Dynamical symmetry algebras of two superintegrable two-dimensional systems

Marquette, Ian; Quesne, Christiane

doi:10.1088/1751-8121/ac9164

Cited by 3 publications

(2 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[3,6,15,27,28] and references therein). Such an approach has been successfully applied to various Lie algebras such as su(3) and gl(3) in order to recover the Smorodinsky-Winternitz systems, as well as the generic model on the sphere S 2 , which connects to the 58 superintegrable systems on conformally flat spaces [29,30]. The approach chosen here is quite different, and is mainly based on the algebraic setting, where integrals are polynomials in the enveloping algebra of a Lie algebra and the (non-commutative) integrability or superintegrability will be deduced from commutation relations of the polynomials in enveloping algebra of s, or either using the simpler relations in terms of the Lie-Poisson or Berezin bracket.…”

Section: Algebraic Integrability and Superintegrabilitymentioning

confidence: 99%

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

Campoamor-Stursberg

Latini

Marquette

et al. 2023

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

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 Sn-1 and the related Racah algebra R(n). In particular, we show explicitly how the models on the 2-sphere and 3-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.

show abstract

Section: Algebraic Integrability and Superintegrabilitymentioning

confidence: 99%

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

Campoamor-Stursberg

Latini

Marquette

et al. 2023

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…[6,27,28,3,15] and references therein). Such an approach has been successfully applied to various Lie algebras such as su(3) and gl(3) in order to recover the Smorodinsky-Winternitz systems, as well as the generic model on the sphere S 2 , which connects to the 58 superintegrable systems on conformally flat spaces [29,30]. The approach chosen here is quite different, and is mainly based on the algebraic setting, where integrals are polynomials in the enveloping algebra of a Lie algebra and the (non-commutative) integrability or superintegrability will be deduced from commutation relations of the polynomials in enveloping algebra of s, or either using the simpler relations in terms of the Lie-Poisson or Berezin bracket.…”

Section: Algebraic Integrability and Superintegrabilitymentioning

confidence: 99%

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

Campoamor-Stursberg¹,

Latini²,

Marquette³

et al. 2022

Preprint

View full text Add to dashboard Cite

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 2-sphere and 3-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.

show abstract

Cylindrical first-order superintegrability with complex magnetic fields

Kubů

Šnobl

2023

Journal of Mathematical Physics

View full text Add to dashboard Cite

This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space E3 in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [Peng et al., Phys. Rev. Lett. 114, 010601 (2015)], they were so far not addressed in the growing literature on superintegrability. Here, we venture into this field by searching for additional first-order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.

show abstract

Dynamical symmetry algebras of two superintegrable two-dimensional systems

Cited by 3 publications

References 43 publications

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

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

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

Cylindrical first-order superintegrability with complex magnetic fields

Contact Info

Product

Resources

About