Speaker: Ian Marquette(La Trobe University)
Time: Tuesday, 2025-May-27 17:00
Location: R1610, SIMIS
Zoom Meeting: 838 524 0106(Passcode: SIMIS)
Abstract:
I will present examples of polynomial algebras that appear in context of quantum superintegrable systems. Those symmetry algebras allow to characterize spectrum and wave functions, in particular for exotic models involving Painlevé transcendents. The talk will motivate the need of other approaches to polynomial algebra beyond explicit differential operator realizations [1,2,3]. I will review recent works were the notion of commutant was applied to simple Lie algebras and their Cartan subalgebras [1,2,3]. This allowed us to present algebraic definition of superintegrability, integrals of motion and symmetry algebra. I will provide details how the Poisson-Lie bracket setting allows to construct indecomposable polynomials via systems of partial differential equations. I will present formula for the indecomposable polynomials for $sl(n)$ that lead to a polynomial algebra of degree n-1. I will present additional explicit formula for the quadratic and cubic algebra in the case of $sl(3)$ and $sl(4)$. I will explain how explicit realization allow recover the Racah algebra as particular case and how they can be quantized. I will point out how this method can be applied to various subalgebras chains g ⊃ g′ with applications to dynamical symmetries and missing label problems. I will discuss the case of the Elliott su(3) ⊃ so(3) and seniority so(5) ⊃ su(2) × u(1) chains [4] which were used in nuclear physics. I will explain how in both cases our approach gives a three generators cubic algebra. I will provide some further comments on the state of those constructions for simple Lie algebras and other subalgebras chains [5]. References: 1. R Campoamor-Stursberg, D Latini, I Marquette, YZ Zhang, Algebraic (super-) integrability from commutants of subalgebras in universal enveloping algebras J. Phys. A: Math. and Theor. 56 (4), 045202 (2023) 2. R Campoamor-Stursberg, D Latini, I Marquette, YZ Zhang, Polynomial algebras from commutants: Classical and quantum aspects of A_3, J. Phys. A: Conf. Series 2667 012037 (2023) 3. R Campoamor-Stursberg, D Latini, I Marquette, J Zhang, YZ Zhang,, Superintegrable systems associated to commutants of Cartan subalgebras in enveloping algebras, arXiv:2406.01958 4. R Campoamor-Stursberg, D Latini, I Marquette, YZ Zhang, Polynomial algebras from Lie algebras reduction chains g ⸧g’ , Ann. of Phys. 459 169496 1-19 (2023) 5. R Campoamor-Stursberg, D Latini, I Marquette, J Zhang, YZ Zhang, Polynomial algebra from Lie algebra reduction chain su(4) ⸧ su(2) x su(2): The supermultiplet model, arXiv:2503.04108
About Speaker:
Prof. Ian Marquette is a distinguished researcher and Senior Lecturer in the Department of Mathematics and Physical Sciences at La Trobe University, bringing over a decade of expertise in mathematical physics. After earning a Ph.D. in Physics from the Université de Montréal (2009), he secured competitive fellowships, including an FQRNT award for work on integrable systems and algebraic structures. His international career spans postdoctoral positions at the University of York (UK) and The University of Queensland (Australia), where he explored exactly solvable models in superconductivity and indecomposable Lie algebra representations. Recognized for his innovative contributions, he received an ARC Discovery Early Career Award (2013–2016) and an ARC Future Fellowship (2019–2024). Since 2024, Prof. Marquette has expanded his research to bridge mathematical physics with broader mathematical disciplines, maintaining a dynamic profile in quantum symmetries, integrability, and interdisciplinary connections.
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