BEGIN:VCALENDAR VERSION:2.0 PRODID:-//132.216.98.100//NONSGML kigkonsult.se iCalcreator 2.20.4// BEGIN:VEVENT UID:20250702T021421EDT-4656V429Tu@132.216.98.100 DTSTAMP:20250702T061421Z DESCRIPTION:Title: Exact solvability and superintegrability : Algebraic con structions.\n\nAbstract: t was discovered how polynomial algebras appear n aturally as symmetry algebra of quantum superintegrable quantum systems. T hey provide insight into their degenerate spectrum\, in particular for mod els involving Painlevé transcendents for which usual approaches of solving ODEs and PDEs cannot be applied. Those algebraic structures extend the sc ope of usual symmetries in context of quantum systems\, but they also been connected to different areas of mathematics such as orthogonal polynomial s. Among them\, the well-known Racah algebra which also admit various gene ralisations.\n \n I will take a different perspective on those algebraic str uctures which is based on Lie algebras\, their related enveloping algebras \, partial Casimir and commutant. I will discuss how such approach differs from using differential operator realizations and why this framework offe rs advantages for their classification. I will point out as well how diffe rent methods from the study of Casimir invariant of non semisimple algebra s which involves solving systems of PDEs can be applied in this context an d greatly facilitate making calculations.The talk will present various exp licit examples\, and in particular the symmetry algebra of the generic sup erintegrable systems on the 2-sphere which can be understood in a purely a lgebraic manner using an underlying Lie algebra.The talk is based on follo wing works:R Campoamor-Stursberg\, I Marquette\, Hidden symmetry algebra a ndconstruction of quadratic algebras of superintegrable systems\, Annals o f Physics 424 168378 (2021)\, arXiv 2020.168378 F Correa\, MA del Olmo\, I Marquette\, J Negro\, Polynomial algebras from su(3) anda quadratically s uperintegrable model on the two sphere\, J.PhysA. Math. and Theor 54 01520 5 (2021) arXiv:2007.11163 R Campoamor-Stursberg\, Ian Marquette\, Quadrati c algebras as commutants of algebraic Hamiltonians in the enveloping algeb ra of Schrodinger algebras\,Annals of Physics. 437 168694 (2022) arXiv 202 1.168694 D Latini\, I Marquette\, YZ Zhang\, Construction of polynomial al gebras from intermediateCasimir invariants of Lie algebras\, J.PhysA Math. and Theor. (2022) arXiv:2204.06840 Rutwig Campoamor-Stursberg\, Ian Marqu ette\, Decomposition of enveloping algebras of simple Lie algebras and the ir related polynomial algebras\, arXiv:2204.11395\n\n \n\nSeminar Physique Mathématique\n CRM-Salle 5340-5380\, Pav. André Aisenstadt\n\n \n\n \n DTSTART:20221004T170000Z DTEND:20221004T180000Z SUMMARY:Ian Marquette\, University of Queensland URL:/mathstat/channels/event/ian-marquette-university- queensland-342426 END:VEVENT END:VCALENDAR