Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

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Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra sl(n) is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of sl(n), this provides an explicit connection with the generic superintegrable model on the (n − 1)-dimensional sphere S n−1 and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of sl(3) and sl(4), respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.
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[1] Perelomov A M 1990 Integrable Systems of Classical Mechanics and Lie Algebras (Basel:Birkhäuser Verlag) [2] Tempesta P, Turbiner A V and Winternitz P 2001 Exact solvability of superintegrable systems J. Math. Phys. 42 4248 [3] Tremblay F, Turbiner A V and Winternitz P 2009 An infinite family of solvable and integrable quantum systems on a plane J. Phys. A: Math. Theor. 42 242001 [4] Kalnins E G, Miller W and Post S 2007 Wilson polynomials and the generic superintegrable system on the 2-sphere J. Phys. A: Math. Theor. 40 11525 [5] Miller W, Post S and Winternitz P 2013 Classical and quantum superintegrability with applications J. Phys. A: Math. Theor. 46 423001 [6] Freidel L and Maillet J M 1991 Quadratic algebras and integrable systems Phys. Lett. A 262 278–84 [7] Ballesteros A and Ragnisco O 1998 A systematic construction of completely integrable Hamiltonians from coalgebras J. Phys. A: Math. Gen. 31 3791 [8] Daskaloyannis C and Tanoudis Y 2010 Quadratic algebras for three-dimensional superintegrable systems Phys. At. Nucl. 73 214–21 [9] Yates L A and Jarvis P D 2018 Hidden supersymmetry and quadratic deformations of the spacetime conformal superalgebra J. Phys. A: Math. Theor. 51 145203 [10] Campoamor-Stursberg R and Marquette I 2022 Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras Ann. Phys. 437 168694 [11] Campoamor-Stursberg R and Marquette I 2021 Hidden symmetry algebra and construction of polynomial algebras of superintegrable systems Ann. Phys. 424 168378 [12] Campoamor-Stursberg R 2022 On some algebraic formulations within universal enveloping algebras related to superintegrability Acta Polytech. 62 16–22 [13] De Bie H, Genest V X, van de Vijver W and Vinet L 2018 A higher rank Racah algebra and the Z2n Laplace-Dunkl operator J. Phys. A: Math. Theor. 51 025203 [14] De Bie H, Iliev P, van de Vijver W and Vinet L 2021 The Racah algebra: an overview and recent results Contemp. Math. 768 3–20 [15] Letourneau P, Vinet L, Vinet S and Zhedanov A 2019 The generalized Racah algebra as a commutant J. Phys.: Conf. Ser. 1194 012034 [16] Correa F, del Olmo M A, Marquette I and Negro J 2020 Polynomial algebras from su(3) and a quadratically superintegrable model on the two-sphere J. Phys. A: Math. Theor. 54 015205 [17] Latini D, Marquette I and Zhang Y-Z 2021 Embedding of the Racah algebra R(n) and superintegrability Ann. Phys. 426 168397 [18] Dixmier J 1974 Alg`Ebres Enveloppantes (Paris: Hermann) [19] Racah G 1950 Sulla caratterizzazione delle rappresentazione irriducibili dei gruppi semisimplici di Lie Rend. Acad. Naz. Lincei, Sci. Fis. Mat. Nat. 8 108–12 [20] Berezin F A 1967 Some remarks about the associated envelope of a Lie algebra Funct. Anal. Appl.1 91–102 [21] Dixmier J 1959 Sur l’algèbre enveloppante d’une algèbre de Lie nilpotente Archiv Math. 10 321–7 [22] Gel’fand I M and Kirillov A A 1968 On the structure of the field of quotients of the enveloping algebra of a semisimple lie algebra Dokl. Akad. Nauk SSSR 180 775–7 [23] Beltrametti E and Blasi A 1966 On the number of Casimir operators associated with any Lie group Phys. Lett. 20 62–64 [24] Campoamor-Stursberg R and de Traubenberg M R 2018 Group Theory in Physics: A Practitioner’s Guide (Singapore: World Scientific) [25] Genest V X, Vinet L and Zhedanov A 2013 The equitable Racah algebra from three su(1, 1) algebras J. Phys. A: Math. Theor. 47 025203 [26] Latini D, Marquette I and Zhang Y-Z 2021 Racah algebra R(n) from coalgebraic structures and chains of R(3) substructures J. Phys. A: Math. Theor. 54 395202 [27] Ovsienko V and Turbiner A V 1992 Plongements d’une algèbre de Lie dans son algèbre enveloppante C. R. Acad. Sci. Paris 314 13–16 [28] Letourneau P and Vinet L 1995 Superintegrable systems, polynomial algebras and quasi-exactly solvable Hamiltonian Ann. Phys. 243 144–68 [29] Kalnins E G and Miller W J 2014 Quadratic algebra contractions and second-order superintegrable systems Anal. Appl. 12 583–612 [30] Marquette I and Quesne C 2022 Dynamical symmetry algebra of two superintegrable twodimensional systems J. Phys. A: Math. Theor. 55 415203 [31] Serre J P 1974 Alg`Ebres de Lie Semi-Simples Complexes (New York: W. A. Benjamin) [32] Calzada J A, Negro J and del Olmo M A 2006 Superintegrable quantum u(3) systems and higher rank factorizations J. Math. Phys. 47 043511 [33] Crampé N, Gaboriaud J, Poulain d’Andecy L and Vinet L 2022 Racah algebras, the centralizer Zn(sl2) and its Hilbert-Poincaré series Ann. Henri Poincaré 23 2657 [34] Crampé N, Frappat L and Ragoucy E Representations of the rank two Racah algebra and orthogonal multivariate polynomials (arXiv:2206.01031) [math.RT] [35] Sharp R T and Lam C S 1969 Internal–labeling problem J. Math. Phys. 10 2033–7 [36] Campoamor-Stursberg R 2011 Internal labelling problem: an algorithmic procedure J. Phys. A: Math. Theor. 44 025234