The general Racah algebra as the symmetry algebra of generic systems on pseudo-spheres

Abstract

We characterize the symmetry algebra of the generic superintegrable system on a pseudo-sphere corresponding to the homogeneous space SO(p, q + 1)/SO(p, q) where p + q = N , N ∈ N. These symmetries occur both in quantum as well as in classical systems in various contexts, so they are quite important in physics. We show that this algebra is independent of the signature (p, q + 1) of the metric and that it is the same as the Racah algebra R(N + 1). The spectrum obtained from R(N + 1) via the Daskaloyannis method depends on undetermined signs that can be associated to the signatures. Two examples are worked out explicitly for the cases SO(2, 1)/SO(2) and SO(3)/SO(2) where it is shown that their spectrum obtained by means of separation of variables coincide with particular choices of the signs, corresponding to the specific signatures, of the spectrum for the symmetry algebra R(3).

Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures

“…The defining relations from P ij are those of the classical analog of the Racah algebra R(n) [45,20]:…”

“…• In Section 3 we specialise to the Lie-Poisson (co)algebra sl(2, R) endowed with the (primitive) coassociative coproduct ∆. After obtaining the corresponding left and right Casimir invariants, we elucidate their connection with the generators of (a Poisson analog) of the generalised Racah algebra R(n) [45,20]. Finally, we proceed towards the main goal of the work and show that the quadratic substructures mentioned above can in fact be interpreted as a classical (Poisson) realisation of some specific Racah subalgebras of rank one.…”

Non-Hermitian superintegrable systems

“…Together with many other systems constructed via the MASAs approach, these models have been extensively studied in the literature, e.g. from the point of view of contractions [14,15], intertwining operators [16][17][18], Wilson polynomials [19], quadratic algebras [20][21][22] and also for an algebraic approach independent of the realisations [23], to name just a few. Although the Marsden-Weinstein scheme has been studied beyond mechanical systems, up to the authors knowledge, there is no a systematic approach using this construction for superintegrable non-Hermitian Hamiltonians with real spectra.…”

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

“…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].…”