Dr. Ian Marquette
University of Queensland, Australia
Construction of polynomial algebras related to superintegrable systems
Over the years, it has been discovered that symmetry algebras of superintegrable systems take the form of polynomial algebras. For examples, the integrals of 2D superintegrable models related to fourth and sixth Painlevé transcendent lead to cubic algebras.
In recent years some progress have been made on particular cases of n-dimensional systems with quadratic integral of motion and their symmetry algebras, referred as higher rank quadratic algebras. Among them deformed Kepler-Coulomb systems, singular oscillator, generic model on n-sphere and on the pseudo sphere have been studied. It has been pointed out how these algebraic structures can allow to obtain the energy spectrum via deformed oscillator and Casimir invariants. However, these approaches rely on explicit realizations as differential operators. I will discuss how an algebraic construction of the symmetry algebra of the generic superintegrable systems on the 2-sphere can be generated from an underlying Lie algebra connected with intertwining operators. We obtain a cubic algebra which can be reduced to a quadratic algebra using Casimir invariants.
Tuesday 04 August 2021, 12PM Brisbane Time.