Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 085, 21 pages      arXiv:2402.08944      https://doi.org/10.3842/SIGMA.2024.085

The Racah Algebra of Rank 2: Properties, Symmetries and Representation

Sarah Post and Sébastien Bertrand
Department of Mathematics, University of Hawai‘i at Mānoa, Honolulu, Hawai‘i, USA

Received March 08, 2024, in final form September 10, 2024; Published online September 22, 2024

Abstract
The goals of this paper are threefold. First, we provide a new ''universal'' definition for the Racah algebra of rank 2 as an extension of the rank-1 Racah algebra where the generators are indexed by subsets and any three disjoint indexing sets define a subalgebra isomorphic to the rank-1 case. With this definition, we explore some of the properties of the algebra including verifying that these natural assumptions are equivalent to other defining relations in the literature. Second, we look at the symmetries of the generators of the rank-2 Racah algebra. Those symmetries allows us to partially make abstraction of the choice of the generators and write relations and properties in a different format. Last, we provide a novel representation of the Racah algebra. This new representation requires only one generator to be diagonal and is based on an expansion of the split basis representation from the rank-1 Racah algebra.

Key words: Racah algebra; representation; symmetry; rank-2.

pdf (447 kb)   tex (27 kb)  

References

1. Askey R., Wilson J., A set of orthogonal polynomials that generalize the Racah coefficients or $6-j$ symbols, SIAM J. Math. Anal. 10 (1979), 1008-1016.
2. Bockting-Conrad S., Huang H.-W., The universal enveloping algebra of $\mathfrak{sl}_2$ and the Racah algebra, Comm. Algebra 48 (2020), 1022-1040, arXiv:1907.02135.
3. Crampé N., Frappat L., Ragoucy E., Representations of the rank two Racah algebra and orthogonal multivariate polynomials, Linear Algebra Appl. 664 (2023), 165-215, arXiv:2206.01031.
4. Crampé N., Gaboriaud J., d'Andecy L.P., Vinet L., Racah algebras, the centralizer $Z_n(\mathfrak {sl}_2)$ and its Hilbert-Poincaré series, Ann. Henri Poincaré 23 (2022), 2657-2682, arXiv:2105.01086.
5. De Bie H., Genest V.X., van de Vijver W., Vinet L., A higher rank Racah algebra and the $\mathbb Z^n_2$ Laplace-Dunkl operator, J. Phys. A 51 (2018), 025203, 20 pages, arXiv:1610.02638.
6. De Bie H., Iliev P., van de Vijver W., Vinet L., The Racah algebra: an overview and recent results, in Lie Groups, Number Theory, and Vertex Algebras, Contemp. Math., Vol. 768, American Mathematical Society, RI, 2021, 3-20, arXiv:2001.11195.
7. Genest V.X., Vinet L., Zhedanov A., The equitable Racah algebra from three $\mathfrak{su}(1,1)$ algebras, J. Phys. A 47 (2014), 025203, 12 pages, arXiv:1309.3540.
8. Granovskii Ya.A., Zhedanov A.S., Nature of the symmetry group of the $6j$-symbol, Soviet Phys. JETP 94 (1988), 1982-1985.
9. Granovskii Ya.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
10. Granovskii Ya.I., Zhedanov A.S., Hidden symmetry of the Racah and Clebsch-Gordan problems for the quantum algebra $\mathfrak{sl}_q(2)$, arXiv:hep-th/9304138.
11. Huang H.-W., Bockting-Conrad S., Finite-dimensional irreducible modules of the Racah algebra at characteristic zero, SIGMA 16 (2020), 018, 17 pages, arXiv:1910.11446.
12. Huang H.-W., Bockting-Conrad S., The Casimir elements of the Racah algebra, J. Algebra Appl. 20 (2021), 2150135, 22 pages, arXiv:1711.09574.
13. Kirillov A.N., Reshetikhin N.Yu., Representations of the algebra ${U}_q(\mathfrak{sl}(2))$, $q$-orthogonal polynomials and invariants of links, in Infinite-Dimensional Lie Algebras and Groups, Adv. Ser. Math. Phys., Vol. 7, World Scientific Publishing, Teaneck, NJ, 1989, 285-339.
14. Lévy-Leblond J.-M., Lévy-Nahas M., Symmetrical coupling of three angular momenta, J. Math. Phys. 6 (1965), 1372-1380.
15. Louck J.D., Recent progress toward a theory of tensor operators in the unitary groups, Amer. J. Phys. 38 (1970), 3-42.
16. Maple 2019, Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.
17. Post S., Racah polynomials and recoupling schemes of $\mathfrak{su}(1,1)$, SIGMA 11 (2015), 057, 17 pages, arXiv:1504.03705.
18. Racah G., Theory of complex spectra. II, Phys. Rev. 62 (1942), 9-10.
19. Racah G., Lectures on Lie groups, in Group Theoretical Concepts and Methods in Elementary Particle Physics, Quantum Physics and Its Applications, Gordon and Breach, New York, 1964, 1-36.