分野別セミナー

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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