6) Novel Insights

Departamento Otto Rutwig Campoamor. Polynomial Algebras and Superintegrable Systems. 

Boletín del IMI, Nº 218 (21 de mayo de 2026), Sección "Novel Insights".

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En esta nueva sección del Boletín del IMI, se invita a la presentación de textos breves sobre problemas abiertos actuales, tanto en Matemáticas Puras como Aplicadas, y su aplicación a diversas ramas de la ciencia. Las contribuciones, que no deberán superar las 600 palabras (incluyendo las referencias), deben presentar una breve descripción del contexto general en el que se enmarca el problema abierto o la conjetura, así como los trabajos recientes más relevantes. Se sugiere que las referencias (idealmente entre 5 y 8) sean de carácter general —como monografías sobre el tema o artículos de revisión— y que resulten accesibles para un público amplio. Las personas que quieran publicar uno de sus textos en el boletín, pueden enviarlo a secreadm.imi@mat.ucm.es.

In this new section of the Bulletin of the IMI, short presentations of currently open problems in either Pure or Applied Mathematics and their application to several branches of science are expected. The contributions, that should not exceed 600 words, including references, should present a brief description of the general context in which the open problem or conjecture is located, as well as relevant recent work. References (optimally 5 to 8) should preferably be of generic nature, i.e., either monographs on the subject or reviews, accesible to a broad readership. People who wish to publish one of their texts in the Bulletin may send it to secreadm.imi@mat.ucm.es.

POLYNOMIAL ALGEBRAS AND SUPERINTEGRABLE SYSTEMS

Otto Rutwig Campoamor Stursberg

Superintegrable and integrable systems constitute a relevant subject within mathematical physics, exhibiting rich algebraic and geometric structures and a wide range of applications [1,2]. They provide a natural connection with the theory of special functions and orthogonal polynomials, such as the so-called Askey-Wilson scheme and the Painlevé transcendents [3]. It is well known that superintegrable systems are intimately related to finitely generated Poisson algebras, a structure that has been observed and analyzed in many problems of applied mathematics and mathematical physics. From a mathematical perspective, the study of classical and quantum superintegrable systems can be traced back to the 1960s (see [1]), generalizing the notion of Liouville-integrable Hamiltonian systems on symplectic manifolds.

In the classical frame, superintegrable systems are usually interpreted as deformations of the Poisson center of a symplectic manifold. The construction of superintegrable systems based on Lie algebras is known to be deeply related to their (generalized) Casimir invariants. Such systems are important for understanding the structural properties of ample hierarchies of systems, such as the Calogero-Moser and Ruijsenaars systems. In addition, lifting a Lie algebra to its universal enveloping algebra, non-commutative polynomial structures that are hidden in the centralizer of certain subalgebras of enveloping algebras emerge naturally, providing an additional approach and providing new geometric insights of these systems (see [4-6] and references therein).

In this context, it has been observed that the Racah algebra and its generalizations [7] constitute a fundamental tool for the classification of superintegrable systems and their connection to special functions. However, the nature of the symmetry algebra of N-dimensional systems still remains an open and difficult problem, and further progress has required an extrapolation of the Racah algebra to higher ranks, showing that all classical and quantum systems with coalgebra symmetry admit the Racah algebra as a subalgebra of their symmetry algebra [8]. It has further been pointed out that higher rank quadratic algebras allow us to derive algebraically the spectra of systems, suggesting the analysis of higher polynomial algebras (of integrals) as an alternative tool for the spectrum analysis in superintegrable systems, hence combining pure analytical and algebraic methods. The formal study of polynomial algebras in the context of (super)integrable systems, which is currently being systematized, constitutes a promising and interdisciplinary subject of research.

References:

[1] A. M. Perelomov. Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I. Birkhäuser Verlag, Basel, 1990.

[2] C. Daskaloyannis. Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems. J. Math. Phys., 42:1100–1119, 2001.

[3] E. G. Kalnins, J. M. Kress and W. Miller jun. Separation of variables and superintegrability. The symmetry of solvable systems. IOP Publishing, Bristol, 2018.

[4] N. Reshetikhin. Degenerately integrable systems. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 433:224–245, 2015.

[5] R. Campoamor-Stursberg and I. Marquette. Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems. Ann. Phys., 424:168378, 2021.

[6] J. C. López Vieyra and A. V. Turbiner. Tremblay–Turbiner–Winternitz system at integer index k: Polynomial algebras of integrals, J. Math. Phys. 67:042101, 2026.

[7] S. Bockting-Conrad and H. Huang. The universal enveloping algebra of sl(2) and the Racah algebra, Comm. Algebra 48:1022–1040, 2020.

[8] D. Latini, I. Marquette Y-Z. Zhang. Embedding of the Racah algebra R(n) and superintegrability, Ann. Phys. 426:168397, 2021.