Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra
Angel Ballesteros, Rutwig Campoamor-Stursberg, Eduardo Fernandez-Saiz, Francisco J. Herranz, Javier de Lucas
aa r X i v : . [ m a t h - ph ] J a n Poisson–Hopf deformations of Lie–Hamilton systems revisited:deformed superposition rules and applications to the oscillatoralgebra
Angel Ballesteros , Rutwig Campoamor-Stursberg , , Eduardo Fern´andez-Saiz ,Francisco J. Herranz and Javier de Lucas Departamento de F´ısica, Universidad de Burgos, E-09001 Burgos, Spain Instituto de Matem´atica Interdisciplinar UCM, Plaza de Ciencias 3, E-28040 Madrid, Spain Departamento de Geometr´ıa y Topolog´ıa, Universidad Complutense de Madrid, Plaza deCiencias 3, E-28040 Madrid, Spain Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5,02-093, Warszawa, PolandE-mail: [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract
The formalism for Poisson–Hopf (PH) deformations of Lie–Hamilton systems, recently pro-posed in [1], is refined in one of its crucial points concerning applications, namely the obten-tion of effective and computationally feasible PH deformed superposition rules for prolongedPH deformations of Lie–Hamilton systems. The two new notions here proposed are a gen-eralization of the standard superposition rules and the concept of diagonal prolongationsfor Lie systems, which are consistently recovered under the non-deformed limit. Using atechnique from superintegrability theory, we obtain a maximal number of functionally inde-pendent constants of the motion for a generic prolonged PH deformation of a Lie–Hamiltonsystem, from which a simplified deformed superposition rule can be derived. As an applica-tion, explicit deformed superposition rules for prolonged PH deformations of Lie–Hamiltonsystems based on the oscillator Lie algebra h are computed. Moreover, by making use thatthe main structural properties of the book subalgebra b of h are preserved under the PHdeformation, we consider prolonged PH deformations based on b as restrictions of thosefor h -Lie–Hamilton systems, thus allowing the study of prolonged PH deformations of thecomplex Bernoulli equations, for which both the constants of the motion and the deformedsuperposition rules are explicitly presented.MSC: 16T05, 17B66, 34A26PACS: 02.20.Uw, 02.20.Sv, 02.60.LjKEYWORDS: Lie system, constant of the motion, diagonal prolongation, superposition rule, Poisson–Hopf algebra, oscillator algebra, Bernoulli differential equations. ontents The original approach to Poisson–Hopf (PH) deformations of Lie–Hamilton systems developed in [1]combined the classical theory of Lie systems with methods from quantum algebras and integrable systems,leading to a novel type of systems of ordinary differential equations with generalized symmetry that,despite being deprived of some of the appealing properties of Lie systems, still allowed for a systematicanalysis of their constants of the motion. In essence, the method is based on the idea of deforming aLie–Hamilton system (LH system in short) with a given Vessiot–Guldberg Lie algebra onto a Hamiltoniansystem depending on a quantum deformation parameter z (or q = e z ), the dynamics of which is describedby a t -dependent vector field taking values in a linear space of vector fields spanning a smooth distributionin the sense of Stefan–Sussmann, with the particularity that the initial LH system and its Vessiot–Guldberg Lie algebra is retrieved when z →
0. This allowed us, among other applications, to provide aunified geometrical description of the PH deformations of the three inequivalent LH systems on the planebased on the Lie algebra sl (2 , R ) (see [2, 3]). Nonetheless, the PH deformation method proposed in [1]was, to a certain extent, still incomplete, as it did not studied the existence and methods of derivationof an extension of the superposition rule concept for PH deformed LH systems, a hereafter called PHdeformed superposition rule or, simply, a deformed superposition rule .In this paper we present the way to implement such a generic procedure, by making use of a powerfultool developed in the context of superintegrable systems possessing a Hopf algebra symmetry [4]. Theconstruction is proved to be complete and valid for the hereafter called prolonged PH deformations ofLH systems, hence providing a generic prescription for the obtention of their deformed superpositionprinciples. In a nutshell, deformed superposition rules are z -parametric families of mappings allowingfor the description of some coordinates of particular solutions of a prolonged PH deformation of a LH ystem in terms of the others. When z →
0, deformed superposition rules and prolonged PH deformationsbecome standard superposition rules and diagonal prolongations [5] for LH systems, respectively. It isremarkable that given a LH system on an n -dimensional manifold M , its prolonged PH deformationsbecome Hamiltonian systems on M m +1 that neither need to be Lie systems nor must consist of severalcopies of the original Lie system for z = 0. This is a remarkable difference with respect to LH systems,whose diagonal prolongations to M m +1 give rise to a LH system consisting of several copies of the initialone and whose constants of the motion, for m large enough, allow for a superposition rule for the initialLH system.The paper is structured as follows. In Section 2 the fundamental properties of LH systems andtheir deformations based on the notion of PH algebras, as developed in [1], are shortly reviewed. Specialattention is devoted to the construction of constants of the motion of both LH systems and their prolongedPH deformations, which are hereafter called prolonged deformations to simplify our terminology. While inSubsection 2.3.1 we recall the method deduced from the coalgebra formalism [6], already used in [1] and [3],in Subsection 2.3.2 we present new material that completes and enlarges the previous work. Specifically,it is shown that for generic prolonged deformations of LH systems, in contrast to what happens for theundeformed systems, constructing the constants of the motion basing solely on the coalgebra formalism [6]may not supply us with a sufficient number of functionally independent constants to establish a deformedsuperposition rule in explicit closed form, as in this case we necessarily have to consider constants of themotion of higher ‘order’ (i.e. the dimension of the underlying tensor product space for the coproduct),which could imply that the resulting expressions are not expressible analytically in a discernible way.This rather subtle phenomenon is due to a symmetry breaking phenomena in the coproduct induced bythe deformation, which also causes that the prolonged deformation of the initial LH system is not, ingeneral, a diagonal prolongation [5]. In turn, this implies that a permutation in the tensor product spacedoes not necessarily transform a constant of the motion for the prolonged deformation into another one.Moreover, it can even happen that some of the constants of the motion of the diagonal prolongation of anundeformed LH system do not have at all a counterpart for a prolonged deformation of the LH system.This can be seen as a gap in the formalism as given in [1], as it somewhat disturbs the correspondencebetween the diagonal prolongations of a LH system and its prolonged deformations through the limitwith respect to the deformation parameter. Based on this observation, we reconsider the problem ofdetermining a sufficient number of constants of the motion for both the diagonal prolongation of a LHsystem and its prolonged deformations, in such a manner that the correspondence through the limit z → m constants of the motion in involution (for each set) that are valid for arbitrary prolonged deformationsto M m +1 , providing a maximal number of (2 m −
1) functionally independent constants of the motion forthe given deformation. From them we can infer a suitable deformed superposition rule selecting thoseconstants of the motion having minimal ‘order’, hence minimizing the analytical computation difficulties.In the classical limit z →
0, the ‘additional’ right-constants of the motion are shown to be obtainable usingthe permutation method from the left-set, which was the one formerly considered in [6]. This new Ansatzrefines in a natural way the results in [1], pointing out the relevance of working simultaneously with left-and right-coproducts in the deformation, as it constitutes a procedure that generically guarantees thatthe resulting functions are constants of the motion of the prolonged deformation of the initial LH system.As an application, Section 3 analyses the LH systems based on the oscillator algebra h . The (unde-formed) diagonal prolongations to h -LH systems, which were already studied in [7], are now consideredby using a different set of constants of the motion, leading to a different albeit equivalent superpositionrule. The purpose of this reformulation is to consistently deduce the superposition rule as the limit z → h [8] is considered and theexplicit derivation of formulae for the corresponding prolonged deformation of h -LH systems is devel-oped. The choice of the nonstandard deformation of h , on the other hand, has an interesting structuralconsequence: the two-dimensional book Lie algebra b is preserved as a Hopf subalgebra, thus allowingus to restrict the prolonged deformation of h -LH systems to this subalgebra and therefore to obtain eformed b -LH systems. As a representative of such systems based on b , we consider the complexBernoulli equations, for which the constants of the motion and the superposition rules are given for bothdeformed and non-deformed versions. In this context, the interpretation of the deformed system as asmall perturbation of the initial one, governed by the deformation parameter z , allows us to establisha connection between non-trivially coupled systems, and one of its equations corresponds to a Riccatiequation. Finally, in Section 5 some conclusions are drawn, several possible future developments of themethod are proposed, and its applications to the analysis of nonlinear systems of differential equationsare commented. This section briefly recalls the fundamental properties of Lie and LH systems. It also reviews the generaldeformation method of LH systems introduced in [1] (see also [3]) based on PH algebras. In contrast to[1, 3], the deformation of LH systems is developed in full generality to point out that the mechanism holdsfor arbitrary smooth manifolds. In particular, we focus on the problem of obtaining a sufficient number ofconstants of the motion for prolonged deformations of LH systems, from which formally explicit deformedsuperposition rules can be deduced, regardless of the initial LH system and the considered deformation. Itshould be emphasized that, albeit in [1] this possibility was outlined for deformed LH systems, no deformedsuperposition rules were explicitly given. In this work we propose an extension of the techniques in [1]that allows us to construct such deformed superposition rules for arbitrary prolonged deformations of LHsystems. Unless otherwise stated, we hereafter assume all structures to be smooth and globally defined.This will simplify the presentation of our results and it will allows us to focus on its main new features.
Let x = { x , . . . , x n } denote the coordinates in an n -dimensional manifold M and consider a non-autonomous system of first-order ordinary differential equations given byd x j d t = f j ( t, x ) , j = 1 , . . . , n, (2.1)for some arbitrary functions f j : R × M → R . Such a system can be described equivalently via a t -dependent vector field X : R × M → T M defined as: X ( t, x ) = n X j =1 f j ( t, x ) ∂∂x j . (2.2)A system of the type (2.1) is called a Lie system [5, 9–17] whenever it admits a fundamental system ofsolutions , i.e., whenever its general solution, x ( t ), can be expressed in terms of m particular solutions { x ( t ) , . . . , x m ( t ) } and n constants { k , . . . , k n } as x ( t ) = Ψ( x ( t ) , . . . , x m ( t ) , k , . . . , k n ) (2.3)for a certain function Ψ : M m × M → M . The expression (2.3) is usually referred to as a superpositionrule of the system (2.1). Within applications to physical phenomena, the wealth of group-theoreticaltechniques developed from the 60’s onwards revived the interest in analyzing systematically the existenceof superposition rules, leading to an extensive geometrical study of Lie systems and superposition rulesand their application to systems at both the classical and quantum levels (see, e.g., [5, 18–24] andreferences therein).The Lie–Scheffers Theorem (see [5, 9, 10, 13]) states that a t -dependent vector field X as in (2.2) de-termines a Lie system if and only if there exist some functions b ( t ) , . . . , b ℓ ( t ) and vector fields X , . . . , X ℓ on M spanning an ℓ -dimensional real Lie algebra V such that X ( t, x ) = ℓ X i =1 b i ( t ) X i ( x ) , ∀ x ∈ M. (2.4) t can then be proved that the system X admits a superposition rule so that the constraint ℓ ≤ nm issatisfied. In these conditions, V is called a Vessiot–Guldberg Lie algebra of X (see also [25–27] for morerecent applications of Vessiot–Guldberg Lie algebras).A Lie system is said to be a Lie–Hamilton system whenever it admits a Vessiot–Guldberg Lie algebra V of Hamiltonian vector fields with respect to a Poisson structure [2, 5–7, 16, 28, 29]. Let us assume thecase of a LH system on M that admits a Vessiot–Guldberg Lie algebra, V , of Hamiltonian vector fieldsrelative to a symplectic form ω . The compatibility condition between the generators X i of V and ω islocally determined by the invariance of ω under the Lie derivative with respect to any generator X i of V , i.e., L X i ω = 0 , i = 1 , . . . , ℓ. (2.5)Now a Hamiltonian function h i is related to the vector field X i through the contraction or inner productof ω with respect to X i : ι X i ω = d h i , i = 1 , . . . , ℓ. (2.6)Recall that every symplectic form allows us to define a Poisson bracket {· , ·} ω : ( f , f ) ∈ C ∞ ( M ) × C ∞ ( M ) → X f f ∈ C ∞ ( M ) , (2.7)where X f is the unique Hamiltonian vector such that ι X f ω = d f for an f ∈ C ∞ ( M ). It follows that( C ∞ ( M ) , {· , ·} ω ) is endowed with a Poisson algebra structure. The space Ham( ω ) of Hamiltonian vectorfields on M relative to ω , which is a Lie algebra with respect to the commutator of vector fields, is relatedto the former by means of the Lie algebra morphism [30]( C ∞ ( M ) , {· , ·} ω ) ϕ −→ (Ham( ω ) , [ · , · ]) (2.8)mapping a function f ∈ C ∞ ( M ) onto the Hamiltonian vector field − X f . The Hamiltonian functions h i ( i = 1 , . . . , ℓ ) coming from (2.6) span, eventually together with a constant function h on M , a finite-dimensional Lie algebra of functions ϕ − ( V ) that is called a Lie–Hamilton algebra (LH algebra), H ω , of X [2, 7]. The remarkable point is that the space C ∞ ( H ∗ ω ) of smooth functions on the dual H ∗ ω of the LH algebra H ω can be endowed with a Hopf algebra structure [31–33]. For our purposes, it suffices to consider the coalgebra structure of the Hopf algebra determined by the coproduct map, as the remaining structuralmaps, namely the counit and antipode, can be deduced from the axioms defining the Hopf algebra. Inparticular, for an associative algebra A , the coproduct ∆ : A → A ⊗ A must be an algebra homomorphismand satisfy the coassociativity condition(Id ⊗ ∆)∆( a ) = (∆ ⊗ Id)∆( a ) , ∀ a ∈ A. (2.9)If A is a commutative Poisson algebra, the coproduct ∆ satisfying (2.9) is required to be a Poisson algebramorphism, so that the Poisson bracket on A ⊗ A is given by { a ⊗ b, c ⊗ d } A ⊗ A = { a, c } ⊗ bd + ac ⊗ { b, d } , ∀ a, b, c, d ∈ A. (2.10)For the case of C ∞ ( H ∗ ω ), the coalgebra structure is determined by the coproduct ∆( f )( x , x ) := f ( x + x ), where x , x ∈ H ω and f ∈ C ∞ ( H ∗ ω ). The details concerning the complete Hopf algebra structurecan be found in [1]; here we just recall that C ∞ ( H ∗ ω ) turns out to be a PH algebra through the Poissonstructure defined by the Kirillov–Kostant–Souriau bracket related to a Lie algebra structure on H ω .Given these algebraic preliminaries, we summarize the notion of PH deformation introduced in [1](see also [3]) in four steps:1. Let X be a LH system of type (2.4) on an n -dimensional manifold M with symplectic form ω ,so that the LH algebra H ω is spanned by a set of functions { h , . . . , h ℓ } ⊂ C ∞ ( M ) satisfying he condition (2.6), with M being a suitable submanifold of M that ensures that each h i is welldefined. Let the Poisson bracket of the functions h i be given by: { h i , h j } ω = ℓ X l =1 C lij h l , i, j = 1 , . . . , ℓ, (2.11)for certain structure constants C lij .2. Consider a PH deformation of C ∞ ( H ∗ ω ), denoted by C ∞ ( H ∗ z,ω ), with deformation parameter z ∈ R (or q = e z ) as the space of smooth functions F z,ij ( h z, , . . . , h z,ℓ ) for a family of functions { h z, , . . . , h z,ℓ } on C ∞ ( M ) with Poisson bracket (with respect to ω ) given by { h z,i , h z,j } ω = F z,ij ( h z, , . . . , h z,ℓ ) , i, j = 1 , . . . , ℓ, (2.12)and satisfying the non-deformed limitslim z → h z,i = h i , lim z → F z,ij ( h z, , . . . , h z,ℓ ) = ℓ X l =1 C lij h l . (2.13)3. Obtain the deformed vector fields X z,i according to the relation (2.6), that is, ι X z,i ω = d h z,i , i = 1 , . . . , ℓ. (2.14)4. And, finally, define the PH deformation X z of the LH system X (2.4) as X z := ℓ X i =1 b i ( t ) X z,i . (2.15)Notice that, by construction, the following non-deformed limits are consistently recovered:lim z → X z,i = X i , lim z → X z = X . (2.16)The essential point to be taken into account is that the deformed vector fields { X z, , . . . , X z,ℓ } obtainedthrough the preceding prescription do not, in general, provide neither a finite-dimensional Lie algebranor a quantum algebra. Actually, they span an involutive Stefan–Sussmann distribution [30, 34, 35] since[ X z,i , X z,j ] = − ℓ X l =1 ∂F z,ij ∂h z,l X z,l , i, j = 1 , . . . , ℓ. (2.17)In other words, the functions { h z, , . . . , h z,ℓ } determine a PH deformation C ∞ ( H ∗ z,ω ) of C ∞ ( H ∗ ω ) withdeformed Poisson brackets (2.12), thus providing a (deformed) Hamiltonian function h z := ℓ X i =1 b i ( t ) h z,i . (2.18)However, the non-autonomous system of first-order ordinary differential equations X z (2.15) does nolonger correspond, in general, to a Lie system, but to a ‘perturbation’ of the initial system (2.1) withrespect to the deformation parameter z , as follows at once from the conditions (2.13) and (2.16) underthe limit z →
0. In this context, it is conceivable to interpret z as a small perturbation parameter.This means that, once the deformed system has been obtained through either X z or h z , a power seriesexpansion in z can be considered, analyzing the behaviour of the deformed Hamiltonian system up to thefirst, second or some higher order, enabling us a comparison with the initial undeformed system. .3 Constants of the motion The coalgebra formalism considered in [36, 37] in the context of integrable systems turned out to bea highly effective tool that allows to prove in a constructive way the complete integrability of systemspossessing coalgebra symmetry, including the explicit construction of the corresponding integrals of themotion. This coalgebra approach was later extended in order to characterize the property of (quasi-maximal) superintegrability [4, 38, 39]. These results covered both non-deformed integrable systems andtheir PH deformations. More recently, the coalgebra formalism was adapted to the framework of LHsystems [6], providing a method to determine t -independent constants of the motion in a more directway than that given by the classical methods [12, 14]. We observe that the constants of the motionof LH systems deduced by this technique are the cornerstone for the obtention of superposition rules.This procedure has been carried out systematically for LH systems on the plane R in [7] as well as ontwo-dimensional spaces of constant curvature (with different signatures of the metric tensor) in [29].At this point, it is of capital importance to realize that the results presented in [6] (and furtherconsidered in [7, 29]) focused on non-deformed LH systems, that is, for cases with a trivial or primitivecoalgebra structure. This approach turns out be unsatisfactory, as its straightforward extension to non-primitive coproducts, which is precisely the case for prolonged deformations of LH systems, provides lessconstants of the motion than in the primitive case. The aim of this section is to enlarge and completesuch previous work, proposing a general procedure for the explicit construction of the constants of themotion for prolonged deformations of LH systems. Let us first briefly summarize the coalgebra approach for constructing t -independent constants of themotion of non-deformed LH systems [6] (see also [1, 7]). Consider the LH algebra H ω of a LH system X (2.4), expressed as a Lie–Poisson algebra with generators { v , . . . , v ℓ } fulfilling the Poisson brackets(see (2.11)): { v i , v j } = ℓ X l =1 C lij v l , i, j = 1 , . . . , ℓ. (2.19)Let S ( H ω ) be the symmetric algebra of H ω (i.e., the associative unital algebra of polynomials in theelements of H ω ) understood as a Poisson algebra, thus with fundamental Poisson brackets (2.19). Underthese conditions, S ( H ω ) can always be endowed with a coalgebra structure with a non-deformed (trivialor primitive) coproduct map ∆ defined by∆ : S ( H ω ) → S ( H ω ) ⊗ S ( H ω ) , ∆( v i ) := v i ⊗ ⊗ v i , i = 1 , . . . , ℓ, (2.20)which is a Poisson algebra homomorphism of (2.19). Notice that the (trivial) counit and antipode canalso be defined giving rise to the non-deformed Hopf structure corresponding to any Lie algebra [31–33].The 2-coproduct ∆ ≡ ∆ (2) can be extended to a third-order coproduct through the coassociativitycondition (2.9): ∆ (3) := (∆ ⊗ Id) ◦ ∆ = (Id ⊗ ∆) ◦ ∆ , ∆ (3) : S ( H ω ) → S ( H ω ) ⊗ S ( H ω ) ⊗ S ( H ω ) ≡ S (3) ( H ω ) , (2.21)∆ (3) ( v i ) = v i ⊗ ⊗ ⊗ v i ⊗ ⊗ ⊗ v i , i = 1 , . . . , ℓ. A k th -order coproduct map can be defined recursively by the rule∆ ( k ) : S ( H ω ) → k times z }| { S ( H ω ) ⊗ . . . ⊗ S ( H ω ) ≡ S ( k ) ( H ω ) , ∆ ( k ) := (cid:0) ( k −
2) times z }| { Id ⊗ . . . ⊗ Id ⊗ ∆ (2) (cid:1) ◦ ∆ ( k − , (2.22) hich is also a Poisson algebra homomorphism for any k ≥ S ( H ω ) can be seen as a function on H ∗ ω so that the coproduct (2.20) in S ( H ω ) canbe extended to ∆ : C ∞ ( H ∗ ω ) → C ∞ ( H ∗ ω ) ⊗ C ∞ ( H ∗ ω ) ⊂ C ∞ ( H ∗ ω × H ∗ ω ) . (2.23)A similar extension holds for the k th -order coproduct defined in (2.22). Therefore, C ∞ ( H ∗ ω ) becomes anon-deformed Poisson coalgebra, and the corresponding extension of the counit and antipode maps turns C ∞ ( H ∗ ω ) into a PH algebra [1].Now consider the LH algebra H ω spanned by the Hamiltonian functions { h , . . . , h ℓ } satisfying thePoisson brackets (2.11). In agreement with equation (2.19), we define the Lie algebra morphism φ : H ω → C ∞ ( M ) , h i := φ ( v i ) , i = 1 , . . . , ℓ, (2.24)where M ⊂ M is chosen in order to ensure that the functions h i , and their PH deformations, to be definedshortly, are well defined. Basing on this result, we construct a family of Poisson algebra morphisms D : C ∞ ( H ∗ ω ) → C ∞ ( M ) , D ( k ) : k times z }| { C ∞ ( H ∗ ω ) ⊗ . . . ⊗ C ∞ ( H ∗ ω ) → k times z }| { C ∞ ( M ) ⊗ . . . ⊗ C ∞ ( M ) ⊂ C ∞ ( M k ) , (2.25)that are given by D ( v i ) = h i ( x ) := h (1) i , D ( k ) (cid:0) ∆ ( k ) ( v i ) (cid:1) = h i ( x ) + · · · + h i ( x k ) := h ( k ) i , i = 1 , . . . , ℓ, (2.26)where x j = { ( x ) j , . . . , ( x n ) j } denotes the coordinates in the j -copy submanifold M ⊂ M within M k .Let us finally assume that C ∞ ( H ∗ ω ) possesses a Casimir invariant C = C ( v , . . . , v ℓ ) , (2.27)that is, an element C that Poisson-commutes with all v i with respect to the Poisson bracket given in(2.19). As proved in [6], it follows that the functions constructed through the family of coproducts (2.22)and Poisson morphisms (2.25) defined by F := D ( C ) , F ( k ) (cid:16) h ( k )1 , . . . , h ( k ) ℓ (cid:17) := D ( k ) h ∆ ( k ) (cid:0) C ( v , . . . , v ℓ ) (cid:1)i , k = 2 , . . . , m + 1 , (2.28)are t -independent constants of the motion for the diagonal prolongation e X m +1 of the LH system X (2.4)to the product manifold M m +1 , i.e., the t -dependent vector field on M m +1 of the form e X m +1 ( t, x , . . . , x m +1 ) := m +1 X k =1 n X j =1 X j ( t, x k ) ∂∂x j = ℓ X i =1 b i ( t ) X h ( m +1) i . (2.29)Note that the functions (2.28) can also be considered as constants of the motion for the LH system X .The right-hand side of expression (2.29) shall be called the prolonged PH deformation of X to M m +1 , orsimply the prolonged deformation . As we shall see shortly, this notion can immediately be extended tonon-primitive coproducts which will invalidate, in general, the equality of the right-hand side of (2.29)with the standard diagonal prolongation of X .Each of the F ( k ) (2.28) can be considered as a function of C ∞ ( M m +1 ). lf all the F ( k ) are non-constant,then they form a set of m functionally independent functions in C ∞ ( M m +1 ) that are in involution. Inaddition, these functions F ( k ) can be used to generate other t -independent constants of the motion bymeans of the prescription [6] F ( k ) ij = S ij (cid:0) F ( k ) (cid:1) , ≤ i < j ≤ k, k = 2 , . . . , m + 1 , (2.30)where S ij denotes the permutation of the variables x i ↔ x j in M m +1 . This can be viewed as a conse-quence of the fact that the diagonal prolongation of X is invariant under such a permutation of variables.It can also be viewed as a consequence of (2.29) and [37, Proposition 1]. ecall that for obtaining a superposition rule depending on m particular solutions, as in (2.3), onesearches for a set, I , . . . , I n , of t -independent constants of the motion on M m +1 for e X m +1 so that [5] ∂ ( I , . . . , I n ) ∂ (( x ) m +1 , . . . , ( x n ) m +1 ) = 0 , (2.31)and the diagonal prolongations e X m , . . . , e X mℓ are linearly independent at a generic point [5]. This allowsus to express the coordinates x m +1 = { ( x ) m +1 , . . . , ( x n ) m +1 } in terms of the remaining coordinates in M m +1 and the constants k , . . . , k n defined by the conditions I = k , . . . , I n = k n . We stress that theset of constants of the motion (2.28) and (2.30) are frequently sufficient to deduce the superposition rulesfor the LH system X (2.4) in a direct way, as it has already been explicitly shown in [7, 29] for somespecific LH systems. Moreover, the existence of a large number of constants of the motion F ( k ) ij (obtainedthrough permutations) rather simplifies the computations, as it allows one to keep the number k low.Actually, in most of the explicit superposition rules worked out in [7, 29], it was sufficient to considerthe function F (2) and its permutations F (2) ij , a fact that helped to avoid long cumbersome computationsenabling to establish in closed form a superposition principle of reasonable simplicity. However, as weshall prove in the sequel, the functions (2.28) and (2.30) will not generically provide us, in the case ofprolonged deformations of a LH system, with a sufficient number of functionally independent constantsof the motion from which a deformed superposition rule could easily be inferred. As it has been already stated, the fact that C ∞ ( H ∗ z,ω ) is a PH deformation of C ∞ ( H ∗ ω ) enables us toapply the coalgebra formalism proposed in [6] to construct t -independent constants of motion for thedeformed LH system X z (2.15) with the deformed Hamiltonian h z as given in (2.18), for which someexamples were presented in [1]. This procedure must however be applied with some caution, as thereare some subtle points that, if not taken into account, may invalidate the conclusions. The key point isto observe that, whenever we are considering a deformed PH algebra, the deformed coproduct ∆ z is nolonger trivial (or primitive) as ∆ (2.20) for all the generators v i . Indeed, the deformation ‘breaks’ thesymmetry within the coproduct, that is, the positions in the tensor product space within the coproductare no longer ‘equivalent’, as happens e.g. for ∆ (3) ( v i ) in (2.21).By the construction in [36, 37], the deformed counterpart of the constants of the motion F ( k ) (2.28)still holds, but it is not ensured that the permutations in (2.30) give rise to t -independent constantsof the motion for the prolonged deformation e X m +1 z to M m +1 of X . Therefore, in the deformed caseone may need to consider a higher number k with respect to the non-deformed system to deduce thedeformed superposition rule, which in turns makes m + 1 to be larger. The drawback of considering anincreased m + 1 is that the complexity of the computations grows exponentially, resulting in an extremelyinvolved derivation of the deformed superposition rule. Fortunately, this difficulty can be circumventedby considering a second set of constants of the motion, additionally to the F ( k ) , which comes from thesuperintegrability property of integrable systems possessing Hopf algebra symmetry [4, 38, 39]. In thefollowing we present the explicit derivation of both sets of deformed constants of the motion.Let H z,ω be the deformed LH algebra of the deformed LH system X z (2.15) with Hamiltonian h z given by (2.18). We take a basis with generators { v , . . . , v ℓ } such that the Poisson brackets are given by(see (2.12)): { v i , v j } z = F z,ij ( v , . . . , v ℓ ) , i, j = 1 , . . . , ℓ. (2.32)Proceeding as in the non-deformed case, we consider the deformed coproduct for the generators v i :∆ z : C ∞ ( H ∗ z,ω ) → C ∞ ( H ∗ z,ω ) ⊗ C ∞ ( H ∗ z,ω ) , (2.33)along with the k th -order deformed coproduct map, ∆ ( k ) z , defined exactly as in (2.22), such that the limitslim z → ∆ z = ∆ , lim z → ∆ ( k ) z = ∆ ( k ) , (2.34) re satisfied.Now recall that the deformed Hamiltonian functions { h z, , . . . , h z,ℓ } fulfil the relations (2.14) and thePoisson brackets (2.12) with respect to the symplectic form ω . We define the map φ z : H z,ω → C ∞ ( M ) , h z,i := φ z ( v i ) , i = 1 , . . . , ℓ, (2.35)where again M ⊂ M is chosen to guarantee that the functions h z,i are properly defined. Next, as in(2.25), we introduce the Poisson algebra morphisms D z : C ∞ ( H ∗ z,ω ) → C ∞ ( M ) , D ( k ) z : k times z }| { C ∞ ( H ∗ z,ω ) ⊗ . . . ⊗ C ∞ ( H ∗ z,ω ) → k times z }| { C ∞ ( M ) ⊗ . . . ⊗ C ∞ ( M ) . (2.36)Now let C z = C z ( v , . . . , v ℓ ) , (2.37)be the Casimir function of C ∞ ( H ∗ z,ω ) with lim z → C z = C . And we define the functions (see (2.26)) D z ( v i ) = h z,i := h (1) z,i , D ( k ) z (cid:0) ∆ ( k ) z ( v i ) (cid:1) := h ( k ) z,i , i = 1 , . . . , ℓ, (2.38)whose explicit form does depend on the initial deformed coproduct ∆ z .The first set of constants of the motion for the prolonged deformation of the LH system X , which isanalogous to the right-hand side of (2.29), i.e. e X m +1 z = ℓ X i =1 b i ( t ) X h ( m +1) z,i , (2.39)is defined by F z := D z ( C z ) , F ( k ) z (cid:16) h ( k ) z, , . . . , h ( k ) z,ℓ (cid:17) := D ( k ) z h ∆ ( k ) z (cid:0) C z ( v , . . . , v ℓ ) (cid:1)i , k = 2 , . . . , m + 1 , (2.40)which is just the deformed counterpart of (2.28). It is worth noting that X h ( m +1) z,i in (2.39) fulfil similarcommutation relations to (2.17): h X h ( m +1) z,i , X h ( m +1) z,j i = − ℓ X l =1 ∂F z,ij ∂h ( m +1) z,l (cid:16) h ( m +1) z, , . . . , h ( m +1) z,ℓ (cid:17) X h ( m +1) z,l , i, j = 1 , . . . , ℓ. (2.41)If all F ( k ) z (2.40) are non-constant functions, they provide a set of m functionally independent functionsin involution [6, 36, 37]. Even if formally these invariants are sufficient to deduce a deformed superpositionrule, it is doubtful that a closed analytical expression can be obtained, as the difficulty of the formulaeincreases exponentially when augmenting the order of the constants of the motion. The crucial differencewith the undeformed case is that the validity of the permutation process (2.30) is not guaranteed anymore, as a consequence of the ‘broken-symmetry’ of the deformed coproduct ∆ z in the tensor productspace. Hence, the deformed prolongation e X m +1 z to M m +1 of X is not, in general, invariant relative tothe interchange of variables x i ↔ x j . In fact, only under the non-deformed limit z →
0, the coproduct∆ z becomes primitive and e X m +1 z (2.39) reduces to e X m +1 (2.29), being the latter symmetric under suchpermutations. Thus, in principle, no additional constants of the motion can be obtained with thisAnsatz for a generically prolonged deformation of a LH system. Nevertheless, following the approach tosuperintegrability of integrable systems with coalgebra symmetry [4, 38, 39], we can construct a secondset of constants of the motion that is valid for any deformed LH system.The essential point is that the k th -order coproduct ∆ ( k ) z is defined on the tensor product space1 ⊗ ⊗ . . . ⊗ k, (2.42) with the numeral j representing the j th -component of the tensor product), as follows at once from thedefinition (2.22). However, instead of using (2.22), it is possible to define another recursion relation forthe k th -order coproduct, as done in [4, 38]:∆ ( k ) zR := (cid:0) ∆ (2) z ⊗ ( k −
2) times z }| { Id ⊗ . . . ⊗ Id (cid:1) ◦ ∆ ( k − zR , k ≥ . (2.43)Since we are considering products in the reversal ordering, it follows that ∆ ( k ) zR lives in the tensor productspace ( m − k + 2) ⊗ ( m − k + 3) ⊗ . . . ⊗ ( m + 1) . (2.44)The maps ∆ ( k ) z and ∆ ( k ) zR are called left- and right-coproducts, respectively. For this reason, we call (2.40)the set of left-constants of the motion for the prolonged deformation e X m +1 z of the LH system, while thecorresponding set of right-constants of the motion is defined by F z ( k ) (cid:16) h ( k ) zR, , . . . , h ( k ) zR,ℓ (cid:17) := D ( k ) zR h ∆ ( k ) zR (cid:0) C z ( v , . . . , v ℓ ) (cid:1)i , k = 2 , . . . , m + 1 , (2.45)where the morphisms D ( k ) zR are defined as in (2.36), but now on the right-tensor product space (2.44), insuch a manner that the functions h ( k ) zR,i are defined by D ( k ) zR (cid:0) ∆ ( k ) zR ( v i ) (cid:1) := h ( k ) zR,i , i = 1 , . . . , ℓ, k = 2 , . . . , m + 1 . (2.46)It is straightforward to verify that, due to the coassociativity property (2.9), the identity ∆ ( m +1) zR ≡ ∆ ( m +1) z holds [4, 38], which imples that F z ( m +1) ≡ F ( m +1) z . Again, if all the F z ( k ) are non-constant,they constitute a set of m functionally independent functions in involution. We stress that functionalindependence among all the integrals follows, by construction, from the different tensor product spaces onwhich they are defined. Furthermore, the two sets F ( k ) z and F z ( k ) altogether provide a maximal number of(2 m −
1) functionally independent constants of the motion which are valid for arbitrary PH deformations.Focusing on those functions having the lowest value of k , a closed analytical expression for the deformedsuperposition rule can be much more easily found that merely considering the set of left-constants of themotion.For completeness in the exposition, we display the constants of the motion corresponding to both setsin Table 1. Under the non-deformed limit z →
0, the left-set F ( k ) z (2.40) reduces to F ( k ) (2.28), whilethe right-set F z ( k ) (2.45) provides constants of the motion F ( k ) for the undeformed LH system that areexpressible in terms of the set of permutations (2.30). LH systems on the manifold M ≡ R were fully classified in [2], basing on a previous classification ofLie algebras of vector fields in the real plane obtained in [40]. It turns out that there are 12 equivalenceclasses of finite-dimensional Lie algebras of Hamiltonian vector fields on R . For most of these planarLH systems, the constants of the motion and the superposition rules were inspected in [7]. The simpleLie algebra sl (2 , R ), that appears three times in the classification, has been studied in detail from boththe non-deformed and deformed viewpoints (see e.g. [1–3, 7]). In this section, we focus on the physicallyrelevant oscillator h -LH systems on R , reviewing the main results and applications, with the aim ofintroducing its Hopf algebra deformation in Section 4, where both deformed constants of the motion anddeformed superposition rules will be determined, as an illustration of the refinement of the deformationprocedure presented above.Let us consider the class I in the classification of real Lie algebras of Hamiltonian vector fields withglobal coordinates x = { x , x } ≡ { x, y } on R obtained in [2]. The Vessiot–Guldberg Lie algebra V isspanned by three generators X = ∂∂x , X = ∂∂y , X = x ∂∂x − y ∂∂y , (3.1) Left- and right-constants of the motion for a prolonged Poisson–Hopf deformation of a Lie–Hamilton system coming from a Casimir C z . By construction, there is a maximal number of (2 m − F z ( m +1) ≡ F ( m +1) z . Set of m left-constants F ( k ) z in involution Tensor product space for the coproduct F (2) z := D (2) z (cid:2) ∆ (2) z ( C z ) (cid:3) ⊗ F (3) z := D (3) z (cid:2) ∆ (3) z ( C z ) (cid:3) ⊗ ⊗ F ( k ) z := D ( k ) z (cid:2) ∆ ( k ) z ( C z ) (cid:3) ⊗ ⊗ . . . ⊗ k ... ... F ( m +1) z := D ( m +1) z (cid:2) ∆ ( m +1) z ( C z ) (cid:3) ⊗ ⊗ . . . ⊗ m ⊗ ( m + 1)Set of m right-constants F z ( k ) in involution Tensor product space for the coproduct F z (2) := D (2) zR (cid:2) ∆ (2) zR ( C z ) (cid:3) m ⊗ ( m + 1) F z (3) := D (3) zR (cid:2) ∆ (3) zR ( C z ) (cid:3) ( m − ⊗ m ⊗ ( m + 1)... ... F z ( k ) := D ( k ) zR (cid:2) ∆ ( k ) zR ( C z ) (cid:3) ( m − k + 2) ⊗ ( m − k + 3) ⊗ . . . ⊗ ( m + 1)... ... F z ( m +1) = F ( m +1) z := D ( m +1) zR (cid:2) ∆ ( m +1) zR ( C z ) (cid:3) ⊗ ⊗ . . . ⊗ m ⊗ ( m + 1) satisfying the Lie brackets[ X , X ] = 0 , [ X , X ] = X , [ X , X ] = − X . (3.2)Hence V is isomorphic to the (1 + 1)-dimensional Poincar´e algebra iso (1 , X (2.4) isgiven by X ( t, x, y ) = b ( t ) ∂∂x + b ( t ) ∂∂y + b ( t ) (cid:18) x ∂∂x − y ∂∂y (cid:19) , (3.3)leading to the following first-order system d x d t = b ( t ) + b ( t ) x, d y d t = b ( t ) − b ( t ) y. (3.4)The generators X i defined in (3.1) become Hamiltonian vector fields h i with respect to the standardsymplectic form ω = d x ∧ d y, (3.5)which, after application of (2.6), are found to be h = y, h = − x, h = xy, h = 1 . (3.6)Note that the addition of a central generator h is required to ensure that the corresponding bracketsclose as a Lie algebra: { h , h } ω = h , { h , h } ω = − h , { h , h } ω = h , { h , ·} ω = 0 . (3.7)It follows that the resulting LH algebra H ω is isomorphic to the centrally extended Poincar´e algebra iso (1 , h . In particular, we consider the usual basis f h = { A − , A + , N, I } corresponding to the ladder, number and central generators, respectively. Underthe identification A − = h , A + = h , N = − h , I = h , (3.8)it is easily verified that the relations (3.7) are brought into the usual form for h : { N, A ± } ω = ± A ± , { A − , A + } ω = I, { I, ·} ω = 0 . (3.9)In the following we shall denote the oscillator LH algebra H ω (3.7) by h ,ω . We now proceed to compute t -independent constants of the motion for the h -LH systems and deducethe corresponding superposition rules.The starting point is to consider the PH algebra C ∞ ( H ∗ ω ) ≡ C ∞ ( h ∗ ,ω ) in a basis { v , v , v , v } satisfying the same Poisson brackets (3.7). Now, besides v , there exists a non-trivial Casimir elementgiven by C = v v + v v . (3.10)From C , applying the morphism D : C ∞ ( h ∗ ,ω ) → C ∞ ( R ) (2.26) to the function F in (2.28), where h i are given in (3.6), we find that the constant of the motion is trivial: F = D ( C ) = h ( x , y ) h ( x , y ) + h ( x , y ) h ( x , y ) = − y x + x y × . (3.11)As the index m + 1 in (2.28) equals 3 (see [7]), we have that k = 2 ,
3. By making use of the morphisms D ( k ) in (2.26) and the coproducts ∆ ( k ) in (2.22), we recursively construct the constants of the motion F (2) and F (3) with the aid of the functions h ( k ) i (2.26): F (2) = D (2) (cid:2) ∆ (2) ( C ) (cid:3) = ( h ( x , y ) + h ( x , y )) ( h ( x , y ) + h ( x , y ))+ ( h ( x , y ) + h ( x , y )) ( h ( x , y ) + h ( x , y ))= − ( y + y )( x + x ) + ( x y + x y )(1 + 1)= ( x − x )( y − y ) . (3.12)In the same way, F (3) is found to be F (3) = D (3) (cid:2) ∆ (3) ( C ) (cid:3) = h (3)1 h (3)2 + h (3)3 h (3)0 = ( x − x )( y − y ) + ( x − x )( y − y ) + ( x − x )( y − y )= (2 x − x − x ) y + (2 x − x − x ) y + (2 x − x − x ) y . (3.13)The elements F (2) and F (3) are left-constants of the motion for the diagonal prolongation e X to ( R ) of the LH system X (3.3). It can be checked that they are in involution in C ∞ (cid:0) ( R ) (cid:1) , that is, theyPoisson-commute with respect to the symplectic form ω = d x ∧ d y + d x ∧ d y + d x ∧ d y . (3.14)From F (2) we obtain two additional constants of the motion through the permutations (2.30) (recall that k = 2 , F (2)13 = S (cid:0) F (2) (cid:1) = ( x − x )( y − y ) , F (2)23 = S (cid:0) F (2) (cid:1) = ( x − x )( y − y ) . (3.15)The remaining transposition is discarded, as F (2)12 = S (cid:0) F (2) (cid:1) ≡ F (2) . As far as the right-constants ofthe motion F ( k ) are concerned, we have that F (2) ≡ F (2)13 and F (3) ≡ F (3) which are also in involution,while F (2)23 does not belong to any of the sets (cid:8) F ( k ) (cid:9) , (cid:8) F ( k ) (cid:9) . he functions (3.12), (3.13) and (3.15) determine four t -independent constants of the motion for e X .They can also be considered as t -independent constants of the motion for X (3.3). Moreover, there existsome constants k i such that F (2) = k , F (2)23 = k , F (2)13 = k , F (3) = F (2) + F (2)23 + F (2)13 = k + k + k ≡ k. (3.16)With these results, we can explicitly derive a superposition rule. We recall that in [7] this was carried outby choosing F (2) and F (2)23 , thus expressing ( x , y ) in terms of ( x , y , x , y ) and k , k . The resultingexpression was further simplified by also introducing k , explicitly x ( x , y , x , y , k , k , k ) = 12 ( x + x ) + k − k ± B y − y ) ,y ( x , y , x , y , k , k , k ) = 12 ( y + y ) + k − k ∓ B x − x ) , (3.17) B = q k + k + k − k k + k k + k k ) , where k = k ( x , y , x , y ) through F (2)13 in (3.15), and such that the following inequality is satisfied: k + k + k ≥ k k + k k + k k ) . (3.18)As we shall prove in Subsection 4.1, the constant F (2)23 will disappear under the deformation, implyingthat we only have to consider the three remaining (left- and right-) constants of the motion F (2) , F (3) ≡ F (3) and F (2) ≡ F (2)13 of (3.16). Furthermore, it will turn out that the latter relation between the right-constant of the motion and the permuted one does not hold any more in this form. Therefore, in orderto obtain a superposition rule that is consistent with the limit z → F (2)23 . To this extent, we startwith F (2) and F (3) , now writing ( x , y ) in terms of ( x , y , x , y ) and the constants k and k (insteadof k ). Next we introduce the constant k to simplify the superposition rule, so that we are led to theexpressions x ( x , y , x , y , k , k, k ) = x + k − k ± B y − y ) ,y ( x , y , x , y , k , k, k ) = y + k − k ∓ B x − x ) , (3.19) B = q(cid:0) k − k + k ) (cid:1) − k k , where again k = k ( x , y , x , y ) through F (2) ≡ F (2)13 , subjected to the constraint (cid:0) k − k + k ) (cid:1) ≥ k k . (3.20)We remark that by introducing k = k + k + k and k = ( x − x )( y − y ) in (3.19), we easily recoverthe formulae (3.17). We observe from (3.9) that the generator N , along with either A + or A − , span a two-dimensionalsubalgebra of h isomorphic to the so-called ‘book’ algebra b , where N can be see as a dilation and A ± as a translation. In the basis of the LH algebra h with commutators (3.2), we choose the subalgebra b as the one generated by X and X : X = ∂∂y , X = x ∂∂x − y ∂∂y , [ X , X ] = − X . (3.21) hen b is seen as a Vessiot–Guldberg Lie algebra, it gives rise to the particular Lie subsystem of (3.3)with b ( t ) ≡
0: d x d t = b ( t ) x, d y d t = b ( t ) − b ( t ) y. (3.22)The symplectic form (3.5) is kept invariant, while the Hamiltonian vector fields (3.6) for b are given by h = − x, h = xy, { h , h } ω = h . (3.23)We recall that b arises within the classification of planar LH systems [2, 7] as the class I r =114 A ≃ R ⋉R ≃ b . Although b does not admit non-constant Casimir invariants, its consideration as a particularcase of the h -LH systems allows us to apply the above results concerning constants of the motion andsuperposition rules, as it was pointed out in [7]. Furthermore, in spite of the apparently naive form of thedifferential equations (3.22), it is worthy to be remarked that b -LH systems emerge in various physicaland mathematical contexts such as [2, 7]: • Generalised Buchdahl equations , which are second-order differential equations appearing in thestudy of relativistic fluids [41, 42] and have also been studied by means of a Lagrangian approachin [43]. • Some particular two-dimensional
Lotka–Volterra systems with t -dependent coefficients [44, 45]. • Complex Bernoulli differential equations with t -dependent real coefficients [46], which are the par-ticular case of the non-autonomous complex Bernoulli differential equations with complex coeffi-cients [47, 48].In what follows, we focus on the third type of b -systems and its PH deformation will be obtained inSubsection 4.2. The two remaining types can also be developed in similar manner, although computationsare rather cumbersome due to the complicated symplectic structure that arises, as well as the change ofvariables required to relate such systems to the expressions (3.21)–(3.23). Let us consider the family of non-autonomous complex Bernoulli differential equationsd w d t = a ( t ) w + a ( t ) w s , s / ∈ { , } , (3.24)where w is a complex function and a ( t ) , a ( t ) are arbitrary real valued t -dependent functions. Introducingthe polar reference w = r e i θ , we obtain that the differential equation (3.24) unfolds as the real first-ordersystem d r d t = a ( t ) r + a ( t ) r s cos[ θ ( s − , d θ d t = a ( t ) r s − sin[ θ ( s − , (3.25)which can be expressed through the t -dependent vector field Y ( t, r, θ ) = a ( t ) Y + a ( t ) Y , (3.26)where Y = r ∂∂r , Y = r s cos[ θ ( s − ∂∂r + r s − sin[ θ ( s − ∂∂θ . (3.27)The corresponding Lie bracket [ Y , Y ] = ( s − Y , (3.28) hows that Y is a Lie system with Vessiot–Guldberg Lie algebra V isomorphic to b .The next step is to determine a symplectic form ω = f ( r, θ )d r ∧ d θ compatible with the vector fields(3.27) by requiring the relation (2.5) to be satisfied. A routine computation shows that ω can be chosenas ω = s − r sin [ θ ( s − r ∧ d θ. (3.29)Therefore Y (3.26) is a LH system whose Hamiltonian functions ¯ h i , deduced by means of the relation(2.6), are given by ¯ h = − θ ( s − , ¯ h = − r s − sin[ θ ( s − . (3.30)The Poisson bracket with respect to the symplectic form (3.29) reads { ¯ h , ¯ h } ω = − ( s − h . (3.31)Now our task consists in establishing the relationship of these results with (3.21)–(3.23). This is doneconsidering the change of variables given by x = r s − sin[ θ ( s − , y = − cos[ θ ( s − s − r s − ,r s − = x s − x y , tan [ θ ( s − s − x y . (3.32)Under these transformations, the symplectic form (3.29) adopts the canonical form (3.5), while therelations amongst vector fields and t -dependent coefficients are given by Y = ( s − X , Y = X , ¯ h = ( s − h , ¯ h = h ,a ( t ) = b ( t ) / ( s − , a ( t ) = b ( t ) . (3.33)With the relations (3.32) at hand, it is straightforward to obtain the constants of the motion for thecomplex Bernoulli differential equations (3.25). The three functions F (2) , F (3) and F (2) ≡ F (2)13 (see(3.12), (3.13) and (3.15) respectively) have the explicit form F (2) = 11 − s (cid:18) r s − sin[ θ ( s − − r s − sin[ θ ( s − (cid:19) (cid:18) cos[ θ ( s − r s − − cos[ θ ( s − r s − (cid:19) ,F (2) = 11 − s (cid:18) r s − sin[ θ ( s − − r s − sin[ θ ( s − (cid:19) (cid:18) cos[ θ ( s − r s − − cos[ θ ( s − r s − (cid:19) , (3.34) F (3) = 11 − s X ≤ i Multiparametric coboundary Lie bialgebras for the oscillator Lie algebra h = { A − , A + , N, I } (3.9) wereclassified in [8] along with their quantum deformations. This exhaustive analysis shows that mathematicaland physical properties of each deformation are in direct correspondence with the generators that remainundeformed, that is, with a primitive (trivial) coproduct (2.20). As the central generator I is alwaysprimitive, one should additionally require either N or a single A ± to be primitive as well. It turns outthat all (multiparametric) deformations with N primitive lead to quantum deformations that are governedby I [8], with N behaving as a ‘secondary’ primitive generator. In the context of LH systems this impliesthat these quantum deformations give rise to ‘trivial’ LH systems (recall that I = h = 1 in (3.8)). Bycontrast, deformations with a primitive A + = − x (or A − = y ) provide non-trivial LH systems, as inthese cases A + plays the role of the ‘main’ primitive generator, with I playing the role of a ‘secondary’one.The simplest (i.e. one-parameter) quantum deformation such that A + is primitive corresponds toconsider the classical r -matrix r = z A + ∧ N, (4.1)which is a solution of the classical Yang–Baxter equation, and where z is the quantum deformationparameter such that q = e z . This element underlies the so-called nonstandard (or Jordanian) quantumoscillator algebra U z ( h ), whose boson representations have been studied in [49, 50].In the LH framework, we start with the Lie algebra h in the basis { v , v , v , v } with Lie brackets(see (3.7)) [ v , v ] = v , [ v , v ] = − v , [ v , v ] = v , [ v , · ] = 0 , (4.2)as well as with the classical r -matrix r = z v ∧ v . (4.3)The Lie bialgebra is provided by the cocommutator map δ that is obtained from the classical r -matrix as δ ( v i ) = [ v i ⊗ ⊗ v i , r ] , (4.4)yielding δ ( v ) = δ ( v ) = 0 , δ ( v ) = z ( v ∧ v + v ∧ v ) , δ ( v ) = z v ∧ v , (4.5)which is just the skew-symmetric part of the first-order term ∆ in z of the full coproduct ∆ z , that is,∆ z ( v i ) = ∆ ( v i ) + ∆ ( v i ) + o [ z ] , ∆ ( v i ) = v i ⊗ ⊗ v i , δ ( v i ) = ∆ ( v i ) − σ ◦ ∆ ( v i ) , (4.6)where σ is the flip operator: σ ( v i ⊗ v j ) = v j ⊗ v i .From the complete quantum algebra U z ( h ) [8, 49], the corresponding Poisson coalgebra structurecan easily be deduced giving rise to the following deformed coproduct and Poisson brackets:∆ z ( v ) = v ⊗ ⊗ v , ∆ z ( v ) = v ⊗ ⊗ v , ∆ z ( v ) = v ⊗ e − zv + 1 ⊗ v + z v ⊗ e − zv v , ∆ z ( v ) = v ⊗ e − zv + 1 ⊗ v , (4.7) { v , v } z = e − zv v , { v , v } z = − v , { v , v } z = 1 − e − zv z , { v , ·} z = 0 , (4.8)such that ∆ z (4.7) satisfies the coassociativity condition (2.9) and is a Poisson algebra homomorphismof the Poisson brackets (4.8). The deformed Casimir turns out to be C z = v (cid:18) e zv − z (cid:19) + v v . (4.9)Now we apply the algorithmic procedure summarized in Subsection 2.2 to construct a PH deformation C ∞ ( h ∗ z ,ω ) of C ∞ ( h ∗ ,ω ), hence deforming the h -LH systems of Section 3. To this extent, we start fromthe functions { h , h , h , h } on C ∞ ( R ) as given in (3.6) with Poisson brackets (3.7), where ω is the anonical symplectic form (3.5). Taking into account the boson representations of U z ( h ) given in [49, 50],we introduce the Hamiltonian functions on C ∞ ( R ) h z, = e zx y, h z, = − x, h z, = (cid:18) e zx − z (cid:19) y, h z, = 1 , (4.10)which satisfy the following Poisson brackets with respect to the same symplectic form (3.5) { h z, , h z, } ω = e − zh z, h z, , { h z, , h z, } ω = − h z, , { h z, , h z, } ω = 1 − e − zh z, z , { h z, , ·} ω = 0 , (4.11)in agreement with the relations (4.8). In the third step, the deformed vector fields X z,i on R are obtainedthrough the relation (2.14), namely X z, = e zx ∂∂x − z e zx y ∂∂y , X z, = ∂∂y , X z, = (cid:18) e zx − z (cid:19) ∂∂x − e zx y ∂∂y . (4.12)Finally, the PH deformation of the h -LH system (3.3) is determined by X z ( t, x, y ) = b ( t ) X z, + b ( t ) X z, + b ( t ) X z, , (4.13)leading to the system of differential equationsd x d t = b ( t ) e zx + b ( t ) (cid:18) e zx − z (cid:19) , d y d t = b ( t ) − (cid:0) b ( t ) + z b ( t ) (cid:1) e zx y. (4.14)It is worth remarking that since ω is the standard symplectic form (3.5), the same system of differentialequations (4.14) can, alternatively, be obtained by computing the usual Hamilton equations from thedeformed Hamiltonian (2.18) with the functions (4.10), h z = b ( t )e zx y − b ( t ) x + b ( t ) (cid:18) e zx − z (cid:19) y + b ( t ) , (4.15)in the form d x d t = ∂h z ∂y , d y d t = − ∂h z ∂x . (4.16)As we have already commented, the deformed vector fields (4.12) span a Stefan–Sussman distribu-tion [30, 34, 35] whose commutation rules (2.17) turn out to be[ X z, , X z, ] = z e − zh z, h z, X z, , [ X z, , X z, ] = X z, , [ X z, , X z, ] = − e − zh z, X z, . (4.17)By introducing the functions (4.10) we obtain that[ X z, , X z, ] = z e zx X z, , [ X z, , X z, ] = X z, , [ X z, , X z, ] = − e zx X z, . (4.18)Note that that the expressions (4.10)–(4.18) reduce to (3.1)–(3.7) in the limit z → 0. It is worth stressingthat, even considering the commutation relations (4.18) up to first-order in the deformation parameter z , the vector fields X z, , X z, , X z, do not close on a finite-dimensional Lie algebra.The remarkable feature of the deformation is the presence of the ‘interacting’ term e zx y in (4.14),when compared with (3.4). This nonlinear interaction or coupling between the two variables can beregarded as a perturbation of the initial system. Indeed, by considering a power series expansion in z ofthe system (4.14) and truncating at the first-order, we obtaind x d t = b ( t ) + (cid:0) b ( t ) + z b ( t ) (cid:1) x + 12 z b ( t ) x + o [ z ] , d y d t = b ( t ) − (cid:0) b ( t ) + z b ( t ) (cid:1) y − z b ( t ) xy + o [ z ] . (4.19)In the first equation, the deformation introduces a quadratic term x , leading to a Riccati equation with t -dependent real coefficients [12], while in the second one, we obtain the nonlinear interaction term xy . .1 Deformed constants of the motion and deformed superposition rules Now we proceed to apply the approach presented in Subsection 2.3.2 in order to obtain three deformedconstants of the motion F (2) z , F z (2) and F (3) z ≡ F z (3) (see Table 1) for the prolonged deformation of h -LHsystems, as in this case (see Subsection 3.1) we have the indices m + 1 = 3 and k = 2 , φ z : h z ,ω → C ∞ ( R ) in(2.35) and D z : C ∞ ( h ∗ z ,ω ) → C ∞ ( R ) in (2.36) lead to D z ( v i ) = h z,i ( x , y ) := h (1) z,i , i = 0 , , , , (4.20)where h z,i are the Hamiltonian functions (4.10) fulfilling (4.11). By introducing this result into theCasimir (4.9), we find that, as expected, the corresponding constant F z of (2.40) is again trivial: F z = D z ( C z ) = h (1) z, e zh (1) z, − z ! + h (1) z, h (1) z, = 0 . (4.21)Now we consider the deformed coproduct ∆ z ≡ ∆ (2) z (4.7) on the tensor product space 1 ⊗ D (2) z (cid:0) ∆ (2) z ( v i ) (cid:1) coming from the morphism D (2) z in (2.36) yielding the functions h (2) z,i (2.38): D (2) z (cid:0) ∆ (2) z ( v ) (cid:1) = h z, ( x , y ) + h z, ( x , y ) = − x − x := h (2) z, ,D (2) z (cid:0) ∆ (2) z ( v ) (cid:1) = h z, ( x , y ) + h z, ( x , y ) = 1 + 1 = 2 := h (2) z, ,D (2) z (cid:0) ∆ (2) z ( v ) (cid:1) = h z, ( x , y )e − zh z, ( x ,y ) + h z, ( x , y ) + z h z, ( x , y )e − zh z, ( x ,y ) h z, ( x , y )= e zx e zx y + e zx y + (e zx − 1) e zx y := h (2) z, ,D (2) z (cid:0) ∆ (2) z ( v ) (cid:1) = h z, ( x , y )e − zh z, ( x ,y ) + h z, ( x , y )= (cid:18) e zx − z (cid:19) e zx y + (cid:18) e zx − z (cid:19) y := h (2) z, . (4.22)These expressions allow us to obtain the left-constant of the motion of (2.40) for k = 2: F (2) z = D (2) z (cid:2) ∆ (2) z (cid:0) C z (cid:1)(cid:3) = h (2) z, e zh (2) z, − z ! + h (2) z, h (2) z, , (4.23)namely F (2) z = (cid:18) − e − zx − e zx z (cid:19) ( y − y ) . (4.24)Similarly, the right-constant of the motion F z (2) defined in (2.45) is deduced, but now working with theright-coproduct ∆ (2) zR on the tensor product space 2 ⊗ h (2) zR,i (2.46) turn out tobe D (2) zR (cid:0) ∆ (2) zR ( v ) (cid:1) = h z, ( x , y ) + h z, ( x , y ) = − x − x =: h (2) zR, ,D (2) zR (cid:0) ∆ (2) zR ( v ) (cid:1) = h z, ( x , y ) + h z, ( x , y ) = 1 + 1 = 2 =: h (2) zR, ,D (2) zR (cid:0) ∆ (2) zR ( v ) (cid:1) = h z, ( x , y )e − zh z, ( x ,y ) + h z, ( x , y ) + z h z, ( x , y )e − zh z, ( x ,y ) h z, ( x , y )= e zx e zx y + e zx y + (e zx − 1) e zx y =: h (2) zR, ,D (2) zR (cid:0) ∆ (2) zR ( v ) (cid:1) = h z, ( x , y )e − zh z, ( x ,y ) + h z, ( x , y )= (cid:18) e zx − z (cid:19) e zx y + (cid:18) e zx − z (cid:19) y =: h (2) zR, , (4.25) iving rise, through the analogous expression to (4.23), to F z (2) = (cid:18) − e − zx − e zx z (cid:19) ( y − y ) . (4.26)The non-deformed limit z → F (2) (3.12) and F (2) ≡ F (2)13 = S ( F (2) ) (3.15). Nevertheless, we stress that in the deformed case, the constant of themotion F (2) z does not remain invariant under the permutation S and, moreover, F z (2) is related to F (2) z through the composition of two permutations which differs from the result obtained merely applying thepermutation S : F (2) z = S (cid:0) F (2) z (cid:1) , F z (2) = S (cid:0) S (cid:0) F (2) z (cid:1)(cid:1) = S (cid:0) F (2) z (cid:1) . (4.27)In addition, there is no deformed constant of the motion that corresponds in the limit z → F (2)23 = S (cid:0) F (2) (cid:1) in (3.15). In fact, it is straightforward to check that the functions obtained from F (2) z by meansof the permutations S , S and S , S (cid:0) F (2) z (cid:1) = (cid:18) − e − zx − e zx z (cid:19) ( y − y ) , S (cid:0) F (2) z (cid:1) = (cid:18) − e − zx − e zx z (cid:19) ( y − y ) ,S (cid:0) F (2) z (cid:1) = (cid:18) − e − zx − e zx z (cid:19) ( y − y ) , (4.28)do not provide any constant of the motion. This shows that the range of application of the permutations S ij in the form (2.30) is rather limited in the deformed case, where only left- and right-constants of themotion can be ensured to be correct.It remains to compute F (3) z ≡ F z (3) , which requires to construct the third-order coproduct ∆ (3) z of(4.7) on the tensor product space 1 ⊗ ⊗ 3. In this case, with m + 1 = 3, ∆ (3) z is obtained by means ofthe coassociativity condition (2.9) (corresponding to (2.22) and (2.43) with k = m + 1 = 3)∆ (3) z = (Id ⊗ ∆ z ) ◦ ∆ z = (∆ z ⊗ Id) ◦ ∆ z = ∆ (3) zR , (4.29)leading to ∆ (3) z ( v l ) = v l ⊗ ⊗ ⊗ v l ⊗ ⊗ ⊗ v l , l = 0 , , ∆ (3) z ( v ) = v ⊗ e − zv ⊗ e − zv + 1 ⊗ v ⊗ e − zv + 1 ⊗ ⊗ v + z (cid:0) v ⊗ e − zv v ⊗ e − zv + v ⊗ e − zv ⊗ e − zv v + 1 ⊗ v ⊗ e − zv v (cid:1) , ∆ (3) z ( v ) = v ⊗ e − zv ⊗ e − zv + 1 ⊗ v ⊗ e − zv + 1 ⊗ ⊗ v , (4.30)provided that ∆ z (1) = 1 ⊗ , ∆ z (e − zv ) = e − zv ⊗ e − zv . (4.31)Then we obtain the Hamiltonian functions on ( R ) given by h (3) z, := D (3) z (cid:0) ∆ (3) z ( v ) (cid:1) = − x − x − x , h (3) z, := D (3) z (cid:0) ∆ (3) z ( v ) (cid:1) = 3 ,h (3) z, := D (3) z (cid:0) ∆ (3) z ( v ) (cid:1) = (cid:0) zx − (cid:1) e z ( x + x ) y + (cid:0) zx − (cid:1) e zx y + e zx y ,h (3) z, := D (3) z (cid:0) ∆ (3) z ( v ) (cid:1) = (cid:18) e zx − z (cid:19) e z ( x + x ) y + (cid:18) e zx − z (cid:19) e zx y + (cid:18) e zx − z (cid:19) y . (4.32)By introducing them into (2.40) we get the third constant of the motion: F (3) z = 1 z (cid:0) − − zx − e zx e zx (cid:1) y + 1 z (cid:0) − zx − e − zx e − zx − zx + e zx e zx (cid:1) y − z (cid:0) − zx − e − zx e − zx (cid:1) y , (4.33) hose limit z → F (3) in the second form written in (3.13).Summing up, the functions (4.32) satisfy the Poisson brackets (4.11) with respect to the symplecticform (3.14) and, with this ω , all the following Poisson brackets vanish ( i = 0 , , , (cid:8) F (2) z , h (3) z,i (cid:9) ω = (cid:8) F z (2) , h (3) z,i (cid:9) ω = (cid:8) F (3) z , h (3) z,i (cid:9) ω = 0 , (cid:8) F (2) z , F (3) z (cid:9) ω = (cid:8) F z (2) , F (3) z (cid:9) ω = 0 . (4.34)Consequently, F (2) z , F z (2) and F (3) z , as given in (4.24), (4.26) and (4.33), are three functionally independentconstants of the motion of the prolonged deformation e X z of h -LH systems to ( R ) .Let us deduce now e X z in an explicit manner. By taking into account that ω (3.14) is the standardsymplectic form, we consider the corresponding deformed Hamiltonian on ( R ) , h (3) z = b ( t ) h (3) z, + b ( t ) h (3) z, + b ( t ) h (3) z, + b ( t ) h (3) z, , (4.35)with the Hamiltonian functions (4.32), and compute the Hamilton equations (similarly to (4.16)), thusfinding that e X z is given by the following system of six differential equations:d x d t = b ( t ) (cid:0) zx − (cid:1) e z ( x + x ) + b ( t ) (cid:18) e zx − z (cid:19) e z ( x + x ) , d y d t = b ( t ) − (cid:0) b ( t ) + 3 zb ( t ) (cid:1) e z ( x + x + x ) y , d x d t = b ( t ) (cid:0) zx − (cid:1) e zx + b ( t ) (cid:18) e zx − z (cid:19) e zx , (4.36)d y dt = b ( t ) − b ( t ) e z ( x + x ) (cid:0) (e zx − y + y (cid:1) − zb ( t ) e z ( x + x ) (cid:0) (3e zx − y + 2 y (cid:1) , d x d t = b ( t ) e zx + b ( t ) (cid:18) e zx − z (cid:19) , d y d t = b ( t ) − b ( t ) e zx (cid:0) (e zx − zx y + (e zx − y + y (cid:1) − zb ( t ) e zx (cid:0) (3e zx − zx y + (2e zx − y + y (cid:1) . Under the non-deformed limit z → 0, the prolonged deformation e X z reduces to the diagonal prolongation e X of the h -LH system X (3.4) to ( R ) which simply corresponds to three copies of X . On thecontrary, it is remarkable that e X z (4.36) is no longer formed by three copies of the deformed h -LHsystem X z (4.14). Therefore, we stress that the constants of the motion of e X z cannot be considered asconstants of the motion of X z .Furthermore, the deformed vector fields X h (3) z,i , which determine e X z by means of the expression (2.39),can directly be deduced from (4.36); these are X h (3) z, = (cid:0) zx − (cid:1) e z ( x + x ) ∂∂x + (cid:0) zx − (cid:1) e zx ∂∂x + e zx ∂∂x − z e z ( x + x + x ) y ∂∂y − z e z ( x + x ) (cid:0) (3e zx − y + 2 y (cid:1) ∂∂y − z e zx (cid:0) (3e zx − zx y + (2e zx − y + y (cid:1) ∂∂y , X h (3) z, = ∂∂y + ∂∂y + ∂∂y , (4.37) X h (3) z, = (cid:18) e zx − z (cid:19) e z ( x + x ) ∂∂x + (cid:18) e zx − z (cid:19) e zx ∂∂x + (cid:18) e zx − z (cid:19) ∂∂x − e z ( x + x + x ) y ∂∂y − e z ( x + x ) (cid:0) (e zx − y + y (cid:1) ∂∂y − e zx (cid:0) (e zx − zx y + (e zx − y + y (cid:1) ∂∂y . It can be checked that they fulfil the relationship (2.14) with respect to the Hamiltonian functions (4.32)and symplectic form (3.14). They satisfy the commutation relations (2.41), which are just those given in h X h (3) z, , X h (3) z, i = 3 z e z ( x + x + x ) X h (3) z, , h X h (3) z, , X h (3) z, i = X h (3) z, , h X h (3) z, , X h (3) z, i = − e z ( x + x + x ) X h (3) z, , (4.38)to be compared with (4.18).The three deformed constants of the motion satisfying (4.34) allow us to deduce a deformed superpo-sition rule for e X z . We keep the notation of Subsection 3.1 and consider the three equations coming from(4.24), (4.26) and (4.33): F (2) z = k , F (3) z = k, F z (2) = k , (4.39)where k , k and k are constants. From the first two equations we can express ( x , y ) in terms of( x , y , x , y ) and the constants k and k . The third equation enables to write the result in a simplifiedmanner, namely e zx = (cid:0) − zx − e zx (cid:1) ( y − y ) + z (cid:0) k − k ± B (cid:1) zk (cid:0) e zx − (cid:1) − zk (cid:0) e zx e zx − (cid:1) + (cid:0) e zx − (cid:1)(cid:0) zx − (cid:1) ( y − y ) ,y = e − zx y + (cid:0) − e − zx (cid:1) y + z (cid:0) k − k ∓ B (cid:1) zx (cid:0) − e − zx − e zx (cid:1) , (4.40) B = q(cid:0) k − k + k ) (cid:1) − k k . The factor B is formally the same given in (3.19), while the constant k , that only appears within B ,should be understood as a function k = k ( x , y , x , y ) through F z (2) (4.26). Therefore, the expressions(4.40) constitute a generic deformed superposition rule corresponding to the prolonged deformation e X z of h -LH systems to ( R ) given by the system of differential equations (4.36). In order to compute theirundeformed limit one should apply in (4.40) the limitslim z → (cid:18) e zx − z (cid:19) = x , lim z → y = y , (4.41)thus recovering the superposition rule (3.19) for the h -LH systems (3.4). One of the remarkable algebraic properties of the nonstandard quantum deformation of the oscillatoralgebra h is that the book subalgebra b remains as a Hopf subalgebra after the deformation, as canbe inferred from the classical r -matrix in (4.3). Therefore, by construction, we obtain a Poisson sub-coalgebra spanned by v and v within the relations (4.7) and (4.8). As a byproduct, from (4.10)–(4.18)we directly get all the ingredients that characterize the resulting deformed b -LH systems, which reduceto the expressions (3.21)–(3.23) under the limit z → 0; these are h z, = − x, h z, = (cid:18) e zx − z (cid:19) y, { h z, , h z, } ω = 1 − e − zh z, z , (4.42) X z, = ∂∂y , X z, = (cid:18) e zx − z (cid:19) ∂∂x − e zx y ∂∂y , [ X z, , X z, ] = − e zx X z, , (4.43)d x d t = b ( t ) (cid:18) e zx − z (cid:19) , d y d t = b ( t ) − b ( t )e zx y. (4.44) onsequently, the prolonged deformation of b -LH systems to ( R ) is straightforwardly achieved bysetting b ( t ) ≡ e X z (4.36), or by only considering the deformed vector fields X h (3) z, and X h (3) z, inthe expressions (4.37) and (4.38). Moreover, the corresponding deformed constants of the motion andsuperposition rules are exactly those given by (4.39) and (4.40) for the prolonged deformation of h -LHsystems. These results can further be applied to all the b -LH systems mentioned in Subsection 3.2.To illustrate the latter point, we briefly present the main results concerning the PH deformation of thecomplex Bernoulli differential equations studied in Subsection 3.3. We keep the symplectic form (3.29),the change of variables (3.32) and the relationships (3.33). It is easily seen that, in these conditions, thedeformed Hamiltonian functions are given by¯ h z, = − cos[ θ ( s − z r s − (cid:18) exp (cid:26) z r s − sin[ θ ( s − (cid:27) − (cid:19) , ¯ h z, = − r s − sin[ θ ( s − , { ¯ h z, , ¯ h z, } ω = ( s − 1) e − z ¯ h z, − z , (4.45)while the corresponding deformed vector fields Y z,i turn out to be Y z, = (cid:18) r cos [ θ ( s − (cid:26) z r s − sin[ θ ( s − (cid:27) + sin [ θ ( s − z r s − (cid:18) exp (cid:26) z r s − sin[ θ ( s − (cid:27) − (cid:19)(cid:19) ∂∂r + sin [ θ ( s − exp n z r s − sin[ θ ( s − o tan[ θ ( s − − cos[ θ ( s − z r s − (cid:18) exp (cid:26) z r s − sin[ θ ( s − (cid:27) − (cid:19) ∂∂θ , Y z, = r s cos[ θ ( s − ∂∂r + r s − sin[ θ ( s − ∂∂θ , [ Y z, , Y z, ] = ( s − 1) exp (cid:26) z r s − sin[ θ ( s − (cid:27) Y z, . (4.46)The deformed Bernoulli system of differential equations adopts the formd r d t = a ( t ) (cid:18) r cos [ θ ( s − (cid:26) z r s − sin[ θ ( s − (cid:27) + sin [ θ ( s − z r s − (cid:18) exp (cid:26) z r s − sin[ θ ( s − (cid:27) − (cid:19)(cid:19) + a ( t ) r s cos[ θ ( s − , d θ d t = a ( t ) sin [ θ ( s − exp n z r s − sin[ θ ( s − o tan[ θ ( s − − cos[ θ ( s − z r s − (cid:18) exp (cid:26) z r s − sin[ θ ( s − (cid:27) − (cid:19) + a ( t ) r s − sin[ θ ( s − , (4.47)where the undeformed limit z → z in order to interpret this result as a perturbation of theinitial Bernoulli differential equations.In spite of the apparently very cumbersome form of the resulting deformed Bernoulli system (4.47), itsprolonged deformation along with the corresponding deformed constants of the motion and superpositionrules can explicitly be derived from the results obtained in the previous section. For the sake of brevity,we merely indicate that F (2) z in (4.24) now becomes F (2) z = 1 z (1 − s ) (cid:18) − exp (cid:26) − z r s − sin[ θ ( s − (cid:27) − exp (cid:26) z r s − sin[ θ ( s − (cid:27)(cid:19) × (cid:18) cos[ θ ( s − r s − − cos[ θ ( s − r s − (cid:19) , and its undeformed limit is given in (3.34). Concluding remarks In this work, a relevant question addressed to but left open in [1], concerning the possibility of deducinga computationally feasible deformed analogue of superposition principles for PH deformations of LHsystems, has been answered in the affirmative. This has been achieved by combining the formalism ofPH deformations with the superintegrability property of systems having coalgebra symmetry. In thisway, two separate sets of constants of the motion have been derived for the prolonged deformations,from which a sufficient number of functionally independent constants of the motion can be extracted,hence making it possible to establish a generic deformed superposition rule of the lowest possible order,regardless on the particular structure of the Hopf algebra deformation. This approach amends andgeneralizes the construction previously proposed in [6] based on permutations in the tensor productspace, which did not take into account the symmetry breaking originated by deformed coproducts, whichprevents that a constant of the motion retains its invariant character after having been transformed bya permutation of the variables. It is worth stressing that in order to develop this refinement, it hasbeen necessary to introduce two new notions: prolonged PH deformations of LH systems and deformedsuperposition rules, which respectively reduce to the usual diagonal prolongations and superposition rulesof the initial LH system under the non-deformed limit of the deformation parameter. Consequently, acomplete correspondence between the characteristic properties of LH systems and their PH deformationshas been established.Along these lines, the LH systems based on the oscillator algebra h (see [7]) and their nonstandarddeformation have been studied, and an explicit deformed superposition rule for their prolonged defor-mation has been obtained. Besides the undeniable physical interest of the oscillator algebra h , anotherremarkable feature has led to this choice for illustrating the generalization of the formalism. The fact thatthe book algebra b is preserved as a Hopf subalgebra after deformation, implies that prolonged deforma-tions of LH systems based on b can easily be obtained through restriction of the prolonged deformationsof h -LH systems. A striking particular case is given by the prolonged deformation of complex Bernoulliequations, for which the method provides a systematic prescription for determining the constants of themotion and a deformed superposition rule. In this context, it is worth to be mentioned that PH defor-mations of LH systems based on b , but seen as a Lie subalgebra of sl (2 , R ), have recently been used forthe description of new SIS epidemic models [51]. This suggests in a natural way to analyze analogousmodels based on b -LH systems obtained as restrictions of prolonged PH deformations of oscillator LHsystems. Even if under the limit z → b is the same, it is expected that the properties ofsuch deformed models should be quite distinct to those studied in [51], due to the different features ofthe sl (2 , R ) and h deformations.The extended formalism here presented gives rise to a number of interesting questions that can beconsidered. A first one concerns a systematic analysis of prolonged PH deformations of LH systems inthe plane, and its eventual identification with dynamical systems appearing in various applications. Thisin particular applies to those systems that can be interpreted as small perturbations of LH systems,and where the deformation formalism may provide a precise insight on the exact role of the deformationparameter with respect to stability or bifurcation properties of the system, as well as concerning thegeometrical and dynamical behaviour of the orbits. At a more profound level, it may be asked if PH de-formations admit some kind of inverse problem. More specifically, it is conceivable that a non-autonomousparametrized nonlinear system of differential equations, without having the structure of a LH system,still allows a description in terms of a t -dependent vector field, the t -independent components of which,although not generating a finite-dimensional Lie algebra, span a distribution in the Stefan–Sussmansense. In these conditions, it would be of interest to know whether a compatible PH structure on anappropriate manifold can be found, so that one or more of the parameters in the system can be identifiedwith deformation parameters, hence allowing the system to be associated with a PH deformation of someLH system that would be recovered by a limiting process. This problem is intimately related to thedevelopment of an unambiguous notion of equivalence classes for PH deformations, possibly focusing oncertain structural properties that until now have not been inspected in full detail, so that some kind ofclassification parallel to that of LH systems may be established. Progress in some of the above-mentionedproblems will hopefully be reported in some future work. cknowledgements A.B. and F.J.H. have been partially supported by Ministerio de Ciencia e Innovaci´on (Spain) undergrants MTM2016-79639-P (AEI/FEDER, UE) and PID2019-106802GB-I00/AEI/10.13039/501100011033(AEI/FEDER, UE), and by Junta de Castilla y Le´on (Spain) under grants BU229P18 and BU091G19.The research of R.C.S. was financially supported by grants MTM2016-79422-P (AEI/FEDER, EU) andPID2019-106802GB-I00/AEI/10.13039/501100011033 (AEI/FEDER, UE). E.F.S. acknowledges a fellow-ship (grant CT45/15-CT46/15) supported by the Universidad Complutense de Madrid. 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