Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0016 Acta Polytechnica 62(1):16–22, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague ON SOME ALGEBRAIC FORMULATIONS WITHIN UNIVERSAL ENVELOPING ALGEBRAS RELATED TO SUPERINTEGRABILITY Rutwig Campoamor-Stursberg Universidad Complutense de Madrid, Instituto de Matemática Interdisciplinar–UCM, Plaza de Ciencias 3, E-28040 Madrid, Spain correspondence: rutwig@ucm.es Abstract. We report on some recent purely algebraic approaches to superintegrable systems from the perspective of subspaces of commuting polynomials in the enveloping algebras of Lie algebras that generate quadratic (and eventually higher-order) algebras. In this context, two algebraic formulations are possible; a first one strongly dependent on representation theory, as well as a second formal approach that focuses on the explicit construction within commutants of algebraic integrals for appropriate algebraic Hamiltonians defined in terms of suitable subalgebras. The potential use in this context of the notion of virtual copies of Lie algebras is briefly commented. Keywords: Enveloping algebras, commutants, quadratic algebras, superintegrability. 1. Introduction Both the study of (quasi-)exactly solvable systems, as well as that of super-integrable systems make an extensive use of the universal enveloping algebras of Lie algebras, either in the context of the so-called hidden algebras or as symmetry algebras of the sys- tem. Of particular interest are those systems that, beyond super-integrability properties, also belong to the class of (quasi-)exactly solvable systems [1–5]. In particular, quadratic subalgebras have been shown to be a powerful tool for classifying and comparing super- integrable systems, as shown in [6], where the scheme of superintegrable systems on a two-dimensional con- formally flat space has been characterized in terms of contractions. Additional examples in higher dimen- sions [7] lead us to suspect that n-dimensional super- integrable systems are somehow associated to (higher rank) polynomials in a suitable enveloping algebra [8], further stimulating the search of alternative algebraic approaches based on the structural properties of en- veloping algebras. Although the precise fundamental properties of enveloping algebras of generic semidirect sums of simple and solvable Lie algebras are still far from being completely understood, a purely formal ansatz applied to the case of the Schrödinger alge- bras Ŝ(n) has recently been shown to provide some interesting features [9]. In this work we comment on some purely alge- braic approaches formulated in the enveloping algebras of Lie algebras for the identification or construction of quadratic algebras that may lead to super-integrable systems, once a suitable realization of the enveloping algebra by first-order differential operators has been chosen. The motivation for this analysis lies primarily on the inspection of super-integrable systems from the point of view of the algebraic properties of first integrals seen as elements of an enveloping algebra, as well as an attempt to determine to which extent these integrals are characterized algebraically by the hidden algebra [10]. This moreover suggests a realization-free description of systems in terms of commutants of al- gebraic Hamiltonians in enveloping algebras [11], in which elements of the coadjoint representation of Lie algebras may be useful to simplify computations. 2. First algebraic reformulation In the context of (quasi)-exactly solvable problems, the Hamiltonians are described as differential operators in p variables that admit an expression as elements in the enveloping algebra of a Lie algebra g, commonly known as the hidden algebra, not necessarily associ- ated to any symmetry algebra of the system. The main requirement is the existence of a representation of g that is invariant for the Hamiltonian, a constraint that allows us to determine its spectrum (either partially or completely) using algebraic methods [12]. So, for ex- ample, the universal enveloping algebra of the simple Lie algebra sl(2, R) and its realization as first-order differential operators on the real line provide a char- acterization of quasi-exactly solvable one-dimensional systems [13]. A second type of systems that uses the structural properties of enveloping algebras is given by super-integrable systems, where both the Hamiltonian and the constants of the motion are interpreted in the enveloping algebra of some Lie algebra g. Merely in- tegrable n-dimensional systems can be interpreted as the image, via a realization Φ by first-order differential operators, of an Abelian subalgebra A of U (g), while super-integrable systems would correspond to non- Abelian extensions of A. The problem under what conditions a system both exhibits super-integrability and (quasi-)exact solvability has been analyzed in de- tail, and large classes of super-integrable systems that are exactly solvable have been found (see [3, 14, 15] and references therein). 16 https://doi.org/10.14311/AP.2022.62.0016 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 On some algebraic formulations . . . A first algebraic formulation, as developed in [10], is motivated by the use of quadratic algebras in the context of super-integrable (and exactly solvable) sys- tems with a given hidden algebra g [3]. To this extent, we consider a Hamiltonian H expressed in terms of a subalgebra m ⊂ g via a realization Φ by differential operators of the Lie algebra g: H = dim m∑ i,j=1 αij Φ(Xi)Φ(Xj ) + dim m∑ k=1 βkΦ(Xk) + γ0, (1) where αij , βk, γ0 are constants and {X1, . . . , Xdim m} is a basis of m. In this context, the Hamiltonian H is obtained as the image of a quadratic element Ha in the universal enveloping algebra U (m) ⊂ U (g). Similarly, the (independent) constants of the motion φ1, . . . , φs can also be rewritten as the image of elements in the enveloping algebra U (g). As differential operators they satisfy the commutators [H, φj ] = 0, 1 ≤ j ≤ s. (2) The commutators [φi, φj ] provide additional (depen- dent) higher-order constants of the motion. A spe- cially interesting case is given whenever the first inte- grals generate a quadratic algebra. Abstracting from the specific realization Φ, and fo- cusing merely on the underlying algebraic formulation, the formal polynomial Ha = dim m∑ i,j=1 αij XiXj + dim m∑ k=1 βkXk + γ0 in the enveloping algebra U (m) of m allows us to re- cover Hamiltonian H of the system once the generators are realized by the differential operators. In analogous form, we can find elements J1, . . . , Js in U (g) that cor- respond, via the realization Φ, to the first integrals φ1, . . . , φs of the system. While for the initial system the relations [H, φk] = 0, 1 ≤ k ≤ s, are ensured, there is no necessity that the polynomials Jk commute with Ha in U (g), although the relation [Ha, Jk] = 0 (mod Φ) (3) is satisfied. Similarly, for the polynomial relations [φi, φj ] = αkℓij φkφℓ + β k ij φk of the first integrals, the commutators in U (g) lead to the relation [Ji, Jj ] = αkℓij JkJℓ + β k ij Jk (mod Φ) (4) If equations (3) and (4) are satisfied for any realiza- tion Φ, then the problem is entirely characterized algebraically by the reduction chain m ⊂ g. It should be observed that this situation is rather exceptional, as the analysis of the exactly solvable systems described in [3] from the point of view of the first algebraic formulation indicates that, in general, the first inte- grals of the system do not correspond, at the level of the enveloping algebra of the hidden algebra, to polynomials that commute with the algebraic Hamil- tonian, showing that the commutativity properties are a consequence of the realization by differential operators. Using the correspondence existing between the rep- resentations of g and those of its enveloping algebra U (g) (see e.g. [11]) and identifying a Lie algebra g with the first-order (left-invariant) differential opera- tors on a Lie group G admitting g as its Lie algebra, it follows that the universal enveloping algebra can be seen as the set of (left-invariant) differential operators on G of arbitrary order. Therefore, if Φ : g → X(Rn) is some realization of the Lie algebra by first-order differential operators, it can be uniquely extended to a realization Φ̂ : U (g) → X(Rn). In this context, this first algebraic reformulation of the system is still strongly related to the representa- tion theory of Lie algebras. More precisely, supposed that Ha ∈ U (m) is an algebraic Hamiltonian defined in the enveloping of some subalgebra m ⊂ g and that the (independent) polynomials J1, . . . , Js gener- ate a quadratic algebra, that is, satisfy the conditions [Ji, Jj ] = αkℓij JkJℓ + β k ij Jk, (5) we consider the (two-sided) ideal I in U (g) generated by the polynomials Qi := [Ha, Ji] , 1 ≤ i ≤ s. The problem is now to analyze whether there ex- ists an equivalence class of (faithful) representations Φ : g → X(Rn) such that for the corresponding ex- tension Φ̂ : U (g) → X(Rn), the image of the ideal I is contained in the kernel ker Φ̂, ensuring that the realized polynomials Φ̂(Qi) correspond to first inte- grals of the Hamiltonian in the given realization. In some sense, this is a special case of an important and still unsolved problem, namely the embedding of a Lie algebra g into the enveloping algebra U (k) of another Lie algebra k, for which currently only the case of embeddings ι : g → U (g) for g semisimple has been completely solved [16], using techniques of deformation theory [17]. We illustrate the preceding procedure consider- ing the six-dimensional non-solvable Lie algebra r ⊂ sl(3, R) with basis {X1, . . . , X6} and nonvanishing commutators [X1, X2] = X1, [X1, X5] = X4, [X2, X5] = X5, [X2, X6] = −X6, [X3, X4] = −X4, [X3, X5] = −X5, [X3, X6] = X6 [X4, X6] = X1, [X5, X6] = X2 − X3. Superintegrable systems based on this hidden algebra r and the vector field realization X1 = ∂t, X2 = t∂t − N 3 , X3 = su∂u − N 3 , X4 = ∂u, X5 = t∂u, X6 = u∂t, (6) 17 Rutwig Campoamor-Stursberg Acta Polytechnica have been extensively studied in [4], where in addition their exact solvability was analyzed. We consider a special case of the generic Hamiltonians studied there. Taking the values s = k = ω = 1, a = b = − 12 and N = 0, we obtain the Hamiltonian h1 and two quadratic integrals h1 = −4t∂2t − 8u∂ 2 tu − 4u∂ 2 u + 4t∂t + 4u∂u, φ1 = 4u(u − t)∂2u, φ2 = 4(t − u) ( ∂2t − ∂t ) . (7) Now h1, φ1, φ2 are the image by the realization of the following polynomials in the enveloping algebra of r: H1 = 4X2(1 − X1) + 8(1 − X3)X1 + 4(1 − X4)X3, P1 = −4X3X5 + 4X 23 − 4X3, Q1 = 4(X2X1 − X6X1 + X6 − X2). At the purely algebraic level we have [H1, P1] ̸= 0, [H1, Q1] ̸= 0, showing that the polynomials P1 and Q1 do not belong to the commutant of H1 in U (r). Therefore, the origin of the quadratic integrals of system (7) is not algebraic, but a consequence of the specific realization (6). If we maintain the algebraic Hamiltonian as given above and search for quadratic polynomials in U (r) commuting with it, we find that only two such opera- tors exist (see [10] for the general case), given by A1 = X4 − X3 − X6 + X1(1 + X3 + X6) + (X3 + X6)X4, B1 = −4X1 − X2 + X6 + X1X2 + X1X3 − X1X6 − X6X4. These polynomials are not independent, as they sat- isfy the relation A1 +B1 + 14 H1 = 0. Now, if we extend the analysis to cubic polynomials in U (r), we find the following operator C1 that commutes with H1: C1 = 3X1 − 2X3 − X5 − 4X6 + 2X1X3 + 4X1X6 + X 23 + X2X4 + X 23 − X3X5 + X3X6 + X6X4 − X6X5 − X1X 23 − X1X3X6 + X2X3X4 + X2X6X4. The operators A1 and C1 generate a finite- dimensional polynomial algebra in U (r), with explicit nonvanishing commutators [A1, C1] = D1, [A1, D1] = D1, [B1, C1] = −D1, [C1, D1] = 1 2 {B1, D1} − 1 2 {A1, D1} − 12A1 − 12A1 + 4B1 + 4C1 − 2{A1, B1}, where {◦, ◦} is the anticommutator. Now, as the operators H1, A1, C1 commute at the algebraic level, for any realization of r by vector fields they give rise to a Hamiltonian system possessing a quadratic and a cubic integral, respectively.1 For the particular realization (6), it follows that the resulting system is actually equivalent to the initial one (7), as the image of the ideal J generated by A1, B1, C1, D1 is properly contained in the ideal spanned by φ1 and φ2, thus being functionally dependent on these integrals. 1Provided that the transformed operators are independent. 3. Commutants in enveloping algebras and coadjoint representations A second algebraic approach, of a more general na- ture, can be proposed considering chain reductions g′ ⊂ g of (reductive) Lie algebras, and analyzing the structure of the commutant of g′ in the enveloping algebra U (g), in order to identify polynomial (in par- ticular, quadratic) subalgebras [9]. In the generic analysis of commutants, elements of the theory the coadjoint representation of Lie algebras can be used, in order to simplify some of the computations in en- veloping algebras. If g is a Lie algebra with generators {X1, . . . , Xn} and commutators [Xi, Xj ] = C kij Xk, the Xi’s are realized in the space C ∞ (g∗) by means of the first-order differential operators: X̂i = C kij xk ∂ ∂xj , (8) where {x1, . . . , xn} are the coordinates of a covector in a dual basis of R {X1, . . . , Xn}. The invariants of g (in particular, the Casimir operators) correspond to the solutions of the following system of partial differential equations: X̂iF = 0, 1 ≤ i ≤ n. (9) For an embedding of Lie algebras f : g′ → g, a basis {X1, . . . , Xr } of the subalgebra can be extended to a basis {X1, . . . , Xn} of g. Therefore, we can con- sider the subsystem formed by the first r equations of (9), corresponding to the generators of the subalgebra g′. The solutions of this subsystem, that in particu- lar encompass the invariants of g′, are usually called subgroup scalars [18]. By means of the standard symmetrization map Λ ( xi1 . . . xip ) = 1 p! ∑ σ∈Sp Xσ(i1) . . . Xσ(ip) (10) polynomial solutions of the subsystem correspond to elements in the enveloping algebra U (g) of g that commute with the subalgebra g′. If we now define an algebraic Hamiltonian H = H (X1, . . . , Xr ) ∈ U (g′), (11) in terms of the subalgebra generators, the commutant CU (g)(H) = {U ∈ U (g) | [H, U ] = 0} certainly includes the solutions of (9) common to the g′-generators, i.e. CU (g)(H) ⊃ { Λ(φ) | X̂1(φ) = · · · = X̂r (φ) = 0 } , where φ(x1, . . . , xn) ∈ C ∞ (g∗). Depending on the structure of g and the subalgebra g′, as well as on the choice of H, two possible cases arise for a polynomial P ∈ CU (g)(H): 18 vol. 62 no. 1/2022 On some algebraic formulations . . . (1.) P commutes with all X1, . . . , Xr . (2.) There is an index k0 with [P, Xk0 ] ̸= 0. Polynomials P in the first case actually commute with the Hamiltonian H, and thus belong to the two-sided ideal ⟨I⟩ generated by the set I = {J1, . . . , Js} of elements corresponding to the symmetrization of in- dependent polynomials satisfying the subsystem of (9) corresponding to g′. For these elements, it fol- lows at once that [Jk, Jℓ] belongs to I. In the general case, the Hamiltonian H does not commute with all Xj -generators, and in order to find the commutant CU (g)(H), we can restrict the analysis to the determi- nation of a basis of the factor module CU (g)(H)/⟨I⟩. Although the problem is computationally cumbersome, certain algorithms in terms of Gröbner bases have been developed that allow its precise determination [19]. A (restricted) systematic procedure that circum- vents the above-mentioned obstruction and allows us to analyze polynomial algebras with respect to a reduc- tion chain g′ ⊂ g can be proposed starting from the polynomials in U (g) that commute with all the gen- erators intervening in the expression of the algebraic Hamiltonian H ∈ U (g′). More precisely, if the Hamil- tonian H is given as a polynomial P (Xi1 , . . . , Xis ) in terms of the generators of the subalgebra g′ with basis {X1, . . . , Xr }, we consider the subsystem of (9) given by X̂ij F (x1, . . . , xn) = 0, 1 ≤ j ≤ s. We then extract a maximal set of independent poly- nomial solutions {Q1, . . . , Qp} of (9), which in the re- ductive case forms an integrity basis for the solutions. Symmetrizing these functions we obtain elements Mj in the commutant CU (g)(H). Starting from the set of polynomials S = {H, M1, . . . , Mp}, we inspect their commutators and determine whether, either adjoining new (dependent) elements to S or discarding some elements of S, a finite-dimensional quadratic algebra A can be found. Although there is some ambiguity in the construction, as there is no quadratic algebra “canonically" associated to the reduction chain g′ ⊂ g, it provides an alternative method that does not re- quire a specific realization by vector fields, as the integrability condition is guaranteed by the commu- tant. This ansatz has been successfully applied in [9] to the enveloping algebra of the Schrödinger algebras Ŝ(n) for arbitrary values of n ≥ 1 and various choices of algebraic Hamiltonian, showing that the construc- tion is formally of use for the analysis of hidden alge- bras that are not reductive. 4. Virtual copies in enveloping algebras In the solution of the embedding problem into envelop- ing algebras for semisimple algebras, the vanishing of the first cohomology group with values in U (g) plays an important role, as it allows to provide a gen- eral solution for the perturbation problem [16]. For nonsemisimple Lie algebras, the application of the pro- cedure is quite complicated for both computational reasons and the currently incomplete understanding of the precise structure of the corresponding envelop- ing algebras. However, for certain types of semidirect sums of simple and solvable Lie algebras, some analo- gous statements may be proposed, providing copies of semisimple Lie algebras in the enveloping algebra of a semidirect sum, up to a polynomial factor. Supposed that s is the Levi subalgebra of a semidi- rect sum g = s−→⊕ Γr, we seek for elements of degree d ≥ 2 in the generators in U (g) that transform accord- ing to the structure tensor of s, up to a (polynomial) factor. The procedure can be summarized as follows: Consider a basis {X1, . . . , Xn} of s with commmuta- tors [Xi, Xj ] = C kij Xk. (12) and extend it to a basis {X1, . . . , Xn, Y1, . . . , Ym} of of the semidirect sum g. We now define operators X ′i = Xi R (Y1, . . . , Ym) + Pi (Y1, . . . , Ym) (13) in U (g), where Pi and R are still undetermined poly- nomials. In order to simplify computations, they can be considered as homogeneous polynomials of degrees k and k − 1 respectively, so that X ′i is homogeneous of degree k. We require that these operators commute with the generators Yk of the radical r, so that the identity [X ′i , Yj ] = 0, 1 ≤ i ≤ n, 1 ≤ j ≤ m is satisfied for all indices. Expanding the latter leads to the expression [X ′i , Yj ] = [XiR, Yj ] + [Pi, Yj ] =Xi [R, Yj ] + [Xi, Yj ] R + [Pi, Yj ] . Taking into account the homogeneity degree of the terms with respect to the generators of s and the rep- resentation space, it follows that Xi [R, Yj ] can be seen as a polynomial of degree (k − 1) in the vari- ables {Y1, . . . , Ym}. On the other hand the terms of [Xi, Yj ] R + [Pi, Yj ] have degree k, allowing us to further separate the commutator as [R, Yj ] = 0, [Xi, Yj ] R + [Pi, Yj ] = 0. (14) From the first equation we conclude that the factor R commutes with all generators Yi, thus defines an invariant of the solvable Lie algebra r. We further require that the operators X ′i transform by the action of s as the generators of the latter algebra, i.e. [X ′i , Xj ] = [Xi, Xj ] ′ := C kij (XkR + Pk) . (15) As this relation must hold for all the generators of the semidirect sum g, further structural constraints on the polynomials R and Pi are obtained. Expanding the left-hand term of condition (15) yields [X ′i , Xj ] = [Xi, Xj ] R − Xi [Xj , R] + [Pi, Xj ] . 19 Rutwig Campoamor-Stursberg Acta Polytechnica As the Yj are the generators of the representa- tion space Γ, it follows that the term [Xi, Xj ] R − Xi [Xj , R] is linear in the generators of s and of de- gree (k − 1) in the Yj ’s, while [Pi, Xj ] does not involve generators of s. Comparing now with the right-hand side of (15), the condition again separates into two parts: [Xi, Xj ] R − Xi [Xj , R] = C kij XkR, [Pi, Xj ] = C kij Pk. (16) Simplifying the first equations shows that Xi [Xj , R] = 0, hence implying that R also commutes with the gen- erators of the Lie algebra. As R corresponds simul- taneously to an invariant polynomial of the radical, it must correspond to an invariant of g that depends only on the generators of its maximal solvable ideal.2 The second equation shows that the polynomials Pi transform according to the adjoint representation of the semisimple Lie algebra s. Supposed that all the conditions are satisfied, we obtain the commutators of the operators X ′i in the enveloping algebra U (g) as[ X ′i , X ′ j ] = [XiR + Pi, Xj R + Pj ] = [XiR + Pi, Xj R] + [XiR + Pi, Pj ] = C kij XkR 2 + C kij PkR + [X ′ i , Pj ] . (17) As the X ′i commute with the Yj , it follows from equation (17) that [X ′i , Pj ] = 0 and therefore that[ X ′i , X ′ j ] = [Xi, Xj ] ′ R, showing that the operators reproduce the commutators of s, up to the invariant factor R. It should be emphasized that R is not nec- essarily a central element, but an invariant of g that solely depends on the generators of the characteristic representation Γ. It follows in particular from this construction that the operators {R, X ′1, . . . , X ′n} generate a finite dimen- sional quadratic algebra A in the enveloping algebra U (g), with commutators [R, X ′i ] = 0, [ X ′i , X ′ j ] = C kij X ′ kR, 1 ≤ i, j, k ≤ n. Under some specific conditions, these so-called vir- tual copies of semisimple Lie algebras in enveloping algebras can be used to construct (formal) Hamiltoni- ans with first integrals given by some of the operators X ′i . Let us outline one possibility, based on the branch- ing rules of representations of semisimple Lie algebras. To this extent, we fix a semisimple subalgebra s′ of the Levi factor s of the semidirect sum g. Further suppose that the adjoint representation ad(s) decomposes, as a representation of s′, as follows ad(s) ↓ ad(s′) + Γ1 + · · · + Γs, (18) 2This fact actually provides information concerning the di- mension of the characteristic representation Γ in the semidirect sum. where Γ = Γ1 + · · · + Γs is the so-called character- istic representation [20]. Suppose that the trivial representation Γ0 of s′ has multiplicity k > 0 in the decomposition (18). This means specifically that we can find k generators { X̃1, . . . , X̃k } of s that commute with the subalgebra s′. Now, by condition (15), for the corresponding operators X̃s (1 ≤ s ≤ k) we have that [ X̃ ′i , Z ] = [ X̃i, Z ]′ = 0, Z ∈ s′, (19) from which it follows that for any algebraic Hamilto- nian H ∈ U (s′) the integrability condition[ X̃ ′i , H ] = [R, H] = 0, 1 ≤ i ≤ k (20) is satisfied. On the other hand, by condition (17), it is straightforward to verify that[ H, [ X̃ ′i , X̃ ′ j ]] = 0. (21) This last identity implies that the terms appearing in the commutator [ X̃ ′i , X̃ ′ j ] also transform according to the trivial representation of the subalgebra s′. We conclude that the set { R, X̃ ′1, . . . , X̃ ′ k } generates a finite-dimensional quadratic algebra in the envelop- ing algebra U (g) that are (formal) first integrals for the Hamiltonian H. Whether or not these integrals are sufficient for guaranteeing (super-)integrability, essentially depends on the subalgebra s′ and the as- sociated branching rule. In any case, the preceding construction determines the maximal number of op- erators X ′i that commute with the Hamiltonian H, independently of any realization of the hidden algebra g by first-order differential operators. For the case where the characteristic representation Γ does not con- tain the trivial representation of the subalgebra s′, i.e., when no generators of s simultaneously commute with the elements of s′, the integrability condition for the operators would not be a consequence of the structure of the enveloping algebra, but the specific consequence of a realization of g, relating this approach with the first algebraic formulation. We finally observe that the construction presented here, that depends essentially on the homogeneity of the operators X ′i , is specially suitable for semidirect sums admitting a nonvanishing centre and the class of one-dimensional non-central extensions of double inhomogeneous Lie algebras [21, 22], while the argu- ment is not valid whenever the Levi factor s and the radical do not have nonconstant invariants in common. Due to this obstruction, it is formally conceivable to propose a generalized construction by skipping the homogeneity assumption. It should however be taken into account that using operators of different degrees in (13) may lead to incompatibilities in the commuta- tors, as equations (14)-(16) cease to hold, and more general constraints depending on the particular de- grees of each Pi would be required. If and under what specific assumptions a solution can be found for a generalized inhomogeneous set of generators (13), is still an unanswered question that is currently being studied in detail. 20 vol. 62 no. 1/2022 On some algebraic formulations . . . 5. Conclusions Two possible approaches to the problem of determin- ing quadratic algebras as subalgebras of the enveloping algebra of a Lie algebra have been commented. The first approach corresponds to an algebraic abstrac- tion of already known systems, which are analyzed purely from the perspective of the Hamiltonian and the integrals as the image by a realization of dif- ferential operators of elements in some enveloping algebra, trying to determine to which extent such integrals are realization-dependent [10]. In a the sec- ond algebraic formulation, commutants of subalgebras g′ ⊂ g in the enveloping algebra of g are considered, from which quadratic algebras formed by polynomi- als that commute with a given algebraic Hamiltonian defined in U (g′) are deduced. In order to simplify the computations in the enveloping algebra, distin- guished elements in the commutant can be deduced from the coadjoint representation. For the subalge- bras found with this method, a realization by vector fields of an appropriate number of variables automat- ically provides a (super-)integrable system for the given Hamiltonian [9]. The method of virtual copies, initially introduced in the context of invariant theory, provides an additional approach that combines ele- ments of the two algebraic formulations, and refers to a number of still open problems, such as the general solution of the embedding problem of Lie algebras into enveloping algebras [16], as well the classification problem of realizations of Lie algebras in terms of differential operators [23]. Whether these approaches are compatible or can be combined with other proce- dures like the quadratic deformations of Lie algebras or the formalism of Racah algebras (see e.g. [8, 24, 25] and references therein) is a problem worthy to be in- spected. We hope to report on some progress in these directions in a near future. Acknowledgements The author is indebted to Alexander V. Turbiner and Ian Marquette for valuable critical comments, to Artur Sergyeyev for providing reference [15], as well as to the referees for several remarks that have greatly improved the presentation. During the preparation of this work, the author was financially supported by the research grant PID2019-106802GB- I00/AEI/10.13039/501100011033 (AEI/ FEDER, UE). References [1] A. M. Perelomov. Integrable Systems of Classical Mechanics and Lie Algebras. Birkhäuser Verlag, Basel, 1990. [2] L. Freidel, J. M. Maillet. Quadratic algebras and integrable systems. Physics Letters B 262(2-3):278–284, 1991. https://doi.org/10.1016/0370-2693(91)91566-E. [3] P. Tempesta, A. V. Turbiner, P. Winternitz. Exact solvability of superintegrable systems. Journal of Mathematical Physics 42(9):4248, 2001. https://doi.org/10.1063/1.1386927. [4] F. Tremblay, A. Turbiner, P. Winternitz. An infinite family of solvable and integrable quantum systems on a plane. Journal of Physics A: Mathematical and Theoretical 42(24):242001, 2009. https://doi.org/10.1088/1751-8113/42/24/242001. [5] P. Letourneau, L. Vinet. Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonian. Annals of Physics 243(1):144–168, 1995. https://doi.org/10.1006/aphy.1995.1094. [6] W. Miller, S. Post, P. Winternitz. Classical and quantum superintegrability with applications. Journal of Physics A: Mathematical and Theoretical 46(42):423001, 2013. https://doi.org/10.1088/1751-8113/46/42/423001. [7] C. Daskaloyannis, Y. Tanoudis. Quadratic algebras for three-dimensional superintegrable systems. Physics of Atomic Nuclei 73:214–221, 2010. https://doi.org/10.1134/S106377881002002X. [8] D. Latini, I. Marquette, Y.-Z. Zhang. Embedding of the Racah algebra R(n) and superintegrability. Annals of Physics 426:168397, 2021. https://doi.org/10.1016/j.aop.2021.168397. [9] R. Campoamor-Stursberg, I. Marquette. Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras. Annals of Physics 437:168694, 2022. https://doi.org/10.1016/j.aop.2021.168694. [10] R. Campoamor-Stursberg, I. Marquette. Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems. Annals of Physics 424:168378, 2021. https://doi.org/10.1016/j.aop.2020.168378. [11] J. Dixmier. Algèbres enveloppantes. Hermann, Paris, 1974. [12] M. A. Shifman, A. V. Turbiner. Quantal problems with partial algebraization of the spectrum. Communications in Mathematical Physics 126:347–365, 1989. https://doi.org/10.1007/BF02125129. [13] A. V. Turbiner, A. G. Ushveridze. Spectral singularities and quasi-exactly solvable quantal problem. Physics Letters A 126(3):181–183, 1987. https://doi.org/10.1016/0375-9601(87)90456-7. [14] M. A. Rodríguez, P. Winternitz. Quantum superintegrability and exact solvability in n dimensions. Journal of Mathematical Physics 43:1309–1322, 2002. https://doi.org/10.1063/1.1435077. [15] A. Sergyeyev. Exact solvability of superintegrable Benenti systems. Journal of Mathematical Physics 48(5):052114, 2007. https://doi.org/10.1063/1.2738829. [16] V. Ovsienko, A. V. Turbiner. Plongements d’une algèbre de Lie dans son algèbre enveloppante. Comptes Rendus de l’Académie des Sciences Paris 314:13–16, 1992. [17] A. Verona. Introducere in coomologia algebrelor Lie. Editura Academiei Rep. Soc. România, Bucuresti, 1974. [18] R. T. Sharp, C. S. Lam. Internal-labeling problem. Journal of Mathematical Physics 10(11):2033–2038, 1969. https://doi.org/10.1063/1.1664799. 21 https://doi.org/10.1016/0370-2693(91)91566-E https://doi.org/10.1063/1.1386927 https://doi.org/10.1088/1751-8113/42/24/242001 https://doi.org/10.1006/aphy.1995.1094 https://doi.org/10.1088/1751-8113/46/42/423001 https://doi.org/10.1134/S106377881002002X https://doi.org/10.1016/j.aop.2021.168397 https://doi.org/10.1016/j.aop.2021.168694 https://doi.org/10.1016/j.aop.2020.168378 https://doi.org/10.1007/BF02125129 https://doi.org/10.1016/0375-9601(87)90456-7 https://doi.org/10.1063/1.1435077 https://doi.org/10.1063/1.2738829 https://doi.org/10.1063/1.1664799 Rutwig Campoamor-Stursberg Acta Polytechnica [19] J. Apel, W. Lassner. An extension of Buchberger’s algorithm and calculations in enveloping fields of Lie algebras. Journal of Symbolic Computation 6(2-3):361–370, 1988. https://doi.org/10.1016/S0747-7171(88)80053-1. [20] W. G. McKay, J. Patera. Tables of Dimensions, Indices and Branching Rules for Representations of Simple Lie Algebras. Marcel Dekker, New York, 1981. [21] R. Campoamor-Stursberg. Intrinsic formulae for the Casimir operators of semidirect products of the exceptional Lie algebra g2 and a Heisenberg Lie algebra. Journal of Physics A: Mathematical and General 37(40):9451–9466, 2004. https://doi.org/10.1088/0305-4470/37/40/009. [22] R. Campoamor-Stursberg, I. Marquette. Generalized conformal pseudo-Galilean algebras and their Casimir operators. Journal of Physics A: Mathematical and Theoretical 52(47):475202, 2019. https://doi.org/10.1088/1751-8121/ab4c81. [23] A. González López, N. Kamran, P. J. Olver. Lie algebras of vector fields in the real plane. Proceedings of the London Mathematical Society 64(2):339–368, 1992. https://doi.org/10.1112/plms/s3-64.2.339. [24] L. A. Yates, P. D. Jarvis. Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra. Journal of Physics A: Mathematical and Theoretical 51(14):145203, 2018. https://doi.org/10.1088/1751-8121/aab215. [25] P. Gaboriaud, L. Vinet, S. Vinet, A. Zhedanov. The generalized Racah algebra as a commutant. Journal of Physics: Conference Series 1194:012034, 2019. https://doi.org/10.1088/1742-6596/1194/1/012034. 22 https://doi.org/10.1016/S0747-7171(88)80053-1 https://doi.org/10.1088/0305-4470/37/40/009 https://doi.org/10.1088/1751-8121/ab4c81 https://doi.org/10.1112/plms/s3-64.2.339 https://doi.org/10.1088/1751-8121/aab215 https://doi.org/10.1088/1742-6596/1194/1/012034 Acta Polytechnica 62(1):16–22, 2022 1 Introduction 2 First algebraic reformulation 3 Commutants in enveloping algebras and coadjoint representations 4 Virtual copies in enveloping algebras 5 Conclusions Acknowledgements References