Event info

Poisson polynomial algebras in superintegrable systems: Structures and representations

Date: Tuesday, 4 June 2024 12:00 - 13:00
Place: Online via Zoom
Speaker: Junze Zhang (University of Queensland, Australia)

Abstract:
A physical system is called superintegrable if it possesses more integrals of motion than degrees of freedom. Such systems play a significant role in many branches of physics and mathematics, especially, their connections to orthogonal polynomials, Painlevé transcendent, Lie theories and symplectic geometries. The existence of superintegrable systems has long and deep historical roots, but their systematic studies are relatively recent. Superintegrability and classification have been studied via constructing symmetry algebras generated by the minimal set of integrals of motion. In the classical case, such algebras usually admit a Poisson structure and therefore can be viewed as polynomial deformations of Lie-Poisson algebras. In quantum analogues, suitable quantization lead to finitely generated associative algebras.

In the first part of the talk, we define superintegrable systems via an example, i.e., 2D harmonic oscillators, and introduce a special type of quadratic Poisson algebra that was studied by Daskaloyannis, Vinet, Zhedanov, etc. Such quadratic algebras, generated by integrals of motions, appear in contexts that are related to second-order superintegrable systems, e.g., in 2D Darboux spaces. We will describe a method that allows us to determine the finite-dimensional representations of these quadratic Poisson algebras and the energy spectrum of the associated Hamiltonians. The second part of the talk is about the construction of superintegrable systems from finite-dimensional Lie algebras and their universal enveloping algebras. Such construction will lead to higher-order polynomial algebras that have played an important role in the classification of superintegrable systems and have deep connections with the special functions. Notice that most of the recent constructions rely on explicit differential operator realizations, homogeneous spaces and Marsden-Weinstein reductions. In the new setting, superintegrability arises from the centralizer subalgebra of a symmetric algebra through reduction chains of Lie algebras. We will have a look at two different examples: one-dimensional and Abelian subalgebras of a 2D conformal algebra so(3, 1) and Cartan centralizers of all the complex semisimple Lie algebras of non-exceptional types. From these constructions, the closure of the Berezin brackets and commutation relations of the polynomial algebraic structures are obtained without relying on explicit realizations or representations. It can also be seen that quadratic and higher-rank Racah types of algebras are embedded in these polynomial algebras via suitable changes of basis.

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The seminar will be held during lunch time, so please feel free to join us while eating your lunch. Especially for graduate students, there are not many opportunities to hear lectures in different fields of mathematics. This seminar is a valuable opportunity for faculty members to give lectures to non-specialists. We hope that you will feel free to participate regardless of your field, whether it is to broaden your mathematical horizons or to get some exposure to English. Motivated third and fourth year undergraduate students are also welcome to participate. Please feel free to drop by.
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