Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Crampé, Nicolas; Frappat, L.; Ragoucy, É.

doi:10.48550/arxiv.2206.01031

Cited by 2 publications

(1 citation statement)

References 40 publications

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“…Thus the connection with the Racah algebra R( 4), considering that several different bases are conceivable to give a presentation (see e.g. [34]), is obtained through the following identifications:…”

Section: Data Availability Statementmentioning

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.

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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 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.

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Section: Data Availability Statementmentioning

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.

View full text Add to dashboard Cite

show abstract

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

Campoamor-Stursberg¹,

Latini²,

Marquette³

et al. 2022

Preprint

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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 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

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Cited by 2 publications

References 40 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

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