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SUMMARY:第65回NITEP談話会
DTSTART;VALUE=DATE-TIME:20261117T060000Z
DTEND;VALUE=DATE-TIME:20261117T070000Z
DTSTAMP;VALUE=DATE-TIME:20261022T061529Z
UID:indico-event-234@indico.nitep.omu.ac.jp
DESCRIPTION:Date : 2026/11/17    15:00 -- 16:00\nPlace: F215\n\n\nSpeak
 er : Ian Marquette    Senior Lecturer\, La Trobe University\, Australia
 \n \n\nTitle : Polynomial algebras from commutants: superintegrability an
 d subalgebra chains\n\nAbstract: \nThe search for superintegrable systems
  connects additional conserved quantities with exact solvability and spect
 ral degeneracy. Their symmetries often form polynomial algebras\, as illus
 trated by the Racah algebra of the generic system on the two-sphere. A par
 allel problem arises in representation theory: restricting a Lie algebra r
 epresentation 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 o
 perators than form also polynomial algebras.\n\nI will describe a common f
 ramework based on commutants: elements of 𝑈(𝔤) commuting with a chos
 en subalgebra 𝔞\, and their counterparts 𝑆(𝔤)𝔞 under the Lie
 –Poisson bracket. Solving invariant differential equations yields\, in t
 he examples considered\, finite generating sets\, polynomial relations and
  central elements before a realisation is chosen. These centralisers suppl
 y conserved quantities for algebraic Hamiltonians built from the subalgebr
 a generators and ambient Casimir operators. Symmetrisation lifts invariant
  generators to the enveloping algebra\, with lower-order corrections enter
 ing their commutation relations.\n\nFor Cartan subalgebras\, invariance be
 comes 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 m
 odels. The rank-three cases exhibit quintic closure for 𝐵3\,𝐶3 and c
 ubic closure for 𝐷3≃𝐴3. For non-Cartan subalgebras\, the Elliott a
 nd seniority chains produce three-generator cubic algebras over their cent
 res\, organising the missing-label operators. Subalgebra-adapted gradings 
 help determine their relations. I will conclude with reductions of coadjoi
 nt orbits and the rank conditions of Mishchenko–Fomenko noncommutative i
 ntegrability\, linking the algebraic construction to geometric superintegr
 able systems.\n\nhttps://indico.nitep.omu.ac.jp/event/234/
LOCATION:
URL:https://indico.nitep.omu.ac.jp/event/234/
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