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Discrete mathematical modelling seminar
Seminar information archive ～12/27｜Next seminar｜Future seminars 12/28～
| Organizer(s) | Tetsuji Tokihiro, Ralph Willox |
|---|
2025/12/18
17:00-18:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Ian Marquette (La Trobe University)
The classification of superintegrable systems with higher-order symmetries and related algebraic structures (English)
Ian Marquette (La Trobe University)
The classification of superintegrable systems with higher-order symmetries and related algebraic structures (English)
[ Abstract ]
Superintegrable systems admit more symmetries than degrees of freedom. The case of maximally superintegrable systems is characterized by 2n-1 integrals for n degrees of freedom. They possess rich mathematical structures and are related to orthogonal polynomials, special functions, and representation theory. The problem of classifying superintegrable systems was solved for quadratically superintegrable Hamiltonians in 2D conformally flat spaces about 20 years ago. The classification of superintegrable systems in higher-dimensional Riemannian spaces or with higher-order integrals is much more involved, and some partial results are known in three dimensions. These systems have attracted interest because they lead to algebraic structures known as polynomial algebras, which also appear in other contexts of mathematical physics.
This talk is devoted to discussing different approaches and recent results related to the classification of superintegrable systems with second-order and higher-order symmetries and the associated algebraic structures. In the direct approach, consisting of solving systems of partial differential equations, compatibility equations can be related to the works of Bureau, Chazy, and Cosgrove on higher-order nonlinear differential equations and Painlevé transcendents. I will present some examples related to the fourth and sixth Painlevé transcendents and demonstrate that their integrals lead to polynomial algebras. We discuss how these algebraic structures can still be used to gain insight into the spectrum.
I will discuss another and more recent approach to classifying superintegrable systems, which build a completely algebraic setting and on higher-degree polynomials in the enveloping algebra of Lie algebras. This allows the construction of algebraic Hamiltonians, integrals, and new perspectives on their associated algebraic structures. This approach offers greater flexibility, as different realizations can be used. This notion of the commutant leads to generalizations of Racah-type algebras.
Superintegrable systems admit more symmetries than degrees of freedom. The case of maximally superintegrable systems is characterized by 2n-1 integrals for n degrees of freedom. They possess rich mathematical structures and are related to orthogonal polynomials, special functions, and representation theory. The problem of classifying superintegrable systems was solved for quadratically superintegrable Hamiltonians in 2D conformally flat spaces about 20 years ago. The classification of superintegrable systems in higher-dimensional Riemannian spaces or with higher-order integrals is much more involved, and some partial results are known in three dimensions. These systems have attracted interest because they lead to algebraic structures known as polynomial algebras, which also appear in other contexts of mathematical physics.
This talk is devoted to discussing different approaches and recent results related to the classification of superintegrable systems with second-order and higher-order symmetries and the associated algebraic structures. In the direct approach, consisting of solving systems of partial differential equations, compatibility equations can be related to the works of Bureau, Chazy, and Cosgrove on higher-order nonlinear differential equations and Painlevé transcendents. I will present some examples related to the fourth and sixth Painlevé transcendents and demonstrate that their integrals lead to polynomial algebras. We discuss how these algebraic structures can still be used to gain insight into the spectrum.
I will discuss another and more recent approach to classifying superintegrable systems, which build a completely algebraic setting and on higher-degree polynomials in the enveloping algebra of Lie algebras. This allows the construction of algebraic Hamiltonians, integrals, and new perspectives on their associated algebraic structures. This approach offers greater flexibility, as different realizations can be used. This notion of the commutant leads to generalizations of Racah-type algebras.
