Racah Algebras, the Centralizer Zn(sl2) and Its Hilbert–Poincaré Series

The higher rank Racah algebra R ( n ) introduced in De Bie et al. (J Phys A Math Theor 51(2):025203, 2017. arXiv:1610.02638 ) is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR ( n ), is then introduced. Using the results from classical invariant...

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Bibliographic Details
Published in:Annales Henri Poincaré 2022, Vol.23 (7), p.2657-2682
Main Authors: Crampé, Nicolas, Gaboriaud, Julien, d’Andecy, Loïc Poulain, Vinet, Luc
Format: Article
Language:English
Subjects:
Algebra
Classical and Quantum Gravitation
Dynamical Systems and Ergodic Theory
Elementary Particles
Mathematical and Computational Physics
Mathematical Methods in Physics
Original Paper
Physics
Physics and Astronomy
Quantum Field Theory
Quantum Physics
Quotients
Relativity Theory
Theoretical
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Summary:The higher rank Racah algebra R ( n ) introduced in De Bie et al. (J Phys A Math Theor 51(2):025203, 2017. arXiv:1610.02638 ) is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR ( n ), is then introduced. Using the results from classical invariant theory, this sR ( n ) algebra is shown to be isomorphic to the centralizer Z n ( s l 2 ) of the diagonal embedding of U ( s l 2 ) in U ( s l 2 ) ⊗ n . This leads to a first and novel presentation of the centralizer Z n ( s l 2 ) in terms of generators and defining relations. An explicit formula of its Hilbert–Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey–Wilson algebra and its higher rank generalizations is discussed.
ISSN:1424-0637
1424-0661
DOI:10.1007/s00023-021-01152-y