Algebraic Structures and Hamiltonians from the Equivalence Classes of 2D Conformal Algebras

Algebraic Structures and Hamiltonians from the Equivalence Classes of 2D Conformal Algebras

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and Marsden–Weinstein reductions. In this paper, we develop an algebraic approach based on the subalgebras of the 2D conformal algebra c(2). This allows us to classify the centralisers of the enveloping algebra of the conformal algebra and construct the corresponding Hamiltonians with integrals in algebraic form. It is found that the symmetry algebras underlying these algebraic Hamiltonians are six-dimensional quadratic algebras. The Berezin brackets and commutation relations of the quadratic algebraic structures are closed without relying on explicit realisations or representations. We also give the Casimir invariants of the symmetry algebras. Our approach provides algebraic perspectives for the recent work by Fordy and Huang on the construction of superintegrable systems in the Darboux spaces.

Superintegrable systems have been studied from different perspectives over the years, encompassing approaches from classical and quantum mechanics, representation theory, and mathematical physics [1–6]. It has been found that symmetry algebra approaches can be used to analyse physical systems more effectively than other approaches [7–11]. It was shown how Lie algebras, their realisations, and reductions allow one to obtain superintegrable Hamiltonians in the context of classical mechanics [12–14]. For instance, in [15–18] various superintegrable systems on spheres and pseudo-spheres have been obtained. However, most of the previous studies rely on the use of explicit differential operator realisations, the Marsden–Weinstein reductions, and connections with special functions [19–21].

Only very recently it was found that an entirely algebraic approach to superintegrable systems can be established by realizing that the integrals of the systems are as close to polynomial algebras in the enveloping algebras of certain Lie algebras. The approach was successfully implemented for su(3) [22] and gl(3) [23] by applying a scheme based on commutants and algebraizations. It is also extended to commutants regarding the Cartan subalgebras of sl(n) [24] and certain non-semisimple algebras [25]. In this paper, we generalise the procedure further and present the construction of commutants concerning subalgebras, the corresponding algebraic superintegrable Hamiltonians, and their integrals of motion. This allows us to propose a new scheme for classifying superintegrable systems by classifying subalgebras and related commutants of a given Lie algebra.

Recently it was realised that conformal algebras play an important role in classifying superintegrable systems in 2-dimensional spaces [2, 26–28]. In the work by Fordy and Huang [29–31], through the use of specific subalgebras, their Casimir operators and differential realisations, it was shown that superintegrable systems in 2D and 3D conformally flat spaces are related to superintegrable systems in Darboux spaces. In particular, the conformal symmetries of the 2D Euclidean metric were applied to construct new free superintegrable systems in the 2D Darboux spaces. In this paper, we will examine various Abelian and non-Abelian subalgebras of the 2D conformal algebra c(2) and propose a new scheme for classifying integrable and superintegrable systems through explicit construction of commutants relative to the subalgebras.