The Racah algebra: An overview and recent
                    results

Bie, Hendrik De; Илиев, Пламен; Vijver, Wouter van de; Vinet, Luc

doi:10.1090/conm/768/15450

Cited by 16 publications

(28 citation statements)

References 42 publications

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“…The vertices of this graph are given by the abelian subalgebras and the edges link two abelian algebras differing by only one generator. The web of the abelian subalgebras given in figure 1 generalizes the result of [9,10] where only the subalgebras of the type (C ij , C ijk ) have been considered. In this latter case, the connection graph reduces to a truncated tetrahedron.…”

Section: A Web Of Abelian Subalgebrasmentioning

confidence: 53%

“…The above presentation of R( 4) is not unique and different other possibilities exist in the literature. Instead of C ij , another set of generators P ij = C ij − C i − C j (for i = j) and P ii = 2C i has been used previously to define the Racah algebra R(4) [9,10,5]. In this paper, we prefer to define the R(4) algebra using all the elements C I with I ⊆ {1, 2, 3, 4} since it simplifies the computations.…”

Section: Definitions and Casimir Elements Of The R(4) Algebramentioning

confidence: 99%

“…The rank two Racah algebra also showed up in the recoupling problem of four copies of su(1, 1) algebra in [38]. Then, the centralizer of the diagonal action on n copies has been studied in [10]. The exact isomorphism between this centralizer and a quotient of the higher rank Racah algebra, called special Racah algebra, has been proven in [5].…”

Section: Introductionmentioning

confidence: 99%

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Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Crampé¹,

Frappat²,

Ragoucy³

2022

Preprint

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The algebraic structure of the rank two Racah algebra is studied in detail. We provide an automorphism group of this algebra, which is isomorphic to the permutation group of five elements. This group can be geometrically interpreted as the symmetry of a folded icosidodecahedron. It allows us to study a class of equivalent irreducible representations of this Racah algebra. They can be chosen symmetric so that their transition matrices are orthogonal. We show that their entries can be expressed in terms of Racah polynomials. This construction gives an alternative proof of the recurrence, difference and orthogonal relations satisfied by the Tratnik polynomials, as well as their expressions as a product of two monovariate Racah polynomials. Our construction provides a generalization of these bivariate polynomials together with their properties.

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Section: A Web Of Abelian Subalgebrasmentioning

confidence: 53%

Section: Definitions and Casimir Elements Of The R(4) Algebramentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Crampé¹,

Frappat²,

Ragoucy³

2022

Preprint

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“…Recently it has been shown that quadratic algebras associated with n-dimensional systems are in general of higher rank [15,16,17,18]. These algebraic structures allow one to obtain useful information on quantum systems and their degenerate spectrum [19].…”

Section: Introductionmentioning

confidence: 99%

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Latini¹,

Marquette²,

Zhang³

2022

Preprint

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We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the gl(n) Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

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“…To obtain these results we relied on the so-called left and right partial Casimir invariants, commonly encountered in the framework of coalgebra symmetry approach to superintegrability [33][34][35], that can be constructed from suitable linear combinations of the generalized Racah R(n) generators [36]. The generalized Racah algebra R(n) previously appeared as the symmetry algebra of the generic superintegrable model on the (n − 1)-sphere (see [22] and references therein) and pseudo-sphere [37]. It was proposed in [38] as a higher-rank generalisation of the rank one Racah algebra R(3), which is in turn the symmetry algebra of the generic superintegrable model on the 2-sphere [39,40].…”

Section: Introductionmentioning

confidence: 99%

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

Latini,

Marquette,

Zhang

2022

Preprint

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We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras g. In this approach, we deal with explicit polynomials in the enveloping algebra of g ⊕ g ⊕ g. We present explicit examples to show how these Lie algebras can display different behaviours and can lead to Abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the Levi factor of a Levi decomposable Lie algebra can be used as a tool to construct "copies"of polynomial algebras. Different schemes to obtain polynomial algebras associated to algebraic Hamiltonians have been proposed in the literature, among them the use of commutants of various type. The present approach is different and relies on the construction of intermediate Casimir invariants in the enveloping algebra U(g ⊕ g ⊕ g).

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The Racah algebra: An overview and recent results

Cited by 16 publications

References 42 publications

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Representations of the rank two Racah algebra and orthogonal multivariate polynomials

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

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