Ian Marquette, University of Queensland
Title:听Exact solvability and superintegrability : Algebraic constructions.
础产蝉迟谤补肠迟:听t was discovered how polynomial algebras appear naturally as symmetry algebra of quantum superintegrable quantum systems. They provide insight into their degenerate spectrum, in particular for models involving Painlev茅 transcendents for which usual approaches of solving ODEs and PDEs cannot be applied. Those algebraic structures extend the scope of usual symmetries in context of quantum systems, but they also been connected to different areas of mathematics such as orthogonal polynomials. Among them, the well-known Racah algebra which also admit various generalisations.
I will take a different perspective on those algebraic structures 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 offers advantages for their classification. I will point out as well how different methods from the study of Casimir invariant of non semisimple algebras which involves solving systems of PDEs can be applied in this context and greatly facilitate making calculations.The talk will present various explicit examples, and in particular the symmetry algebra of the generic superintegrable systems on the 2-sphere which can be understood in a purely algebraic manner using an underlying Lie algebra.The talk is based on following works:R Campoamor-Stursberg, I Marquette, Hidden symmetry algebra andconstruction of quadratic algebras of superintegrable systems, Annals of Physics 424 168378 (2021), arXiv 2020.168378 F Correa, MA del Olmo, I Marquette, J Negro, Polynomial algebras from su(3) anda quadratically superintegrable model on the two sphere, J.PhysA. Math. and Theor 54 015205 (2021) arXiv:2007.11163 R Campoamor-Stursberg, Ian Marquette, Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrodinger algebras,Annals of Physics. 437 168694 (2022) arXiv 2021.168694 D Latini, I Marquette, YZ Zhang, Construction of polynomial algebras from intermediateCasimir invariants of Lie algebras, J.PhysA Math. and Theor. (2022) arXiv:2204.06840 Rutwig Campoamor-Stursberg, Ian Marquette, Decomposition of enveloping algebras of simple Lie algebras and their related polynomial algebras, arXiv:2204.11395
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Seminar Physique Math茅matique
CRM-Salle 5340-5380, Pav. Andr茅 Aisenstadt
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