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Dirac operators for the Dunkl angular momentum algebra

Kieran Calvert and Marcelo Gonçalves De Martino (UGent)
(2022) SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS. 18.
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Abstract
We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.
Keywords
Geometry and Topology, Mathematical Physics, Analysis, Dirac operators, Calogero-Moser angular momentum, rational Cherednik alge-bras, COHOMOLOGY, REPRESENTATIONS

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MLA
Calvert, Kieran, and Marcelo Gonçalves De Martino. “Dirac Operators for the Dunkl Angular Momentum Algebra.” SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, vol. 18, 2022, doi:10.3842/sigma.2022.040.
APA
Calvert, K., & Gonçalves De Martino, M. (2022). Dirac operators for the Dunkl angular momentum algebra. SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 18. https://doi.org/10.3842/sigma.2022.040
Chicago author-date
Calvert, Kieran, and Marcelo Gonçalves De Martino. 2022. “Dirac Operators for the Dunkl Angular Momentum Algebra.” SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS 18. https://doi.org/10.3842/sigma.2022.040.
Chicago author-date (all authors)
Calvert, Kieran, and Marcelo Gonçalves De Martino. 2022. “Dirac Operators for the Dunkl Angular Momentum Algebra.” SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS 18. doi:10.3842/sigma.2022.040.
Vancouver
1.
Calvert K, Gonçalves De Martino M. Dirac operators for the Dunkl angular momentum algebra. SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS. 2022;18.
IEEE
[1]
K. Calvert and M. Gonçalves De Martino, “Dirac operators for the Dunkl angular momentum algebra,” SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, vol. 18, 2022.
@article{8756588,
  abstract     = {{We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.}},
  articleno    = {{040}},
  author       = {{Calvert, Kieran and Gonçalves De Martino, Marcelo}},
  issn         = {{1815-0659}},
  journal      = {{SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS}},
  keywords     = {{Geometry and Topology,Mathematical Physics,Analysis,Dirac operators,Calogero-Moser angular momentum,rational Cherednik alge-bras,COHOMOLOGY,REPRESENTATIONS}},
  language     = {{eng}},
  pages        = {{18}},
  title        = {{Dirac operators for the Dunkl angular momentum algebra}},
  url          = {{http://doi.org/10.3842/sigma.2022.040}},
  volume       = {{18}},
  year         = {{2022}},
}
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