Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras



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Campoamor-Stursberg, Rutwig and Latini, Danilo and Marquette, Ian and Zhang, Yao-Zhong (2023) Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras. Journal of physics A: Mathematical and general, 56 (4). 045202. ISSN 0305-4470

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Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra sl(n) is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of sl(n), this provides an explicit connection with the generic superintegrable model on the (n − 1)-dimensional sphere S n−1 and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of sl(3) and sl(4), respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

Item Type:Article
Uncontrolled Keywords:Commutants; Lie algebras; Enveloping algebras; Racah algebras; Superintegrability
Subjects:Sciences > Mathematics > Algebra
ID Code:76853
Deposited On:02 Mar 2023 08:44
Last Modified:02 Mar 2023 08:51

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