Date : 2026/11/17 15:00 -- 16:00
Place: F215
Speaker : Ian Marquette Senior Lecturer, La Trobe University, Australia
Title : Polynomial algebras from commutants: superintegrability and subalgebra chains
Abstract:
The search for superintegrable systems connects additional conserved quantities with exact solvability and spectral degeneracy. Their symmetries often form polynomial algebras, as illustrated by the Racah algebra of the generic system on the two-sphere. A parallel problem arises in representation theory: restricting a Lie algebra representation to a subalgebra can produce repeated components that Casimir labels do not distinguish. In Elliott's 𝔰𝔲(3)⊃𝔰𝔬(3) model, requiring a missing-label operator. The different sets of missing-label operators than form also polynomial algebras.
I will describe a common framework based on commutants: elements of 𝑈(𝔤) commuting with a chosen subalgebra 𝔞, and their counterparts 𝑆(𝔤)𝔞 under the Lie–Poisson bracket. Solving invariant differential equations yields, in the examples considered, finite generating sets, polynomial relations and central elements before a realisation is chosen. These centralisers supply conserved quantities for algebraic Hamiltonians built from the subalgebra generators and ambient Casimir operators. Symmetrisation lifts invariant generators to the enveloping algebra, with lower-order corrections entering their commutation relations.
For Cartan subalgebras, invariance becomes a zero-weight condition on root monomials. Directed cycles organise the type-𝐴 generators, and suitable realisations and Marsden-Weinstein reduction recover higher-rank Racah algebras and superintegrable sphere models. The rank-three cases exhibit quintic closure for 𝐵3,𝐶3 and cubic closure for 𝐷3≃𝐴3. For non-Cartan subalgebras, the Elliott and seniority chains produce three-generator cubic algebras over their centres, organising the missing-label operators. Subalgebra-adapted gradings help determine their relations. I will conclude with reductions of coadjoint orbits and the rank conditions of Mishchenko–Fomenko noncommutative integrability, linking the algebraic construction to geometric superintegrable systems.