalgebras of Riemannian type, as introduced by Huang and Pandžić, serve as a powerful framework for studying unitarizable supermodules. This paper explores the relationships among the Dirac operator, Dirac cohomology, and unitarizable supermodules specifically within the context of the basic classical Lie superalgebras of type $A(m\vert n)$. The first part examines the structural properties of Dirac cohomology and unitarizable supermodules, including how the Dirac operator captures unitarity, a Dirac inequality, and the uniqueness of the supermodule determined by its Dirac cohomology. Additionally, we calculate the Dirac cohomology for unitarizable simple supermodules. The second part focuses on applications: we give a novel characterization of unitarity, relate Dirac cohomology to nilpotent Lie superalgebra cohomology, derive a decomposition of formal characters, and introduce a Dirac index.

Dirac Cohomology and Unitarizable Supermodules over Lie Superalgebras of Type $A(m\vert n)$

TL;DR

This work extends the Dirac-operator framework to basic classical Lie superalgebras of type to illuminate unitarizable supermodules. It establishes a Dirac-inequality criterion for unitarity, computes Dirac cohomology for unitarizable simple modules, and proves that Dirac cohomology determines infinitesimal characters via a super-Vogan-like conjecture. The paper develops Dirac induction, connects Dirac cohomology with Kostant cohomology, and introduces a Dirac index that matches the Euler characteristic in the unitarizable setting; it also provides two complementary formulas for -characters via Kostant cohomology and Dirac index. Overall, Dirac theory yields a structural, computational, and character-theoretic toolkit for understanding unitarizable -supermodules and their -constituents, with potential applications to representation theory and harmonic analysis on Lie supergroups.

Abstract

Dirac operators and Dirac cohomology for Lie superalgebras of Riemannian type, as introduced by Huang and Pandžić, serve as a powerful framework for studying unitarizable supermodules. This paper explores the relationships among the Dirac operator, Dirac cohomology, and unitarizable supermodules specifically within the context of the basic classical Lie superalgebras of type . The first part examines the structural properties of Dirac cohomology and unitarizable supermodules, including how the Dirac operator captures unitarity, a Dirac inequality, and the uniqueness of the supermodule determined by its Dirac cohomology. Additionally, we calculate the Dirac cohomology for unitarizable simple supermodules. The second part focuses on applications: we give a novel characterization of unitarity, relate Dirac cohomology to nilpotent Lie superalgebra cohomology, derive a decomposition of formal characters, and introduce a Dirac index.

Paper Structure

This paper contains 42 sections, 79 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Vue d'ensemble
  3. Results
  4. Structure of the paper
  5. Notation
  6. Preliminaries
  7. Lie superalgebras of type
  8. Special unitary Lie superalgebras
  9. Lie superalgebras and
  10. Unitarizable supermodules
  11. Unitarizable supermodules.
  12. Harish-Chandra supermodules
  13. Algebraic realization and Shapovalov form
  14. Verma supermodules
  15. Infinitesimal characters
  16. ...and 27 more sections

Key Result

Theorem 1.2.1

Let $\langle \cdot,\cdot \rangle_{M}$ be a positive definite Hermitian form on a ${\mathfrak{g}}$-supermodule $M$ with $\langle M_{\bar{0}},M_{\bar{1}}\rangle_{M}=0$. Then the following statements are equivalent In particular, on unitarizable ${\mathfrak{g}}$-supermodules $M$ a Dirac inequality holds: for all $v\in M\otimes M({\mathfrak{g}_{\bar{1}}}).$

Theorems & Definitions (121)

  • Theorem 1.2.1
  • Theorem 1.2.2
  • Theorem 1.2.3
  • Theorem 1.2.4
  • Theorem 1.2.5
  • Lemma 2.2.1
  • Definition 3.1.1
  • Definition 3.1.2: jakobsen1994full
  • Proposition 3.1.3
  • Lemma 3.1.4
  • ...and 111 more