A geometric approach to Lie systems: formalism of Poisson-Hopf algebra deformations
UU NIVERSIDAD C OMPLUTENSE DE M ADRID P H . D. T HESIS
A geometric approach to Lie systems:formalism of Poisson–Hopf algebradeformations (Un enfoque geométrico a los sistemas de Lie:formalismo de las deformaciones de álgebras dePoisson–Hopf)
Author:
EduardoF
ERNÁNDEZ -S AIZ
Supervisors:
Dr. Otto RutwigCampoamor-Stursberg,Dr. Francisco José HerranzZorrilla
A thesis submitted in fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics
Madrid, 2020 a r X i v : . [ m a t h - ph ] J a n ii Declaration of Authorship D. Eduardo F
ERNÁNDEZ -S AIZ , estudiante en el Programa de Doctorado Investigación Matemática,de la Facultad de Ciencias Matemáticas de la Universidad Complutense de Madrid, como autor dela tesis presentada para la obtención del título de Doctor y titulada: A geometric approach to Liesystems: formalism of Poisson-–Hopf algebra deformations (Un enfoque geométrico a los sistemasde Lie: formalismo de las deformaciones de álgebras de Poisson–Hopf), y dirigida por: Dr. OttoRutwig Campoamor Stursberg y Dr. Francisco José Herranz Zorrilla.DECLARO QUE:La tesis es una obra original que no infringe los derechos de propiedad intelectual ni los derechosde propiedad industrial u otros, de acuerdo con el ordenamiento jurídico vigente, en particular, laLey de Propiedad Intelectual (R.D. legislativo 1/1996, de 12 de abril, por el que se aprueba el textorefundido de la Ley de Propiedad Intelectual, modificado por la Ley 2/2019, de 1 de marzo, regular-izando, aclarando y armonizando las disposiciones legales vigentes sobre la materia), en particular,las disposiciones referidas al derecho de cita.Del mismo modo, asumo frente a la Universidad cualquier responsabilidad que pudiera derivarsede la autoría o falta de originalidad del contenido de la tesis presentada de conformidad con elordenamiento jurídico vigente.En Madrid, a 30 de septiembre de 2020Fdo.: Eduardo Fernández Saiz
By homely gifts and hindered WordsThe human heart is toldOf Nothing -“Nothing” is the forceThat renovates the World -
Emily Dickinsoni
Abstract
The notion of quantum algebras is merged with that of Lie systems in order to establish a new for-malism called Poisson–Hopf algebra deformations of Lie systems. The procedure can be naturallyapplied to Lie systems endowed with a symplectic structure, the so-called Lie–Hamilton systems.This is quite a general approach, as it can be applied to any quantum deformation and any un-derlying manifold. One of its main features is that, under quantum deformations, Lie systems areextended to generalized systems described by involutive distributions. As a consequence, a quan-tum deformed Lie system no longer has an underlying Vessiot–Guldberg Lie algebra or a quantumalgebra one, but keeps a Poisson–Hopf algebra structure that enables us to obtain, in an explicitway, the t -independent constants of the motion from quantum deformed Casimir invariants, whichare potentially useful in a further construction of the generalized notion of superposition rules.We illustrate this approach by considering the non-standard quantum deformation of sl ( ) appliedto well-known Lie systems, such as the oscillator problem or Milne–Pinney equation, as well asseveral types of Riccati equations. In this way, we obtain their new generalized (deformed) coun-terparts that cover, in particular, a new oscillator system with a time-dependent frequency anda position-dependent mass. Based on a recently developed procedure to construct Poisson–Hopfdeformations of Lie–Hamilton systems [13], a novel unified approach to nonequivalent deforma-tions of Lie–Hamilton systems on the real plane with a Vessiot–Guldberg Lie algebra isomorphicto sl ( ) is proposed. This, in particular, allows us to define a notion of Poisson–Hopf systems independence of a parameterized family of Poisson algebra representations [14]. Such an approach isexplicitly illustrated by applying it to the three non-diffeomorphic classes of sl ( ) Lie–Hamilton sys-tems. Furthermore t -independent constants of motion are given as well. Our methods can be em-ployed to generate other Lie–Hamilton systems and their deformations for other Vessiot–GuldbergLie algebras and their deformations. In addition, we study the deformed systems obtained fromLie–Hamilton systems associated to the oscillator algebra h , seen as a subalgebra of the 2-photonalgebra h . As a particular application, we propose an epidemiological model of SISf type thatuses the solvable Lie algebra b as subalgebra of sl ( ) , by restriction of the corresponding quantumdeformed systems. Keywords:
Lie system, Poisson–Hopf algebra, Poisson coalgebra, quantum deformation.ii
Resumen
La noción de álgebras cuánticas se fusiona con la de sistemas de Lie para establecer un nuevo for-malismo, las deformaciones del álgebra de Poisson–Hopf de los sistemas de Lie. El procedimientopuede aplicarse a sistemas de Lie dotados de una estructura simpléctica, los denominados sistemasde Lie–Hamilton. Este es un enfoque bastante general, ya que se puede aplicar a cualquier defor-mación cuántica y a cualquier variedad subyacente. Una de sus principales características es que,bajo deformaciones cuánticas, los sistemas de Lie se extienden a distribuciones involutivas genera-lizadas. Como consecuencia, un sistema de Lie deformado cuánticamente ya no tiene un álgebrade Vessiot–Guldberg Lie subyacente o un álgebra cuántica, sino que mantiene una estructura deálgebra de Poisson–Hopf que permite obtener, de manera explícita, las constantes del movimiento t -independientes a partir de los invariantes de Casimir deformados, que son potencialmente útilesen una construcción adicional de la noción generalizada de reglas de superposición. Ilustramoseste enfoque considerando la deformación cuántica no estándar de sl ( ) aplicada a sistemas de Lieconocidos, como el problema del oscilador o la ecuación de Milne–Pinney, así como varios tipos deecuaciones de Riccati. De esta manera, se obtienen sus análogos generalizados (deformados) quedan lugar, en particular, a un nuevo sistema de tipo oscilatorio con una frecuencia dependiente deltiempo y una masa dependiente de la posición. Basándonos en este procedimiento, desarrolladorecientemente en [13], se presentan de modo unificado las deformaciones no equivalentes de los sis-temas de Lie–Hamilton en el plano real con un álgebra de Lie de Vessiot–Guldberg isomorfa a sl ( ) .Esto, en particular, nos permite definir una noción de sistemas de Poisson–Hopf en dependencia deuna familia parametrizada de representaciones de álgebras de Poisson [14]. Este enfoque se ilustraexplícitamente aplicándolo a las tres clases no difeomórficas de los sistemas sl ( ) Lie–Hamilton. Sedan las constantes de movimiento independientes de t . Nuestros métodos se pueden emplear paragenerar otros sistemas de Lie–Hamilton y sus deformaciones para otras álgebras de Lie de Vessiot–Guldberg y sus deformaciones. Análogamente, se estudian los sistemas deformados a partir de lossistemas de Lie–Hamilton basados en el álgebra del oscilador h , vista como subálgebra del álgebrade Lie h , la llamada 2-photon algebra. Como aplicación adicional, se estudian modelos epidemi-ológicos del tipo SISf obtenidos como deformación cuántica de sistemas de Lie–Hamilton basadosen el álgebra resoluble en b , pero vista como subálgebra de sl ( ) . Palabras clave: sistema de Lie, álgebra de Poisson–Hopf, coálgebra de Poisson, deformación cuán-tica.iii
Breve extracto de la tesis
Introducción y objetivos
Desde su formulación original por Lie [100], los sistemas no autónomos de primer orden de ecua-ciones diferenciales ordinarias que admiten una regla de superposición no lineal, los llamados sis-temas de Lie, se han estudiado extensamente (ver [38, 43, 47, 154] y sus referencias). El teoremade Lie [100] establece que todo sistema de ecuaciones diferenciales de primer orden es un sistemade Lie si, y solo si, puede describirse como una curva en un álgebra de Lie de dimensión finita decampos vectoriales, la denominada álgebra de Lie de Vessiot–Guldberg. Aunque ser un sistema deLie es más una excepción que una regla [41], se ha demostrado que los sistemas de Lie son de graninterés dentro de las aplicaciones físicas y matemáticas (ver [47] y sus referencias). Los sistemas deLie que admiten un álgebra de Lie de Vessiot–Guldberg de campos vectoriales hamiltonianos, enrelación con una estructura de Poisson, los sistemas de Lie–Hamilton, han encontrado incluso másaplicaciones que los sistemas de Lie estándar sin esta estructura geométrica asociada [9, 15, 49]. Lossistemas de Lie–Hamilton admiten un álgebra de Lie de dimensión finita adicional de funcioneshamiltonianas, un álgebra de Lie–Hamilton, que permite la determinación algebraica de reglas desuperposición y constantes del movimiento [29].La mayoría de los enfoques de los sistemas de Lie se basan en la teoría de las álgebras de Lie ylos grupos de Lie [119]. Sin embargo, el éxito de los grupos cuánticos [51, 105] y el formalismode coálgebra dentro del análisis de sistemas superintegrables [8, 26], y el hecho de que las álgebrascuánticas aparezcan como deformaciones de las álgebras de Lie sugirieron la posibilidad de ampliarla noción y técnicas de los sistemas de Lie–Hamilton más allá de la teoría de Lie. En esta memoriase propone un enfoque en esta dirección (capítulo 3, [13]), donde se da un método para construirsistemas de Lie–Hamilton deformados cuánticamente (sistemas LH en resumen) mediante el for-malismo de coálgebra y álgebras cuánticas. La idea subyacente es utilizar la teoría de los gruposcuánticos para deformar sistemas de Lie y sus estructuras geométricas asociadas. Más exáctamente,la deformación transforma un sistema LH con su álgebra de Lie de Vessiot–Guldberg en un sis-tema hamiltoniano cuya dinámica está determinada por un conjunto (finito) de generadores de unadistribución generalizada de Stefan–Sussmann, sección 3.1.3. Mientras tanto, el álgebra inicial deLie–Hamilton (álgebra LH en resumen) se identifica con un álgebra de Poisson–Hopf. Las estruc-turas deformadas permiten la construcción explícita de constantes de movimiento t -independientesmediante técnicas de álgebras cuánticas para el sistema deformado.Este trabajo tiene como objetivo ilustrar el enfoque introducido en [13, 14] para construir defor-maciones de los sistemas LH. Esto abarca, fundamentalmente, los siguientes objetivos: desarrollarel formalismo basado en deformaciones de estructuras de Poisson–Hopf, ya que estas estructuraspermiten una sistematización adicional que abarca los sistemas LH no equivalentes entre sí, corre-spondientes a las álgebras isomórficas de LH, y ofrecer un procedimiento general para la obtenciónde nuevos sistemas LH.Específicamente, se aporta un algoritmo para la consecución de deformaciones sistemas LH, lo cualpone de manifiesto la importancia del formalismo previamente desarrollado. Además, mostramosque las deformaciones de Poisson–Hopf de los sistemas LH basadas en un álgebra LH isomorfas a sl ( ) (teorema 4.1), se pueden describir genéricamente, facilitando así las funciones hamiltonianasdeformadas y los campos vectoriales hamiltonianos deformados asociados; una vez estudiado elsistema no deformado.Igualmente, se proporciona un nuevo método para construir sistemas LH con un álgebra LH iso-morfa a un álgebra g de Lie fija, sección 4.3. Nuestro enfoque se basa en el uso de la foliaciónsimpléctica en g inducida por el corchete de Kirillov–Konstant–Souriau, teorema 4.3. Como casoparticular, se muestra explícitamente cómo nuestro procedimiento explica la existencia de tres tiposde sistemas LH en el plano relacionado con un álgebra LH isomorfa a sl ( ) . Esto se debe al hechode que cada uno de los tres tipos diferentes corresponde a uno de los tres tipos de hojas simplécticasen sl ( ) . Análogamente, se puede generar el único tipo de sistemas LH en el plano admitiendo unálgebra de Lie de Vessiot–Guldberg isomorfa a so ( ) .xNuestra sistematización nos permite dar directamente el sistema deformado de Poisson–Hopf apartir de la clasificación de sistemas LH [9, 29], sugiriendo además una noción de sistemas Poisson–Hopf Lie basados en una familia z -parametrizada de morfismos del álgebra de Poisson (definición4.2 en la sección 4.2). Resultados más relevantes y estructura
Presentamos un procedimiento genérico que nos permite introducir la noción de deformación cuán-tica de sistemas LH, y basados en la noción de distribuciones involutivas en el sentido de Stefan–Sussman. La existencia de principios de superposición no lineal para sistemas no autónomos deprimer orden de ecuaciones diferenciales ordinarias constituye una propiedad estructural que surgenaturalmente del enfoque desde la teoría de grupos para las ecuaciones diferenciales iniciado porLie, en el contexto del desarrollo del programa geométrico basado en grupos de transformación, asícomo de la clasificación analítica de ecuaciones diferenciales desarrollada por Painlevé y Gambier,entre otros, dando lugar a lo que hoy se conoce como la teoría de los sistemas de Lie [100, 149].Recordamos que, más allá de los sistemas superintegrables [8, 26], las coálgebras se han aplicadorecientemente a deformaciones bi-hamiltonianas integrables de sistemas de Lie–Poisson [24] y adeformaciones integrables de los sistemas de Rössler y Lorenz [12]. Este trabajo propone una gene-ralización de los sistemas de Lie en esta línea. La idea es considerar sistemas de Lie deformadoscuánticamente que poseen una estructura de álgebra de Poisson–Hopf que reemplaza el álgebra deLie de Vessiot–Guldberg del sistema inicial, lo que nos permite una construcción de constantes delmovimiento t -independientes expresadas en términos de Casimires invariantes deformados.La estructura de la tesis es la siguiente. Los capítulos 1 y 2 están dedicados a la introducción delos principales aspectos de los sistemas LH y las álgebras de Poisson–Hopf, así como una sínte-sis de sus propiedades fundamentales. El formalismo general para construir deformaciones deltipo Poisson–Hopf de sistemas LH [13] se analiza con detalle en el capítulo 3. En él se presentaun procedimiento algorítmico esquematizado para determinar tanto las deformaciones como lasconstantes del movimiento.En el capítulo 4 se presentan la deformaciones (no estándar) de sistemas basados en el álgebra deLie simple sl ( ) . Un enfoque unificado de las deformaciones de los sistemas Poisson–Hopf Lie conun álgebra LH isomorfa a g se estudia en la sección 4.3.1. Dicho procedimiento se ilustra explíci-tamente, aplicándolo a las tres clases no difeomorfas de sistemas LH sl ( ) en el plano, obteniendoasí su correspondiente deformación. En el capítulo 5 se estudian deformaciones de ecuacionesdiferenciales relevantes. En primer lugar, se consideran las deformaciones de la ecuación de Milne–Pinney, de las cuales se derivan nuevos sistemas de tipo oscilatorio con una masa dependiente dela posición. Como segundo tipo relevante, se estudian las ecuaciones de Riccati, específicamentelas ecuaciones compleja y acoplada. En el capítulo 6 se considera el problema de las deformacionescuánticas de sistemas LH basados en el álgebra del oscilador h , con énfasis especial en su estruc-tura como subálgebra de la llamada 2-photon álgebra h . Esto nos permite deducir deformacionesdel oscilador armónico amortiguado (sección 6.3). El álgebra resoluble afín b , vista como subálge-bra de sl ( ) , se aplica en el capítulo 7 para proponer un nuevo modelo epidemiológico alternativo.Las técnicas desarrolladas con anterioridad se utilizan para construir y analizar un modelo SISfdeformado, del que se obtienen las constantes del movimiento. Finalmente, en las conclusiones re-sumimos los resultados obtenidos y comentamos vías futuras de trabajo o actualmente en procesode ejecución. Conclusiones
En este trabajo hemos propuesto una noción de deformación de sistemas LH basada en las álgebrasde Poisson–Hopf. Este enfoque difiere radicalmente de otros métodos empleados en la teoría desistemas de Lie [15, 41, 47, 49, 154], dado que las deformaciones no corresponden formalmente asistemas de Lie, sino a una noción extendida que precisa de una estructura de Hopf, de modo queel sistema sin deformar se obtiene por un paso al límite en el cual el parámetro de deformacióndesaparece. La introducción de una estructura de Poisson–Hopf permite una generalización deestos sistemas, en el sentido de que el álgebra finito-dimensional de Vessiot–Guldberg se reemplazapor una distribución involutiva en el sentido de Stefan–Sussman (capítulo 3).En el capítulo 4 se estudia el análogo de las deformaciones cuánticas de sl ( ) , estableciendo ex-plícitamente las constantes del movimiento para los sistemas deformados cuánticamente. Los tressistemas de Lie planos no equivalentes basados en el álgebra de Lie sl ( ) se describen de modounificado, lo que proporciona una bella interpretación geométrica de estos sistemas y sus corres-pondientes deformaciones cuánticas. El capítulo 5 está dedicado al análisis de sistemas específi-cos de ecuaciones diferenciales y su contrapartida deformada. Consideramos en primer lugar laecuación de Milne–Pinney, cuyas deformaciones nos proporcionan nuevos sistemas de tipo oscila-torio con la particularidad de que la masa de la partícula es dependiente de la posición, y donde seobtienen explícitamente las constantes del movimiento. Merece la pena observar que la deforma-ción estándar o de Drinfel’d–Jimbo de sl ( ) no lleva a un oscilador del tipo mencionado, ya que, eneste caso, la deformación viene descrita por sinhc ( zqp ) en lugar de sinhc ( zq ) . Esto se deduce de lacorrespondiente realización simpléctica dada en [14]. Este hecho justifica que hayamos escogido ladeformación no-estándar de sl ( ) , para obtener aplicaciones físicas. A pesar de ello, la deformaciónde Drinfel’d–Jimbo podría proporcionar información suplementaria para la ecuación de Milne–Pinney, dando lugar a sistemas no equivalentes a los estudiados en la memoria. En cualquier caso,el método nos sugiere un enfoque alternativo de los sistemas con masa no constante, para los cualeslos métodos clásicos son de limitado alcance. Un segundo tipo que se ha estudiado corresponde alas ecuaciones compleja y acoplada de Riccati, que se han analizado exhaustivamente en la lite-ratura. Para ellos se obtienen la versión deformada y las constantes del movimiento. Los resultadosprincipales de los capítulos 3-5 han sido publicados en [13] y [14]. En el capítulo 6 nos centramosen sistemas de tipo oscilatorio obtenidos a partir de los sistemas LH deformados basados en elálgebra de Lie h , vista como subálgebra de la llamada 2-photon algebra h . En particular, estas de-formaciones se obtienen como restricción de los correspondientes sistemas deformados para h . Unejemplo ilustrativo de este tipo de sistemas viene dado por el oscilador amortiguado deformado.Resta determinar un principio de superposición para tales sistemas, un problema actualmente enejecución. En el capítulo 7 se usa el álgebra resoluble afín b , vista como subálgebra de sl ( ) , paraobtener sistemas deformados aplicables en el contexto de los modelos epidemiológicos. Esta ideaconstituye una novedad, dado que los métodos empleados habitualmente en este contexto son denaturaleza estocástica. Los resultados de este capítulo has sido enviados recientemente para supublicación.Existe una plétora de posibilidades y aplicaciones que emergen del formalismo de deformacionesde Poisson–Hopf. Aunque los resultados han sido principalmente considerados en el plano, parael cual existe una clasificación explícita de sistemas LH [9, 15], el método es válido para variedadesarbitrarias y álgebras de Vessiot–Guldberg de dimensiones mayores. Un estudio sistemático deestos sistemas sin duda dará lugar a nuevas propiedades de los sistemas deformados que merecenser analizadas con detalle. En particular, las propiedades dinámicas de sistemas específicos deecuaciones diferenciales pueden estudiarse mediante estas técnicas, donde se espera que nuevaspropiedades sean descubiertas.En relación con la actual pandemia COVID-19, podemos preguntarnos si existe una descripción entérminos de los modelos SISf. Este modelo es una primera aproximación para procesos de infecciónprimarios, en los cuales se consideran dos tipos de estados en la población: los infectados y los sus-ceptibles de infección. El modelo no contempla la posibilidad de adquirir inmunidad. Parece quela COVID-19 está sujeta a ciertos tipos de inmunidad, aunque sólo hasta un treinta por ciento dela población. En este sentido, un modelo SIR que considera individuos inmunes no es un modeloapropiado para esta situación. Sería interesante tener un modelo que contemple individuos in-munes y no inmunes simultáneamente. Actualmente estamos buscando un modelo hamiltonianoestocástico que contemple estas variables.i Trabajos futuros
Una de las cuestiones principales a resolver es si el enfoque de Poisson–Hopf proporciona unmétodo efectivo para deducir un análogo deformado de los principios de superposición para sis-temas LH deformados. Asimismo, sería interesante saber si una tal descripción puede aplicarsesimultáneamente a sistemas no equivalentes, como una extrapolación de la noción de contraccióna los sistemas de Lie. Otros aspecto relevante es la posibilidad de obtener una descripción unifi-cada de estos sistemas a partir de ciertos sistemas "canónicos" fijos, lo que implicaría una primerasistematización de los sistemas LH desde una perspectiva más amplia que la de las álgebras de Liefinito-dimensionales. Algunas vías de trabajo futuro pueden resumirse como sigue:• En la clasificación de los sistemas LH en el plano juega un papel central la llamada 2-photonálgebra h , ya que es el álgebra de máxima dimensión que puede aparecer con las propiedadesde un álgebra LH. El estudio de sus deformaciones cuánticas es, por tanto, una cuestión fun-damental para completar el análisis de las deformaciones de los sistemas LH en el plano. Cabeseñalar que existen esencialmente dos posibilidades diferentes para estas deformaciones, de-pendiendo de la estructura de dos subálgebras prominentes, el álgebra h y sl ( ) , que danlugar a sistemas y deformaciones con diferentes propiedades. El primer caso, basado en el ál-gebra del oscilador h , ha sido parcialmente considerado en el capítulo 6. Sin embargo, restaaún obtener una regla de superposición efectiva, cuya implementación estamos analizandoactualmente. El análisis debe completarse identificando clases particulares de sistemas deecuaciones diferenciales que puedan deformarse mediante este procedimiento, y que puedaninterpretarse como perturbaciones del sistema inicial. El segundo caso, basado en la extensiónde los resultados obtenidos para sl ( ) al álgebra h , es estructuralmente muy distinto debidoa la naturaleza de la deformación cuántica. Esperamos que nuevos sistemas con propiedadesde interés surgan de este análisis. Desde el punto de vista de las aplicaciones, estos sistemastienen muchas propiedades interesantes, tales como nuevos sistemas del tipo Lotka–Volterrao sistemas de tipo oscilatorio con masas y frecuencias dependientes de la posición y el tiempo,pero cuya dinámica pueda caracterizarse mediante la existencia de un procedimiento para ladeterminación exacta de las constantes del movimiento y las reglas de superposición. Enanálisis completo de los sistemas LH deformados basados en el álgebra h está actualmenteen proceso, para ser enviado próximamente para su publicación.• Por otro lado, cabe observar que, actualmente, no existe una clasificación de los sistemas LHpara las dimensiones n ≥
3. Un problema de interés que surge en este contexto es analizar laposibilidad de generar nuevos sistemas LH, tanto clásicos como deformados, mediante la ex-tensión de los sistemas en el plano, en combinación con las proyecciones de las realizacionesde campos vectoriales. En este contexto, se sabe que las proyecciones de realizaciones delálgebra de Lie asociadas con una representación lineal dan lugar a realizaciones no lineales.Analizando la cuestión desde la perspectiva de las álgebras funcionales (hamiltonianas), esconcebible la existencia de estructuras simplécticas compatibles que dan lugar a sistemas LHen dimensiones superiores, así como una dependencia de dichas formas simplécticas. Crite-rios de este tipo pueden combinarse con deformaciones cuánticas de álgebras de Lie conoci-das, para obtener nuevas aplicaciones de estas en el contexto de ecuaciones diferenciales.• Como complemento al modelo SISf basado en b como subálgebra de sl ( ) , es razonable de-sarrollar el modelo correspondiente a la misma álgebra, pero vista como una subálgebra de h . Nuevamente, el carácter esencialmente distinto de la deformación nos lleva a sistemas conpropiedades muy diferentes, lo que sugiere comparar ambos modelos con detalle, analizandolas soluciones numéricas deducidas de ambos enfoques. Un primer paso en esta direcciónestá actualmente en proceso.• Desearíamos asimismo extender nuestro estudio a modelos epidemiológicos más complica-dos, aunque a primera vista no hayamos localizado nuevos sistemas de Lie, al menos en suforma usual. Sospechamos que la descripción hamiltoniana de los modelos compartimentalespueden asociarse a sistemas de Lie, como muestra el ejemplo desarrollado. En particular, debeanalizarse con más detalle como las soluciones del modelo deformado (7.47) permiten recu-perar las soluciones del modelo sin deformar, cuando el parámetro de deformación tiende acero. Se precisa un análisis más detallado para determinar si tales modelos integrables mode-lizan procesos distintos a los infecciosos. En particular, nos interesaría saber si es posibleii modelizar la dinámica subatómica mediante hamiltonianos deformados del tipo (7.46). Existeuna teoría estocástica de los sistemas de Lie desarrollada en [95] que podría ser otro punto departida para tratar estos sistemas. En el presente trabajo tuvimos la suerte de encontrar unateoría con fluctuaciones que coincidían con una expansión estocástica, pero esto es más unaexcepción que una regla. De hecho, parece que la forma más factible de proponer modelosestocásticos es utilizar la teoría estocástica de Lie en lugar de esperar un destello de suertecon las fluctuaciones. Como hemos dicho, encontrar soluciones particulares no es en absolutotrivial. La búsqueda analítica es una tarea muy atroz. Creemos que para ajustar solucionesparticulares en el principio de superposición, es posible que sea necesario calcular estas solu-ciones particulares numéricamente. En [124] se pueden idear algunos métodos numéricosespecíficos para soluciones particulares de sistemas de Lie.• Finalmente, a partir de la ecuación de Chebyshev, se ha demostrado que el punto de simetríasde Noether de esta ecuación se puede expresar para n arbitrario en términos de los polinomiosde Chebyshev T n ( x ) , U n ( x ) de primer y segundo tipo, respectivamente. Además, se ha ob-servado que la realización genérica del álgebra de simetría de puntos de Lie sl ( R ) puedeampliarse a ecuaciones diferenciales ordinarias de segundo orden lineales más generales y lassoluciones se pueden expresar en términos de funciones trigonométricas o hiperbólicas. Enparticular, los conmutadores de las simetrías de puntos genéricos muestran que varias de lasrelaciones algebraicas de las soluciones generales surgen realmente como consecuencia de lasimetría. Las mismas conclusiones son válidas para la estructura de la subálgebra de cincodimensiones de las simetrías de Noether. Se ha demostrado que la realización de los genera-dores de simetría sigue siendo válida para ecuaciones diferenciales de tipo hipergeométrico, loque nos permite obtener realizaciones de sl ( R ) en términos de funciones hipergeométricasen general y varios polinomios ortogonales en particular, como los polinomios de Chebyshevo Jacobi. Otro hecho notable que surge de este análisis es que los términos forzados son siem-pre independientes de las "velocidades" y (cid:48) , y (cid:48) . Esto es nuevamente una consecuencia de larealización genérica elegida, y la cuestión de si otras realizaciones genéricas en términos de lasolución general de la EDO (o sistema) permiten determinar términos forzosos que dependenexplícitamente de las derivadas, e incluso conducen a ecuaciones diferenciales autónomas(sistemas), surge de manera natural. En este contexto, sería deseable obtener una realizaciónde sl ( R ) que no solo permita describir genéricamente las simetrías de punto y Noether delos polinomios de Jacobi, sino que también se aplique a las ecuaciones diferenciales asociadasa la familias restantes de polinomios ortogonales, específicamente los polinomios de Laguerrey Hermite. Esto permitiría construir más ecuaciones y sistemas no lineales que posean unasubálgebra de simetrías de Noether, los generadores de las cuales se den en términos de estospolinomios ortogonales.iii Agradecimientos “Entre los pecados mayores que los hombres cometen, aunque algunos dicen que es la soberbia, yo digo que esel ser desagradecido, ateniéndome a lo que suele decirse: que de los desagradecidos está lleno el infierno. Estepecado, en cuanto me ha sido posible, he procurado yo huir desde el instante que tuve uso de razón, y si nopuedo pagar las buenas obras que me hacen con otras obras, pongo en su lugar los deseos de hacerlas, ycuando estos no bastan, las publico, porque quien dice y publica las buenas obras que recibe, también lasrecompensara con otras, si pudiera; porque por la mayor parte los que reciben son inferiores a los que dan, yasí es Dios sobre todos, porque es dador sobre todos, y no pueden corresponder las dádivas del hombre a las deDios con igualdad, por infinita distancia, y esta estrecheza y cortedad en cierto modo la suple elagradecimiento.”
M. de Cervantes,
El ingenioso hidalgo don Quijote de la Mancha
Esta tesis ha sido realizada gracias al contrato predoctoral (CT45/15-CT46/15) otrogado por la Uni-versidad Complutense de Madrid, a la Universidad de Burgos por su financiación durante el primeraño y a los diversos proyectos de investigación: MTM2016-79422-P (AEI/FEDER, EU) y MTM2013-48320- P.Ha sido un duro y satisfactorio período de aprendizaje, por lo que me gustaría agradecer a todaslas personas que enriquecieron de una forma u otra esta etapa de mi vida.Antes de nada, deseo agradecer a mis dos directores de tesis por su inmensa generosidad e in-agotable paciencia, no habría llegado tan lejos. He aprendido más de lo que imaginaba junto avosotros tanto en el plano personal como en el profesional. Por la confianza y dedicación que mehabéis brindado cada día.Quiero agradecer a la Universidad de Burgos, en especial a Ángel Ballesteros por su paciencia,sus siempre acertadas palabras y su buen hacer, pues me otorgaron una gran confianza para con-tinuar mi camino; así como a todo el departamento de Física por hacerme sentir en casa, y comoolvidarme del "Patillas" y del variopinto elenco de personas que lo frecuentan. A la Universidadde Varsovia por su gran acogida durante mi estancia de la mano de Javier de Lucas, quién tanamablemente compartió conocimientos y numerosas discusiones que sin duda han ayudado a miformación matemática. No tengo más que palabras buenas, llenas de cariño y grandes recuerdos,para describir a las personas que componen la Red de Geometría, Mecánica y Control y los mo-mentos que me han regalado, solo mencionaré en representación de todos los componentes de estegrupo a Cristina por su amistad y apoyo en todo momento, siento mucho no dar más nombres enpero mi mala memoria no haría justicia a los omitidos, ha sido una suerte haberos conocido. A laUniversidad Complutense de Madrid por mi formación, y por haber sido mi hogar durante tan-tos años, sin olvidar a las grandes personas que componen el servicio de limpieza y cafetería porapoyarnos y hacernos la vida más fácil. A José M. Ancochea quien me ofreció la oportunidad dedisfrutar la experiencia de compartir asignatura. En especial a las personas que empezaron siendocompañeros y compañeras de clase y continuaron siendo grandes amistades como David, Patri,Javi, Judit, Irene y Amanda; así como los que dejaron su huella más tarde, gracias Jorge, Luismi,Rober, Luis, Paco, Pedro, Juan... Y tantas otras personas que de diferentes maneras me ayudaron arecorrer esta senda.ivMis amigos de toda la vida, como agradecer a Rober, Alex, Fran, Carlos, Bea, Cris y compañia, todolo que me habéis enseñado sobre la vida y lo que hemos compartido juntos... no tiene precio elapoyo que me ofrecéis cada día, recordandome siempre que es mejor caer que no intentar volar.Esas conversaciones de madrugada, viviendo nuestros sueños aprovechando aquellas noches deinsomnio. Al dojo de Aikido Miguel Hernández por aportarme la serenidad y el equilibrio mentalque necesitaba. Gracias, Alba, por mostrarme que incluso siguiendo los senderos oscuros y salvajesentre las rocas florece el brillo de un nuevo día.A mi familia, que merecen una mención especial pues sois un apoyo y un ejemplo a seguir, graciasa mi madre, a mi hermano y a Yuki por vuestra paciencia, compañia y comprensión siempre seréislos pilares más importantes en el laberinto de la vida.Por último, por haber sido el precursor de esta aventura en la que me embarqué al empezar el grado,gracias a Paco Conejero, desde que te conocí supe que debía seguir este camino. Te encargaste derecordarmelo y animarme en cada taza de café.Muchas gracias a todas las personas que se cruzaron en mi vida, espero no ofender a las que noaparecen de manera explícita pues no es por falta de cariño o agradecimiento. Siempre podéisllamarme y os invitaré a una cerveza como compensación. Eternamente agradecido,Edu.v
Contents
Declaration of Authorship iiiAbstract/Resumen viBreve extracto de la tesis viiiAgradecimientos xiiiIntroduction 1
I Formalism of Poisson–Hopf Algebra Deformations 5
II Applications to the Theory of Quantum Poisson–Hopf Algebras 33 sl ( ) -related Systems 35 sl ( ) . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Poisson–Hopf deformations of sl ( ) Lie–Hamilton systems . . . . . . . . . . . . . . . 374.3 Deformations of sl ( ) Lie–Hamilton systems in R . . . . . . . . . . . . . . . . . . . . 41 h h . . . . . . . . . . . . . . . . . . . . 556.2 Poisson–Hopf deformation of h Lie–Hamilton systems . . . . . . . . . . . . . . . . . 566.3 Poisson–Hopf deformation of h Lie–Hamilton systems from h . . . . . . . . . . . . 57 III Appendix 81
A Lie algebras: Elementary properties 83
A.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
B The hyperbolic sinc function 85
B.1 The hyperbolic sinc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
C Orthogonal systems and symmetries of ODEs 87
C.1 Point symmetries of ordinary differential equations . . . . . . . . . . . . . . . . . . . 87C.2 Functional realization of sl ( R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90C.3 Orthogonal functions as solutions to the ODE (C.27) . . . . . . . . . . . . . . . . . . . 96C.4 Non-linear deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References 109 vii
List of Figures sl ∗ ( ) given by the surfaces with constantvalue of the Casimir for the Poisson structure in sl ∗ ( ) (left) and its deformation(right). Such submanifolds are symplectic submanifolds where the Poisson bivectors Λ and Λ z admit a canonical form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1 The position-dependent mass (5.18) for different values of the deformation parameter z . . . . 505.2 The deformed oscillator potential (5.19) for different values of the deformation parameter z . . 507.1 The first particular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.2 The second particular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.3 The second particular solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.4 Superposition rule for exact solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.5 Superposition rule for linear approximation . . . . . . . . . . . . . . . . . . . . . . . . 747.6 Superposition rule with two particular solutions . . . . . . . . . . . . . . . . . . . . . 75B.1 The hyperbolic sinc function versus the hyperbolic cosine function and the derivative of theformer versus the hyperbolic sine function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86C.1 y ( x ) for the non-linear ODE (C.117) and the linearizable ODE. . . . . . . . . . . . . . 104C.2 Solutions y ( x ) (dashed) and y ( x ) for the system (C.140)-(C.141). . . . . . . . . . . . 107C.3 Plane trajectory of the solutions ( y ( x ) , y ( x )) of system (C.140)-(C.141). . . . . . . . 108ix List of Tables sl ( ) . For each class, it is displayed, in this order, a basis ofvector fields X i , Hamiltonian functions h i , symplectic form ω , the constants of motion F and F ( ) as well as the corresponding specific LH systems. . . . . . . . . . . . . . . 424.2 Poisson–Hopf deformations of the three classes of sl ( ) -LH systems written in Table1. The symplectic form ω is the same given in Table 1 and F ≡ F z . . . . . . . . . . . . 44xi A mi madre y a mi hermano. . .
Introduction A Lie system is a nonautonomous system of first-order ordinary differential equations whose generalsolution can be written as a function (a so-called superposition rule ) depending on a certain numberof particular solutions and some significant constants [58, 101, 149]. Superposition rules constitutea structural property that emerges naturally from the group-theoretical approach to differentialequations initiated by Sophus Lie, Vessiot, and Guldberg, within the context of the development ofthe geometric program based on transformation groups, as well as from the analytic classification ofdifferential equations developed by Painlevé and Gambier, among others. Indeed, Lie proved thatevery Lie system can be described by a finite-dimensional Lie algebra of vector fields, a
Vessiot–Guldberg Lie algebra [101], and Vessiot used Lie groups to derive superposition rules [149].In the frame of physical problems, it was not until the 80’s when the power of superposition rulesand Lie systems was fully recognized [154], motivating a systematic analysis of their applicationsin classical dynamics and their potential generalization to quantum systems (see [43, 44, 47, 154]and references therein).Although Lie systems, as well as their refinements and generalizations, represent a valuable aux-iliary tool in the integrability study of physical systems, it seems surprising that the methods em-ployed have always remained within the limitations of Lie group and distribution theory, withoutconsidering other frameworks that have turned out to be a very successful approach to integrabil-ity, such as quantum groups and Poisson–Hopf algebras [2, 8, 26, 51, 105]. We recall that, beyondsuperintegrable systems [8, 26], Poisson coalgebras have been recently applied to integrable bi-Hamiltonian deformations of Lie–Poisson systems [24] and to integrable deformations of Rösslerand Lorenz systems [12].This work presents a novel generic procedure for the Poisson–Hopf algebra deformations of
Lie–Hamilton (LH) systems , namely Lie systems endowed with a Vessiot–Guldberg Lie algebra of Hamil-tonian vector fields with respecto to a Poisson structure [49]. LH systems possess also a finite-dimensional Lie algebra of functions, a so-called
LH algebra , governing their dynamics [49]. Theproposed approach is based on the Poisson coalgebra formalism extensively used in the context ofsuperintegrable systems, together with the notion of involutive distributions in the sense of Stefan–Sussman (see [120, 148, 152] for details). The main point will be to consider a Poisson–Hopf algebrastructure that replaces the LH algebra of the non-deformed LH system, thus allowing us to ob-tain an explicit construction of t -independent constants of the motion, that will be expressed interms of the deformed Casimir invariants. Moreover, the deformation will generally transform theVessiot–Guldberg Lie algebra of the LH system into a set of vector fields generating an integrabledistribution in the sense of Stefan–Sussman. Consequently, the deformed LH systems are not, ingeneral, Lie systems anymore.The novel approach is presented in the chapter 3, where the basic properties of LH systems andPoisson–Hopf algebras are reviewed (for details on the general theory of Lie and LH systems, chap-ters 1 and 2; the reader is referred to [9, 15, 29, 37, 38, 41, 42, 43, 44, 47, 49, 81, 85, 86, 101, 154]). Toillustrate this construction, we consider the Poisson–Hopf algebra analogue of the so-called non-standard quantum deformation of sl ( ) [17, 116, 137] together with its deformed Casimir invariant,chapter 4.Afterwards, relevant examples of deformed LH systems that can be extracted from this deformationare given. Firstly, the non-standard deformation of the Milney–Pinney equation is presented inpart II, where this deformation is shown to give rise to a new oscillator system with a position-dependent mass and a time-dependent frequency (chapter 5), whose (time-independent) constantsof the motion are also explicitly deduced. In secction 5.2 several deformed (complex and coupled)Riccati equations are obtained as a straightforward application of the formalism here presented. We List of Tables would like to stress that, albeit these applications are carried out on the plane, thus provinid us adeeper insight in the proposed formalism, the method here presented is by no means constraineddimensionally, and its range of applicability goes far beyond the particular cases considered here.Since its original formulation by Lie [101], Lie systems, have been studied extensively (see [38, 43,47, 84, 120, 149, 154] and references therein). The Lie theorem [44, 101] states that every system offirst-order differential equations is a Lie system if and only if it can be described as a curve in afinite-dimensional Lie algebra of vector fields, referred to as
Vessiot–Guldberg Lie algebra .Although being a Lie system is rather an exception than a rule [42], Lie systems have been shown tobe of great interest within physical and mathematical applications (see [47] and references therein).Surprisingly, Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fieldsrelative to a Poisson structure, the
Lie–Hamilton systems , have found even more applications thanstandard Lie systems with no associated geometric structure [9, 15, 29, 49]. Lie–Hamilton systemsadmit an additional finite-dimensional Lie algebra of Hamiltonian functions, a
Lie–Hamilton algebra ,that allows us to deduce an algebraic determination of superposition rules and constants of themotion of the system [29].Apart from the theory of quasi-Lie systems [42] and superposition rules for nonlinear operators,most approaches to Lie systems rely strongly on the theory of Lie algebras and Lie groups. How-ever, the success of quantum groups [51, 105] and the coalgebra formalism within the analysis ofsuperintegrable systems [8, 26], and the fact that quantum algebras appear as deformations of Liealgebras, suggested the possibility of extending the notion and techniques of Lie–Hamilton systemsbeyond the range of application of the Lie theory. An approach in this direction was recently pro-posed in [14], where a method to construct quantum deformed Lie–Hamilton systems (LH systemsin short) by means of the coalgebra formalism and quantum algebras was given.The underlying idea is to use the theory of quantum groups to deform Lie systems and their asso-ciated structures. More exactly, the deformation transforms a LH system with its Vessiot–GuldbergLie algebra into a Hamiltonian system whose dynamics is determined by a set of generators of aSteffan–Sussmann distribution. Meanwhile, the initial Lie–Hamilton algebra (LH algebra in short)is mapped into a Poisson–Hopf algebra. The deformed structures allow for the explicit constructionof t -independent constants of the motion through quantum algebra techniques for the deformedsystem.We illustrate how the approach introduced in [14] to construct deformations of LH systems viaPoisson–Hopf structures allows a further systematization that encompasses the nonequivalent LHsystems corresponding to isomorphic LH algebras. Specifically, we show that Poisson–Hopf de-formations of LH systems based on a LH algebra isomorphic to sl ( ) can be described generically,hence providing the deformed Hamiltonian functions and the corresponding deformed Hamilto-nian vector fields, once the corresponding counterpart of the non-deformed system is known. Thisprovides a new method to construct LH systems with a LH algebra isomorphic to a fixed Lie algebra g . The approach is based on the symplectic foliation in g ∗ induced by the Kirillov–Konstant–Sourioubracket on g . As a particular case, it is explicitly shown how our procedure explains the existenceof three types of LH systems on the plane related to a LH algebra isomorphic to sl ( ) . This is dueto the fact that each one of the three different types corresponds to one of the three types of sym-plectic leaves in sl ∗ ( ) . In analogy, one can generate the only LH system on the plane admitting aVessiot–Guldberg Lie algebra isomorphic to so ( ) . This systematization enables us to give directlythe Poisson–Hopf deformed system from the classification of LH systems [9, 29], further suggestinga notion of Poisson–Hopf Lie systems based on a z -parameterized family of Poisson algebra mor-phisms. Our methods seem to be extensible to study also LH systems and their deformations onother more general manifolds.The structure of the thesis goes as follows. Chapters 1 and 2 are devoted to review the main aspectsof LH systems and Poisson–Hopf algebras. The general procedure to construct Poisson–Hopf alge-bra deformations of LH systems [14], and other properties of the underlying formalism are givenin Chapter 3. In 4, the (non-standard) Poisson–Hopf algebra deformation of LH systems based onthe simple Lie algebra sl ( ) are analyzed in detail, while the unified approach to deformations ofPoisson–Hopf Lie systems with a LH algebra isomorphic to a fixed Lie algebra g is treated in Sec-tion 4.3.1. The procedure is explicitly illustrated, considering the three non-diffeomorphic classesof sl ( ) -LH systems on the plane, from which the corresponding deformations are described in ist of Tables h is considered, with special emphasis of its structure as a subalgebra ofthe so-called 2-photon algebra h . This in particular leads to quantum deformations of the dampedharmonic oscillator. In Chapter 7 we use the solvable Lie subalgebra b of sl ( ) to propose a newand alternative epidemiological model. The techniques discussed in previous chapters are appliedto analyze a defomed SISf model, which in particular is obtained by restriction of the corresponding sl ( ) -deformed system, and for which the constants of the motion are explicitly constructed. Finally,in the Conclusions we summarize the results and outline some future work to be accomplished oralready in progress. Part I
Formalism of Poisson–Hopf AlgebraDeformations
Lie systems, besides their undeniable interest within Geometry, play a relevant role in many appli-cations in Biology, Cosmology, Control Theory, Quantum Mechanics, among other disciplines. Aspecially interesting case is when a systems of first-order ordinary differential equations, which isthe prototype of Lie system, can be endowed with a compatible symplectic structure, leading to thenotion of Lie–Hamilton systems, of special interest within the frame of Classical Mechanics [58, 71].Lie–Hamilton systems and their fundamental properties are conveniently described in terms of a t -dependent vector field that describes the dynamics. An illustrative example for this type of systemsis given by the second-order Riccati equation. General properties of Lie systems, as well as additional geometrical applications, can be e.g. foundin [47, 152].
Definition 1.1.
A superposition rule for a system X defined on an n-dimensional manifold M is a map Φ : M k × M −→ Msuch that x ( t ) : = Φ ( x ( ) ( t ) , . . . , x ( k ) ( t ) ; λ ) is a general solution of the system X , where x i ( t ) are particularsolutions and λ is a point of the manifold M, corresponding to the initial condition of the Cauchy problem. Let M be an n -dimensional manifold and let π i be the projections π : TM −→ M with π ( x , v ) : = x and π : R × M −→ M with π ( t , x ) : = x . A smooth map X : Ω ⊆ R × M → TM , where Ω is anopen subset of R × M , is called a t-dependent vector field if the diagram Ω X (cid:47) (cid:47) π (cid:33) (cid:33) TM π (cid:15) (cid:15) M is commutative, i.e. π ◦ X = π in Ω . We observe that defining Ω t : = { x ∈ M / ( t , x ) ∈ Ω } foreach t ∈ R , Ω t is not empty and recovers the usual notion of vector field, denoted by X t . The notionof t -dependent vector field is very useful in the geometric theory of Lie systems and will play arelevant role in the formalism that will be developed. Remark.
It follows that if X is a t -dependent vector field, then it is equivalent to a linear morphism X : C ∞ ( M ) → C ∞ ( R × M ) defined as X ( f )( t , x ) : = ( X t f )( x ) , for all ( t , x ) ∈ R × M , such that itsatisfies the Leibniz rule in C ∞ ( M ) at each point of R × M . Chapter 1. Lie Systems and Poisson–Hopf Algebras
Let (cid:101) X be a vector field over R × M such that ι (cid:101) X dt = and ( (cid:101) X π ∗ f )( t , x ) = X ( f )( t , x ) . If the t -dependent vector field X is given by X = n ∑ j = X j ( t , x ) ∂∂ x j , (1.1)in local coordinates, then (cid:101) X has the expression (cid:101) X = ∂∂ t + n ∑ j = X j ( t , x ) ∂∂ x j ,called the autonomization of X . An integral curve of the t -dependent vector field X is a map γ : R → R × M such that π ◦ γ , where γ corresponds to an integral curve of (cid:101) X . It follows in particular thatfor any t with γ ( t ) = ( t , x ( t )) the components of x ( t ) satisfy the following first order system: dx j dt = X j ( t , x ) , i =
1, . . . n . (1.2)This is the so-called associated system of X . It is straightforward to verify that an arbitrary systemof first-order ordinary differential equations of the type (1.2) determines a t -dependent vector field(1.1), the integral curves of which satisfy the system. This establishes a one-to-one correspondencebetween systems of type (1.2) and t -dependent vector fields.If A is a family of vector fields on an n -dimensional manifold M , then Lie ( A ) denotes the Lie algebraspanned by the vector fields and their successive commutators A , [ A , A ] , [ A [ A , A ]] , [ A , [ A , [ A , A ]]] , . . .where [ A , B ] is a shorthand notation for the brackets { [ X , Y ] / X ∈ A and Y ∈ B} . It follows fromthe construction that the Lie algebra Lie ( A ) is the smallest Lie algebra of vector fields containing theset A . The main result concerning the classical theory of Lie-systems is given by the Lie–Schefferstheorem (for more details see [101]). Theorem 1.2 (Lie–Scheffers Theorem) . A system of first-order ordinary differential equations (1.2) admitsa superposition rule if and only if there exist smooth functions β j ( t ) such that associated t-dependent vectorfield has the form X ( t , x ) = (cid:96) ∑ j = β j X j ( x ) , (1.3) and such that the vector fields X j (in a manifold M) span an (cid:96) -dimensional real Lie algebra V X : = Lie ( { X j / j =
1, . . . , (cid:96) } ) . The Lie algebra of vector fields V X is usually called a Vessiot–Guldberg Lie algebra , where in additionthe following numerical constraint must be satisfied: dim ( V X ) ≤ m · n = dim ( M ) . (1.4)The scalar m corresponds to the number of particular solutions of the system that are required forestablishing a superposition rule. The relation (1.4) is also known as the Lie condition . It shouldbe emphasized that a given system may admit different superposition rules, as a Vessiot–Guldbergalgebra is not an invariant of the system, as happens e.g. with the Lie algebra of Lie-point sym-metries. Thus, a specific system may admit nonisomorphic Vessiot–Guldberg algebras, and hencedifferent superposition rules.
Definition 1.3.
A first-order system of differential equations that admits a superposition rule is a Lie system. Let M be a smooth manifold, X ∈ X ( M ) and ω ∈ Ω p ( M ) , then ι X ω is the contraction of a differential form ω withrespect to the vector field X . .1. Lie systems Example 1.4.
Let X be a t -dependent vector field over R that expressed in local coordinates is givenby X = − t X + X + η X , (1.5)where X = y ∂∂ y , X = y ∂∂ x , X = x ∂∂ y , X = x ∂∂ x + y ∂∂ y . (1.6)These vector fields span a Vessiot–Guldberg Lie algebra V X isomorphic to gl ( ) , with domain R x (cid:54) = (for more details see Appendix B) and the associated system takes the form dxdt = y , dydt = − t y + η x . (1.7)Hence, this system is a Lie system. Definition 1.5.
A Lie system X is, furthermore, a Lie–Hamilton system [9, 15, 29, 47, 49, 81] if it admits aVessiot–Guldberg Lie algebra V of Hamiltonian vector fields relative to a Poisson structure. This amounts tothe existence, around each generic point of M, of a symplectic form, ω , such that: L X i ω = for a basis X , . . . , X (cid:96) of V (cf. [9]). Then each vector field X i admits a Hamiltonian function h i given by therule: ι X i ω = d h i , (1.9) where ι X i ω stands for the contraction of the vector field X i with the symplectic form ω . Since ω is non-degenerate, every function h induces a unique associated Hamiltonian vector field X h . This fact gives rise to a Poisson bracket on C ∞ ( M ) given by {· , ·} ω : C ∞ ( M ) × C ∞ ( M ) (cid:51) ( f , f ) (cid:55)→ X f f ∈ C ∞ ( M ) , (1.10)turning ( C ∞ ( M ) , {· , ·} ω ) into a Lie algebra. The space Ham ( ω ) of Hamiltonian vector fields on M relative to ω is also a Lie algebra relative to the commutator of vector fields. Moreover, we have thefollowing exact sequence of Lie algebras morphisms (see [148])0 (cid:44) → R (cid:44) → ( C ∞ ( M ) , {· , ·} ω ) φ −→ ( Ham ( ω ) , [ · , · ]) π −→
0, (1.11)where π is the projection onto 0 and φ maps each f ∈ C ∞ ( M ) into the Hamiltonian vector field X − f .In view of the sequence (1.11), the Hamiltonian functions h , . . . , h (cid:96) and their successive Lie bracketswith respect to (1.10) span a finite-dimensional Lie algebra of functions contained in φ − ( V ) . ThisLie algebra is called a Lie–Hamilton (LH) algebra H ω of X (see [49, 81] and references therein).Let X be a system on an n -dimensional manifold M . A function f ∈ C ∞ ( TM ) is called a constant ofthe motion of the system X if it is a first integral of the vector field (cid:101) X , i.e., (cid:101) X f = The space of constant of the motion for a system form a K − algebra. Chapter 1. Lie Systems and Poisson–Hopf Algebras
Damped harmonic oscillator
Consider a t -dependent one-dimensional damped harmonic oscillator of the form dxdt = a ( t ) x + b ( t ) p + f ( t ) , dpdt = − a ( t ) p − c ( t ) x − d ( t ) , ( x , p ) ∈ T ∗ x R , (1.12)for arbitrary t -dependent functions a ( t ) , b ( t ) , c ( t ) , d ( t ) , f ( t ) . The system (1.12) is associated with the t -dependent vector field X t = f ( t ) X − d ( t ) X + a ( t ) X + b ( t ) X − c ( t ) X ,where X = ∂∂ x , X = ∂∂ p , X = x ∂∂ x − p ∂∂ p , X = p ∂∂ x , X = x ∂∂ p ,are such that (cid:104) X , X (cid:105) (cid:39) R and (cid:104) X , X , X (cid:105) (cid:39) sl ( ) . Moreover (cid:104) X , . . . , X (cid:105) (cid:39) sl ( R ) (cid:110) R and X becomes a Lie system related to a Vessiot–Guldberg Lie algebra V do isomorphic to sl ( R ) (cid:110) R .It can be easily shown that V do consists of Hamiltonian vector fields with respect to the symplec-tic form ω = d x ∧ d p on T ∗ R . In fact, the Hamiltonian functions associated to the vector fields X , . . . , X are given by h = p , h = − x , h = xp , h = p , h = − x ,respectively. The functions h , . . . , h along with h span a Lie algebra H ω (cid:39) h with respect to thestandard Poisson bracket on T ∗ R .The constants of the motion for the damped harmonic oscillator equations can be obtained by ap-plying the coalgebra formalism introduced in [15]. Using the Casimir invariants of the underlyingLie algebra, it follows that these constants of the motion of the Lie system (1.12) are given by [29] F ( ) = F ( ) = F ( ) = (cid:16) ( x ( ) − x ( ) ) p ( ) + ( x ( ) − x ( ) ) p ( ) + ( x ( ) − x ( ) ) p ( ) (cid:17) .By permutation of the indices corresponding to the variables of the non-trivial invariant F ( ) , wecan find additional constants of the motion: F ( ) = (cid:16) ( x ( ) − x ( ) ) p ( ) + ( x ( ) − x ( ) ) p ( ) + ( x ( ) − x ( ) ) p ( ) (cid:17) , F ( ) = (cid:16) ( x ( ) − x ( ) ) p ( ) + ( x ( ) − x ( ) ) p ( ) + ( x ( ) − x ( ) ) p ( ) (cid:17) .In order to derive a superposition rule, we just need to obtain the value of p ( ) from the equation k = F ( ) , where k is a real constant; and then plug this value into the equation k = F ( ) toobtain [29] x ( ) = x ( ) + ( x ( ) − x ( ) ) √ k + ( x ( ) − x ( ) ) √ k (cid:113) F ( ) , p ( ) = √ k x ( ) − x ( ) + p ( ) + √ k ( p ( ) − p ( ) ) (cid:113) F ( ) + √ k ( p ( ) − p ( ) )( x ( ) − x ( ) ) (cid:113) F ( ) ( x ( ) − x ( ) ) .Clearly, the superposition rule obtained above is merely one among many possible superpositionrules, that could be derived considering other choices of the constants of the motion. .1. Lie systems A second-order Riccati equation in Hamiltonian form
Another relevant application of Lie–Hamilton systems is given by the class of Riccati equations,that have been analyzed in detail in [49]. The most general class of second-order Riccati equationsis given by the family of second-order differential equations of the form d xdt + ( f ( t ) + f ( t ) x ) dxdt + c ( t ) + c ( t ) x + c ( t ) x + c ( t ) x =
0, (1.13)with f ( t ) = (cid:113) c ( t ) , f ( t ) = c ( t ) (cid:112) c ( t ) − c ( t ) dc dt ( t ) , c ( t ) (cid:54) = t -dependentnon-natural regular Lagrangian of the form L ( t , x , v ) = v + U ( t , x ) ,with U ( t , x ) = a ( t ) + a ( t ) x + a ( t ) x and a ( t ) , a ( t ) , a ( t ) being certain functions related to the t -dependent coefficients of (1.13), see [53]. Therefore, p = ∂ L ∂ v = − ( v + U ( t , x )) , (1.14)and the image of the Legendre transform F L : ( t , x , v ) ∈ W ⊂ R × T R (cid:55)→ ( t , x , p ) ∈ R × T ∗ R ,where W = { ( t , x , v ) ∈ R × T R | v + U ( t , x ) (cid:54) = } , is the open submanifold R × O where O = { ( x , p ) ∈ T ∗ R | p < } . The Legendre transform is not injective, as ( t , x , p ) = F L ( t , x , v ) for v = ± √− p − U ( t , x ) . Nevertheless, it can become so by restricting it to the open set W + = { ( t , x , v ) ∈ R × T R | v + U ( t , x ) > } . In such a case, v = √− p − U ( t , x ) and we can define over R × O the t -dependent Hamiltonian h ( t , x , p ) = p (cid:18) √− p − U ( t , x ) (cid:19) − (cid:112) − p = − (cid:112) − p − p U ( t , x ) .Its Hamilton equations read dxdt = ∂ h ∂ p = √− p − U ( t , x ) = √− p − a ( t ) − a ( t ) x − a ( t ) x , dpdt = − ∂ h ∂ x = p ∂ U ∂ x ( t , x ) = p ( a ( t ) + a ( t ) x ) . (1.15)Since the general solution x ( t ) of every second-order Riccati equation (1.14) can be recovered fromthe general solution ( x ( t ) , p ( t )) of its corresponding system (1.15), the analysis of the latter providesinformation about general solutions of second-order Riccati equations.The relevant point is to observe that the system (1.15) is actually a Lie system as shown in [48].Indeed, consider the vector fields over O of the form X = √− p ∂∂ x , X = ∂∂ x , X = x ∂∂ x − p ∂∂ p , X = x ∂∂ x − xp ∂∂ p , X = x √− p ∂∂ x + (cid:112) − p ∂∂ p . (1.16)2 Chapter 1. Lie Systems and Poisson–Hopf Algebras
Their non-vanishing commutation relations read [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , (1.17)and therefore span a five-dimensional Lie algebra V of vector fields. The t -dependent vector field X t associated to the system (1.15) is given by [48] X t = X − a ( t ) X − a ( t ) X − a ( t ) X . (1.18)In view of expressions (1.17) and (1.18), the system (1.15) is a Lie system. Note also that a similarresult would have been obtained by restricting the Legendre transform over the open set W − = { ( t , x , v ) ∈ R × T R | v + U ( t , x ) < } .Next, the vector fields (1.16) are additionally Hamiltonian vector fields relative to the Poisson bivec-tor Λ = ∂ / ∂ x ∧ ∂ / ∂ p on O . Indeed, they admit the Hamiltonian functions h = − (cid:112) − p , h = p , h = xp , h = x p , h = − x (cid:112) − p , (1.19)which span along with h = Λ (see [48] for details).The above action enables us to write the general solution ξ ( t ) of system (1.15) in the form ξ ( t ) = Φ ( g ( t ) , ξ ) , where ξ ∈ O and g ( t ) is the solution of the equation dgdt = − (cid:16) X R ( g ) − a ( t ) X R ( g ) − a ( t ) X R ( g ) − a ( t ) X R ( g ) (cid:17) , g ( ) = e , (1.20)on G , with the X R α being a family of right-invariant vector fields over G whose vectors X R α ( e ) ∈ T e G close on the same commutation relations as the X α (cf. [47]).Let us now apply to Lie systems (1.20) the reduction theory for Lie systems. Since T e G (cid:39) R ⊕ s sl ( R ) , a particular solution of a Lie system of the form (1.20) but over SL ( R ) , which amounts tointegrating (first-order) Riccati equations (cf. [47]), provides us with a transformation which mapssystem (1.20) into an easily integrable Lie system over R . In short, the explicit determination of thegeneral solution of a second-order Riccati equation reduces to solving Riccati equations.In order to determine a superposition rule for the system (1.15), it suffices to consider two commonfunctionally independent first-integrals for the diagonal prolongations (cid:101) X , (cid:101) X , (cid:101) X , (cid:101) X , (cid:101) X to a certainT ∗ R ( m + ) , provided that these prolongations to T ∗ R m are linearly independent at a generic point.In the present case, it can be easily verified that m =
4. The resulting first-integrals (see [29, 49, 133])are explicitly given by F = ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) + ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) + ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) , F = ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) + ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) + ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) , F = ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) + ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) + ( x ( ) − x ( ) ) (cid:112) p ( ) p ( ) .Note that given a family of solutions ( x ( i ) ( t ) , p ( i ) ( t )) , with i =
0, . . . , 3, of (1.15), then d ¯ F j / dt = (cid:101) X t F j = j =
0, 1, 2 and ¯ F j = F j ( x ( ) ( t ) , p ( ) ( t ) , . . . , x ( ) ( t ) , p ( ) ( t )) .In order to derive a superposition rule, it remains to obtain the value of p ( ) from the equation k = F , where k is a real constant. Proceeding along these lines and from the results given in [29],we get (cid:113) − p ( ) = k + ( x ( ) − x ( ) ) √ p ( ) p ( ) ( x ( ) − x ( ) ) (cid:112) − p ( ) + ( x ( ) − x ( ) ) (cid:112) − p ( ) , .2. Poisson–Hopf algebras k = F to have x ( ) = k Γ ( x ( ) , p ( ) , x ( ) , p ( ) ) + k Γ ( x ( ) , p ( ) , x ( ) , p ( ) ) − F x ( ) (cid:112) − p ( ) k ( (cid:112) − p ( ) − (cid:112) − p ( ) ) + k ( (cid:112) − p ( ) − (cid:112) − p ( ) ) − (cid:112) − p ( ) F , p ( ) = − (cid:104) k / F ( (cid:113) − p ( ) − (cid:113) − p ( ) ) + k / F ( (cid:113) − p ( ) − (cid:113) − p ( ) ) + (cid:113) − p ( ) (cid:105) ,where Γ ( x ( i ) , p ( i ) , x ( j ) , p ( j ) ) = (cid:112) − p ( i ) x ( i ) − (cid:112) − p ( j ) x ( j ) . The above expressions give us a super-position rule Φ : ( x ( ) , p ( ) , x ( ) , p ( ) , x ( ) , p ( ) ; k , k ) ∈ T ∗ R × R (cid:55)→ ( x ( ) , p ( ) ) ∈ T ∗ R for thesystem (1.15). Finally, since every x ( i ) ( t ) is a particular solution for (1.13), the map Υ = τ ◦ Φ furnishes the general solution of second-order Riccati equations in terms of three generic particu-lar solutions x ( ) ( t ) , x ( ) ( t ) , x ( ) ( t ) of (1.13), the corresponding p ( ) ( t ) , p ( ) ( t ) , p ( ) ( t ) and two realconstants k , k . The Hopf structure [2, 112, 144] is introduced through relevant examples, such as the universalenveloping algebra of a Lie algebra. The Hopf algebra structure originally appeared in the contextof group cohomology, from which its use has been expanded to constitute nowadays an essentialtool for the algebraic analysis. The next important definition will be the notion of Poisson structureand its compatibility with the Hopf algebras. In this section all vector spaces are defined over C ,where V ⊗ W denotes the tensor product of the two vector spaces V and W . Definition 1.6.
An algebra is a pair ( A , m ) , where A is a linear space and m : A ⊗ A → A is a bilinearmap. Furthermore, ( A , m ) is an algebra with unit if there is an element in A such that m ( a ) = m ( a , 1 ) forall a ∈ A. The algebra is associative if for arbitrary x , y , z ∈ A the identity m ( m ( x , y ) , z ) = m ( x , m ( y , z )) holds. If the map m is the inner product on the tensor product, it is possible define the unit map η : C → A such that η ( ) = A . Then, the properties of algebra are m ◦ ( η ◦ id A ) = id A = m ◦ ( id A ◦ η ) m ◦ ( m ⊗ id A ) = id A = m ◦ ( id A ⊗ m ) . Definition 1.7.
A coassociative coalgebra with counit is a linear space A endowed with two linear maps: thecoproduct ∆ : A → A ⊗ A and the counit (cid:101) : A → C such that ( id ⊗ ∆ ) ◦ ∆ = ( ∆ ⊗ id ) ◦ ∆ , (1.21) ( id ⊗ (cid:101) ) ◦ ∆ = id = ( (cid:101) ⊗ id ) ◦ ∆ , (1.22) moreover, the coalgebra is cocommutative if the following diagram is commutativeA ∆ (cid:47) (cid:47) ∆ (cid:15) (cid:15) A ⊗ A id ⊗ ∆ (cid:15) (cid:15) A ⊗ A ∆ ⊗ id (cid:47) (cid:47) A ⊗ A ⊗ A .4 Chapter 1. Lie Systems and Poisson–Hopf Algebras
Coalgebras can be described essentially by a dualization process. More specifically, from the pointof view of the commutative diagrams, they are obtained reversing the direction of the correspond-ing maps.In view of this definitions, a bialgebra ( A , m , ∆ ) is a linear space where ( A , m ) is an algebra and ( A , ∆ ) is a coalgebra, such that it is verified ∆ ( xy ) = ∆ ( x ) ∆ ( y ) , ∆ ( ) = ⊗
1, (1.23) (cid:101) ( xy ) = (cid:101) ( x ) (cid:101) ( y ) for all x , y ∈ A , i.e. the coproduct and the counit are algebra homomorphisms.The existence of a compatible coproduct with the product in an algebra allows the representationof A over vector the space V ⊗ V (cid:48) , once the representations of A over V and V (cid:48) are known. Therepresentation theory can be framed in this context, the map D : A ⊗ V → V is a representation of A on V if it satisfies D ◦ ( id A ⊗ D ) = D ◦ ( m ⊗ id V ) , D ◦ ( η ⊗ id V ) = id V ,i.e. D is a representation consistent with the inner product and unit. Definition 1.8.
A bialgebra ( A , m , ∆ ) is a Hopf algebra if there exist a linear map γ : A → A, so-calledantipode, such that m ◦ ( id A ⊗ γ ) ◦ ∆ = η ◦ (cid:101) = m ◦ ( γ ⊗ id A ) ◦ ∆ . (1.24)Let ( A , m , ∆ ) be a bialgebra, if the antipode map exists, then it is unique. Hence, the Hopf algebrastructure compatible with the bialgebra is unique. Furthermore, the antipode can be defined as thelinear map such that the following diagram is commutative A ⊗ A id A ⊗ γ (cid:47) (cid:47) A ⊗ A m (cid:27) (cid:27) A ∆ (cid:27) (cid:27) (cid:101) (cid:47) (cid:47) ∆ (cid:67) (cid:67) K η (cid:47) (cid:47) AA ⊗ A γ ⊗ id A (cid:47) (cid:47) A ⊗ A m (cid:67) (cid:67) (1.25)where K is R or C . The following proposition puts together the most relevant properties of Hopfalgebras. Proposition 1.9.
Let ( A , m , ∆ ) be a Hopf algebra. Then for all x , y ∈ A it is verified γ ( ) = (cid:101) ( γ ( x )) = (cid:101) ( x ) , γ ( xy ) = γ ( y ) γ ( x ) , ( ∆ ◦ γ )( x ) = (( γ ⊗ γ ) ◦ σ ◦ ∆ )( x ) , where σ ( x ⊗ y ) = y ⊗ x is a permutation. Hence, γ is an antihomomorphism and anticohomomorphism.Moreover, the antipode map can be non-invertible. .2. Poisson–Hopf algebras Example 1.10 (Lie algebra) . [88] Let g be a Lie algebra, then the universal enveloping algebra of g , denoted U ( g ) , is a general associative algebra that is obtained through the following quotient U ( g ) = T ( g ) / I ,where I is an ideal spanned by { XY − YX − [ X , Y ] / X , Y ∈ g } and the tensor algebra of g , T ( g ) denotes the graded algebra (cid:76) k T k ( g ) . Given a generic element X ∈ g such that ∆ ( X ) = ⊗ X + X ⊗ ∆ ( ) = ⊗ (cid:101) ( X ) = (cid:101) ( ) = γ ( X ) = − X ,then we can extend the mappings to U ( g ) . This endowes the universal enveloping algebra with aHopf algebra structure . Remark (Friedrichs’ theorem) . Only the generators of a Lie algebra can be primitive elements of theuniversal enveloping algebra. For more details see [125].
Theorem 1.11 (Poincaré–Birkhoff–Witt) . [92] Let g be a Lie algebra and { X , . . . , X n } a basis of g , thenthe set { X k · · · X k n n / k j ∈ N } is a basis of the universal enveloping algebra U ( g ) . Definition 1.12.
A Poisson algebra is a vector space A over field K endowed with two bilinear maps:m : A ⊗ A → A , commutative, and the Poisson bracket {· , ·} : A ⊗ A → A , where ( A , m ) is an as-sociative algebra and ( A , {· , ·} ) is a Lie algebra. Then, the Poisson bracket obeys the Jacobi identity and it isantisymmetric. Let M be an n -dimensional manifold and consider its ring of smooth functions C ∞ ( M ) . Accordingto the last definition, a bivector Λ ∈ X ( M ) induces a Poisson bracket on C ∞ ( M ) and it is an anti-symmetric biderivation. If ( x , . . . , x n ) are a local coordinates on M , a bivector Λ turns out to be Λ = λ ij ( x ) ∂∂ x i ∧ ∂∂ x j , i < j , λ ij ∈ C ∞ ( M ) ,hence the bracket takes the form { f , g } : = Λ ( f , g ) = λ ij ( x ) ∂ f ∂ x i ∂ g ∂ x j , (1.26)for all f , g ∈ C ∞ ( M ) . Note also that, in a general context, the last condition is the vanishing of theSchouten—Nijenhuis bracket [[ Λ , Λ ]] = λ ij can be degenerate, thenthis Poisson structure exists in odd-dimensional manifolds.In this theoretical environment, a function C ∈ C ∞ ( M ) is a Casimir element of the Poisson algebra ( C ∞ ( M ) , Λ ) if it satisfies { C , f } = f ∈ C ∞ ( M ) , i.e. the Casimir element belongs to the center of the Poisson algebra. U ( g ) is an algebra together with a morphism of Lie algebras i : g → U ( g ) such that F : g → g (cid:48) is a morphism of Liealgebras. Then, there is a single morphism (cid:101) F : U ( g ) → g (cid:48) that verifies (cid:101) F ◦ i = F . The elements X ∈ g are called primitive elements of the Hopf algebra if they can be written as X ⊗ + ⊗ X . The Poisson bracket induces a derivation in the product m . Chapter 1. Lie Systems and Poisson–Hopf Algebras
Remark (Racah 1951) . It was first shown by Racah that for semisimple Lie algebras s , the numberof functionally independent Casimir invariants (operators) coincide with the the rank of s , i.e, thedimension of a Cartan subalgebra. Example 1.13. [14] Consider the Lie algebra sl of traceless real 2 × { e , e , e } with commutation relations [ e , e ] = e , [ e , e ] = e , [ e , e ] = e .The elements of sl can be considered as linear functions v , v , v on sl ∗ , respectively. In this case,the corresponding Poisson bracket is given by { v , v } = v , { v , v } = v , { v , v } = v .This amounts to the Poisson bivector Λ = v ∂∂ v ∧ ∂∂ v + v ∂∂ v ∧ ∂∂ v + v ∂∂ v ∧ ∂∂ v .The Poisson structure on g ∗ admits a Casimir given by the function C = v v − v .The surfaces S k , where C takes a constant value k , are one-sided hyperboloids for k >
0, two-sidedhyperboloids when k >
0, and cones for k =
0. The Poisson bivector on the neighbourhood of ageneric point of such surfaces reads Λ g = v ∂∂ v ∧ ∂∂ v ,in the coordinate system { v , v , C } . As the canonical Poisson bivector on R (cid:39) T ∗ R is given by Λ R = ∂∂ x ∧ ∂∂ y ,it turns out that Λ and Λ g locally describe the same Poisson bivector, but in different coordinates.Hence, there exits a Poisson algebra morphism φ : C ∞ ( S k ) → C ∞ ( R ) given by φ ( v ) = x , φ ( v / v ) = y , φ ( C z ) = k .Therefore, φ ( v ) = k / x + y x , where k is an arbitrary constant. Example 1.14.
Consider the Lie–Hamilton algebra iso ( ) = (cid:104) e , e , e , e (cid:105) , where { e , e , e , e } is abasis with commutation relations [ e , e ] = e , [ e , e ] = e , [ e , e ] = − e and e spans the center of iso ( ) . Considering the elements of the above basis as linear functions { v , v , v , v } on the dual to iso ( ) , the corresponding Poisson bivector reads Λ = v ∂∂ v ∧ ∂∂ v + v ∂∂ v ∧ ∂∂ v − v ∂∂ v ∧ ∂∂ v .This Poisson algebra admits two Casimirs C : = v , C : = v v − v − v which allow us to restrict the Poisson bracket to the surfaces S κ , κ where the Casimirs take constantvalues κ , κ . These are symplectic manifolds where the Poisson bivector Λ can be mapped into a .2. Poisson–Hopf algebras v , v , C , C leads to the Poisson bivector in theform Λ = ∂∂ ( v / v ) ∧ ∂∂ v .Hence, this leads to a representation of the Poisson bracket in terms of functions φ ( v / v ) = x , φ ( v ) = y , φ ( C ) = k , φ ( C ) = k .In consequence, a possible morphism of Poisson algebras is given by φ ( v ) = k , φ ( v ) = k x , φ ( v ) = y , φ ( v ) = k + k x + y k . Example 1.15.
Consider now the Lie algebra so ( ) = (cid:104) e , e , e (cid:105) , where the basis { e , e , e } satisfiesthe commutation relations [ e , e ] = e , [ e , e ] = e , [ e , e ] = e .Following the same methods sketched in the previous example, let us consider the Poisson algebra ( C ∞ ( so ( ) ∗ ) , {· , ·} ) . In this case, one obtains that the Poisson bivector associated with this Poissonmanifold reads Λ = v ∂∂ v ∧ ∂∂ v + v ∂∂ v ∧ ∂∂ v + v ∂∂ v ∧ ∂∂ v and C = v + v + v becomes a Casimir function for this Poisson algebra. Therefore, the symplectic foliation for theLie algebra is given by surfaces parametrized by the value of C . Moreover, the Poisson bivectorbecomes a symplectic structure on each leaf admitting a canonical form. This canonical form can beobtained by writing Λ in the coordinate system v , v , C , then Λ = (cid:113) k − v − v ∂∂ v ∧ ∂∂ v .To write the above in a canonical form, it is enough to introduce new variables v = r cos ϕ v = r sin ϕ . This is natural as the symplectic leaves are spheres. Consequently, Λ = √ k − r r ∂∂ r ∧ ∂∂ϕ = ∂∂ ( −√ k − r ) ∧ ∂∂ϕ .Hence, it is enough to define the Poisson morphism as φ ( − (cid:112) k − r ) = x , φ ( ϕ ) = y .Undoing the previous changes of coordinates, we obtain h = − (cid:112) k − x cos ( y ) , h = (cid:112) k − x sin ( y ) , h = x ,where h i : = φ ( v i ) for i =
1, 2, 3.
Remark (Schur’s lemma) . [138] Let g be a Lie algebra and φ : g → End ( V ) a representation of g . If this representation is irreducible, then φ can be extended to a representation of the universalenveloping algebra U ( g ) . In view of this result, any x in the center of U ( g ) has as its image φ ( x ) amultiple of the identity.8 Chapter 1. Lie Systems and Poisson–Hopf Algebras
Consider the Poisson algebras ( A , {· , ·} A ) and ( B , {· , ·} B ) . A linear map f : A → B is a homomor-phism of Poisson algebras if f ( xy ) = f ( x ) f ( y ) , f ( { x , y } ) = { f ( x ) , f ( y ) } ,for all x , y ∈ A . In this context, we define the Poisson structure on the tensor product A ⊗ B as { x ⊗ y , x ⊗ y } A ⊗ B : = { x ! , x } A ⊗ y y + x x ⊗ { y , y } B .Moreover, this Poisson bracket is the only Poisson structure such that, when it is projected ontoeither A or B , the original Poisson brackets are recovered [146]. Example 1.16 (Symmetric coalgebra) . The symmetric algebra S ( g ) of a (finite dimensional) Lie al-gebra g is the smallest commutative algebra containing g . The second tensor power g ⊗ g of theLie algebra is the space of real valued bilinear maps on the dual space. Recursively, the k th tensorpower g ⊗ k is the space of real valued k-linear maps. Taking the direct sum of the tensor powers ofall orders, we arrive at the tensor algebra T ( g ) of g . Here, the multiplication is T ( g ) × T ( g ) −→ T ( g ) , ( v , u ) (cid:55)→ v ⊗ u . (1.27)We consider a basis { x , . . . , x r } of the Lie algebra g . The space generated by the elements x i ⊗ x j − x j ⊗ x i (1.28)is an ideal, denoted by R . The quotient space T ( g ) / R is called a symmetric algebra and denotedby S ( g ) . The elements of S ( g ) can be regarded as polynomial functions on g ∗ . Therefore, this spacecan be endowed with an appropriate Poisson bracket that makes S ( g ) into a Poisson algebra. It canbe shown that a coalgebra structure can always be defined on S ( g ) introducing the comultiplication ∆ : S ( g ) → S ( g ) ⊗ S ( g ) , ∆ ( x ) = x ⊗ + ⊗ x , ∀ x ∈ g ⊂ S ( g ) , (1.29)which is a Poisson algebra homomorphism. This makes S ( g ) into a Poisson-Hopf algebra. Further-more, in the light of the coassociatity condition ∆ ( ) : = ( ∆ ⊗ Id ) ◦ ∆ = ( Id ⊗ ∆ ) ◦ ∆ , (1.30)we can define the third-order coproduct ∆ ( ) : S ( g ) → S ( g ) ⊗ S ( g ) ⊗ S ( g ) , ∆ ( ) ( x ) = x ⊗ ⊗ + ⊗ x ⊗ + ⊗ ⊗ x (1.31)for all x ∈ g , where g is understood as a subset of S ( g ) . The m th-order coproduct map can bedefined, recursively, as ∆ ( m ) : S ( g ) → S ( m ) ( g ) , ∆ ( m ) : = ( ( m − ) − times (cid:122) (cid:125)(cid:124) (cid:123) Id ⊗ . . . ⊗ Id ⊗ ∆ ( ) ) ◦ ∆ ( m − ) , m ≥
3, (1.32)which, clearly, is also a Poisson algebra homomorphism.
Definition 1.17.
Let ( A , m , ∆ ) be a Hopf algebra and ( A , {· , ·} ) a Poisson structure for A, then the triple ( A , m , ∆ , {· , ·} ) is a Poisson–Hopf algebra if the coproduct ∆ is a homomorphism of Poisson algebras, i.e. ∆ ( { x , y } A ) = { ∆ ( x ) , ∆ ( y ) } A ⊗ A , for all x , y ∈ A. If the antipode map does not exist, the triple is a Poisson bialgebra [51, 60, 146]. If M is a smooth manifold such that its ring of functions is a Poisson algebra ( C ∞ ( M ) , {· , ·} ) , then M is called a Poisson manifold [148]. Clearly, if {· , ·} is a Poisson bracket, then { f , ·} defines a vectorfield X f ∈ X ( M ) for all f ∈ C ∞ ( M ) . The vector field X f is well-defined and satisfies the commutator { f , g } = X f g = − X g f = dg ( X f ) = − d f ( X g ) .Such a vector field X f is called a Hamiltonian vector field . In these conditions, it can be easily verifiedthat a Poisson bivector is a skew-bilinear form on T ∗ ( M ) .In particular, if X f , X g ∈ X ( M ) are Hamiltonian vector fields, they satisfy the identity [ X f , X g ]( h ) = { f , { g , h }} − { g , { f , h }} = {{ f , g } , h } = X { f , g } ( h ) .In order words, for all functions f , g ∈ C ∞ ( M ) , the Hamiltonian vector fields satisfy the commutatorcondition [ X f , X g ] = X { f , g } . (2.1) Lemma 2.1. If ( M , {· , ·} ) is a Poisson manifold, then L X f Λ =
0, (2.2) for all f ∈ C ∞ ( M ) . Whenever ( M , ω ) is a symplectic manifold, the preceding equation (2.2) turns out to adopt theparticular L X f ω = f ∈ C ∞ ( M ) . Remark.
Let ( M , g ) be a Riemannian manifold with Levi-Civita connection ∇ and Λ be a bivectoron M . If there exists a tensor field T ijk on M such that T ijk = T ikj and ∇ k Λ ij + T ihk Λ hj + T jhk Λ ih = Λ is a Poisson bivector, i.e. Λ defines a Poisson structure on the Riemannianmanifold.If ( M , ω ) is a symplectic manifold, then ω is a closed non-degenerate 2-form. This determines on M a Poisson structure through the prescription { f , g } : = ω ( X f , X g ) ,where X f and X g are defined by (1.9).0 Chapter 2. Poisson Manifolds and Quantum Groups
Definition 2.2.
Let ( M , Λ ) be a Poisson manifold. If the set D p ( M ) is defined as D p ( M ) : = { v ∈ T p ( M ) / ∃ f ∈ C ∞ ( M ) , X f ( p ) = v } , then the set of linear subspaces D ( M ) = {D p ( M ) } is called a general distribution. Moreover, it is calledthe characteristic distribution of the Poisson structure whenever for all p ∈ M there are vector fields X f , . . . , X f k ∈ D ( M ) such that for each point in M they span the subspace D p ( M ) . Example 2.3.
Let { f , f , f } be the set of Hamiltonian functions f = x , f = x + zy , f = ω = dx ∧ dy the canonical symplectic form in the plane. Then the Hamiltonian vector fields aregiven by X f = − ∂∂ y , X f = − x ∂∂ y + z ∂∂ x , X f = D = < X f , X f , X f > has rank 2 if and only if z (cid:54) =
0, asdet (cid:18) − z − x (cid:19) = z .A point ( x , y , z ) ∈ M such that z (cid:54) = regular , otherwise it is called singular .In this context, it is possible to establish a simple criterion that ensures the complete integrability ofa distribution (the proof can be found e.g. in [28, 148]). Definition 2.4.
Let D ( M ) be a general distribution on a Poisson manifold, M. The distribution is invariantif there are vector fields on M such that for all p ∈ M, these vector fields span D p at p, and such that for allt ∈ R and p ∈ M the exponential map ( exp t X ) ∗ ( D p ( M )) = D exp t X ( p ) ( M ) . is defined. Theorem 2.5 (Stefan–Sussmann Frobenius theorem) . A general distribution D ( M ) is completely inte-grable if and only if it is invariant. If the manifold M is additionally a Poisson manifold, the leaves of D ( M ) are called symplectic leaves ,with D ( M ) being referred to as a symplectic foliation on M . Theorem 2.6.
Let ( M , Λ ) be a Poisson manifold and D ( M ) its characteristic distribution. Then D ( M ) iscompletely integrable and the Poisson structure defines a symplectic structure on each leaf of the characteristicdistribution. For completeness, we briefly recall some elementary notions about cohomology groups.
Definition 2.7.
Let ( M , Λ ) be a Poisson manifold, and δ Λ the coboundary operator such that δ Λ ( X ) =[ Λ , X ] ∈ X k + ( M ) for all X ∈ X k ( M ) . Then, the cohomology groups are defined asH k Λ ( M ) : = ker ( δ Λ : X k ( M )) → X k + ( M ) im ( δ Λ : X k − ( M )) → X k ( M )) . (2.3) .2. Poisson–Lie groups and Lie bialgebras Remark.
For low values of k , the cohomology groups have a concise geometrical interpretation: k=0: If X f ∈ X ( M ) is a Hamiltonian vector field, then δ Λ ( f ) = X f . Hence, H Λ ( M ) is the set offunctions f ∈ C ∞ ( M ) such that { f , ·} =
0, i.e., the Casimir functions of the Poisson structure.Hence, H Λ ( M ) coincides with the center of C ∞ ( M ) . k=1: Let X ∈ X ( M ) be a vector field. If X is a infinitesimal automorphism of the Poisson struc-ture, then L X Λ = δ Λ X =
0. Therefore, H Λ ( M ) is the quotient of the space of infinitesimalautomorphisms of the Poisson manifold by the space of the Hamiltonian vector fields. k=2: This group has the class of the element Λ defined by [ Λ , Λ ] =
0. If this class is 0, there is avector field X such that δ Λ ( X ) = L X Λ . Poisson manifolds with this property are called exactPoisson manifolds.If ω is a k − form on M , then the identity δ Λ ( (cid:93) ( ω )) = (cid:93) ( d ω ) is satisfied . It follows that the map (cid:93) in-duces a homomorphism of H kdR ( M ) → H k Λ ( M ) , where H kdR ( M ) denotes the de Rham cohomology.If the bivector is non-degenerate, then the map is an isomorphism. Definition 2.8.
Let G be a Lie group. If ( C ∞ ( G ) , ∆ , {· , ·} ) is a Poisson–Hopf algebra, then G is a Poisson–Lie group. Let G be a Poisson–Lie group with its Poisson structure defined by the bivector Λ . For all f ∈ C ∞ ( G ) we have a vector field locally determined by X f : = { f , ·} = X ij ∂ f ∂ x i ∂∂ x j .As the right-invariant vector fields { R i } are a basis for T g ( G ) for arbitrary elements g ∈ G , we con-clude that there exist functions α i such that X f = α i R i . Hence, all Poisson structures of a Poisson–Liegroup can be written in local coordinates as Λ = λ ij R i ∧ R j ,where λ ij ∈ C ∞ ( G ) and i < j .Let g be a Lie algebra of a Poisson–Lie group G . Then, the Lie algebra has a Lie bracket associatedwith the group structure. Moreover, there is a Lie structure in g ∗ due to the linearization of thePoisson structure in G . The Lie structure in g ∗ is well-defined as [ v , v ] : = ( d { f , f } ) | e (2.4)for all f , f ∈ C ∞ ( G ) , where {· , ·} is the Poisson bracket and v i : = d f i | e . Hence, the cocommutatoris defined by means of the following relation δ ([ X , Y ]) = [ δ ( X ) , 1 ⊗ Y + Y ⊗ ] + [ ⊗ X + X ⊗ δ ( Y )] (2.5)for all X , Y ∈ g , such that if Λ R : G → g ⊗ g is the right-translation of the Poisson bivector on G ,then δ : g → g ⊗ g is the tangent map δ : = T e Λ R . Thus, δ ∗ defines a Lie bracket on g ∗ , such that δ ∗ ( v ⊗ v (cid:48) ) = [ v , v (cid:48) ] , and v , v (cid:48) ∈ g ∗ . In this context, the Lie algebra ( g , δ ) is called a Lie bialgebra .If there is an element ρ ∈ g ⊗ g , such that δ ( ρ ) = [ ⊗ X + X ⊗ ρ ] , then the Lie bialgebra ( g , δ , ρ ) is called a coboundary Lie bialgebra . The element ρ is the classical r-matrix of g . The map (cid:93) : T ∗ ( M ) → T ( M ) is a homomorphism, such that for all β , α ∈ T ∗ ( M ) is defined by β ( (cid:93) ( α )) = Λ ( α , β ) . If thebivector Λ is non-degenerate then the map (cid:93) is an isomorphism. Chapter 2. Poisson Manifolds and Quantum Groups
Let ( g , δ , ρ ) be a coboundary Lie bialgebra. If { X , . . . , X n } is a basis of g then the classical r-matrixturns out to be ρ = ρ ij X i ⊗ X j . By virtue of the elements ρ + : = ( ρ i , j + ρ j , i ) X i ⊗ X j , ρ : = ρ ij X i ⊗ X j ⊗
1, (2.6) ρ : = ρ ij X i ⊗ ⊗ X j , ρ : = ρ ij ⊗ X i ⊗ X j , (2.7) ( g , δ , ρ ) is a Lie bialgebra if, and only if, the following identities are satisfied: ( ad g ⊗ ad g )( ρ + ) =
0, (2.8)and the modified classical Yang-Baxter equation holds ( ad g ⊗ ad g ⊗ ad g )[[ ρ , ρ ]] =
0, (2.9)where [[ · , · ]] is the Schouten-–Nijenhuis bracket [[ ρ , ρ ]] : = [ ρ , ρ ] + [ ρ , ρ ] + [ ρ , ρ ] .The first condition guarantees the antisymmetry of δ ∗ , while the last equation ensures that the Jacobiidentity in g ∗ is satisfied. Definition 2.9.
Two Lie bialgebras, ( g , δ ) and ( g , δ ) are equivalent if there is a Lie automorphism Θ of g such that δ = ( Θ − ⊗ Θ − ) ◦ δ ◦ Θ , that is, the next diagram g δ (cid:47) (cid:47) Θ (cid:15) (cid:15) g ⊗ g Θ ⊗ Θ (cid:15) (cid:15) g δ (cid:47) (cid:47) g ⊗ g is commutative.Remark. (see [146])• If g is a semisimple Lie algebra, then all its Lie bialgebras are coboundary Lie bialgebras.• If g is an abelian Lie algebra, then there is a non-coboundary Lie bialgebra ( g , δ ) , and if the Liealgebra g is not abelian, then there is a non-trivial coboundary Lie bialgebra ( g , δ ) .• In general, the Lie bialgebra ( g , δ ) can be a coboundary Lie bialgebra, while its dual bialgebra, ( g ∗ , δ ∗ ) is not a coboundary Lie bialgebra.• Let g be a Lie algebra, the commutation relations [ X i , X j ] = c kij X k , [ v i , v j ] = λ kij v k , [ v i , X j ] = c ijk v k − λ jik X k ,span a Lie algebra over g ⊗ g ∗ called a Manin–Lie algebra , g (cid:46)(cid:47) g ∗ . In this section, we shall consider the algebra A = C ∞ ( M ) with M being a Poisson manifold. Albeitquantum deformations can be defined in a rather general context, we restrict ourselves to the syudyon Poisson manifolds. For the general case, see e.g. [7, 51, 60, 148]. .2. Poisson–Lie groups and Lie bialgebras Definition 2.10.
The algebra A z is a z-parametric deformation of the algebra A, if A z is a formal seriesalgebra A [[ z ]] such that the quotient A / zA z is isomorphic to A. Furthermore, in terms of the product, A z is a quantization of the Poisson algebra ( A , m , {· , ·} ) if thereis an associative ∗ z -product such that it is a deformation of m given by f ∗ z g : = f g + z { f , g } + o ( z ) , f ∗ z a = a ∗ z f for all a ∈ C and f , g ∈ A . If there is a homomorphism of Poisson algebras Φ suchthat Φ ( f ) ∗ z Φ ( g ) = Φ ( f ∗ z g ) ,then the ∗ z -product is invariant. It follows that the limit { f , g } = lim z → z ( f ∗ z g − g ∗ z f ) recovers the classic bracket, implying the relation [ f , g ] : = f ∗ z g − g ∗ z f = z { f , g } + o ( z ) . (2.10)Let ( A , m , ∆ ) be a Hopf algebra and ( A , {· , ·} ) be a Poisson structure for A . Then ( A z ∆ z ) is a quan-tum deformation of ( A , m , ∆ , {· , ·} ) if there is a coproduct ∆ z such that ∆ z ( f ∗ z g ) = ∆ z ( f ) ∗ z ∆ z ( g ) , (2.11)for all f , g ∈ A . Remark.
Let ω be a bilinear form on the vector space C ∞ ( M ) of a Poisson Manifold M and let z be aparameter. Hence, a deformation of ω is a formal power series. If ω determines a Poisson bracket,a deformation of the Poisson–Lie bracket will be denoted as ω z ( f , g ) = { f , g } z . Definition 2.11.
A Hopf algebra ( A z , m z , ∆ z ) is a quantization of a co-Poisson–Hopf algebra ( A , m , ∆ , δ ) if A z is a deformation of a Hopf algebra ( A , m , ∆ ) and there is a ∗ z -co-product defined by ∆ z ( X ) = ∆ ( X ) + z δ ( X ) + o ( z ) (2.12) compatible with a product m z .Remark. If g is a Lie bialgebra over R or C , then admits a quantization.Within the Hopf algebras, quantum groups represent a specially interesting class that have shownto be of capital importance in Geometry [51, 105]. For example, as algebraic groups are well de-scribed by its Hopf algebra of functions, the deformed version of the latter Hopf algebra describesa quantized version of the algebraic group, which generally does not correspond anymore to analgebraic group. Definition 2.12.
Let g be a Lie algebra, ( U z ( g ) , m z , ∆ z ) is a quantum algebra if it is a quantization of aco-Poisson–Hopf algebra ( U ( g ) , m , ∆ , δ ) . Chapter 2. Poisson Manifolds and Quantum Groups
Example 2.13.
Let us consider sl ( ) with the standard basis { J , J + , J − } satisfying the commutationrelations [ J , J ± ] = ± J ± , [ J + , J − ] = J .In this basis, the Casimir operator reads C = J + ( J + J − + J − J + ) . (2.13)Considering the non-standard (triangular or Jordanian) quantum deformation U z ( sl ( )) of sl ( ) [116](see also [17] and references therein), we are led to the following deformed coproduct ∆ z ( J + ) = J + ⊗ + ⊗ J + , ∆ z ( J l ) = J l ⊗ e zJ + + e − zJ + ⊗ J l , l ∈ {− , 3 } and the commutation rules [ J , J + ] z = ( zJ + ) J + , [ J + , J − ] z = J , [ J , J − ] z = − J − ch ( zJ + ) − ch ( zJ + ) J − .Here shc denotes the cardinal hyperbolic sinus function defined bysinhc ( ξ ) : = sinh ( ξ ) ξ , for ξ (cid:54) = ξ = U z ( g ) related to a semi-simple Lie algebra g admits anisomorphism of algebras U z ( g ) → U ( g ) (see [51, Theorem 6.1.8]). This allows us to obtain a Casimiroperator of U z ( sl ( )) from C in (2.13) as (see [51] for details) C z = J + shc ( zJ + ) J + J − + J − J + shc ( zJ + ) +
12 ch ( zJ + ) ,As expected, this coincides with the expression formerly given in [17].If G is a Lie group with Lie algebra g , then we know that there is a duality between the universalenveloping algebra U ( g ) and C ∞ ( G ) [146]. The function ρ : g → End C ∞ ( G ) defined by ( ρ ( X ) φ )( g ) : = ddt φ ( exp tX g ) (cid:112) t = ,where X ∈ g , φ ∈ C ∞ ( G ) and g ∈ G , can be extended to a homomorphism of U ( g ) into End C C ∞ ( G ) .There is an right invariant action R : G → C ∞ ( G ) such that R g ( φ )( h ) = φ ( gh ) , and such that theLeibniz rule is fulfilled. Then, it is possible to define a bilinear map (cid:104)· , ·(cid:105) of C ∞ ( G ) ⊗ U ( g ) onto C (cid:104) φ , x (cid:105) = ( p ( x ) φ )( e ) , (2.14)where e is the neutral element of G . Hence, the map defined by C ∞ ( G ) −→ U ( g ) ∗ (2.15) φ (cid:55)−→ (cid:104) φ , ·(cid:105) (2.16)is an immersion, according to the Gelfand-–Naimark theorem [69]. We conclude that the ring offunctions C ∞ ( G ) can be considered as a dual of U ( g ) . It is not true that both Hopf algebras, C ∞ ( G ) and U ( g ) , are duals of each other, due to some problems that arise in infinitedimensional Lie algebras [2, 51]. .2. Poisson–Lie groups and Lie bialgebras Remark (Quantum group) . It ought to be observed that currently there is no universal and uni-fied definition of quantum groups [7, 51], albeit all definitions explicitly refer to the Hopf algebrastructure. Alternative approaches are e.g. given by the following:• Approach in Noncommutative Geometry: as a deformation of algebraic groups. Here matrixgroups are subjected to satisfy certain algebraic identities.• The Faddeev theory [65]: it uses solutions of the quantum Yang–Baxter equation (YBE). Thisapproach is the preferred one in Quantum Field Theory.In the following we will use the notion of quantum group as introduced by Drinfel’d in [60]: Aquantum group is a non-commutative Hopf algebra (deformation of the universal enveloping alge-bra of a Lie algebra) that gives rise to a Poisson–Hopf algebra ( C ∞ ( G ) , {· , ·} , ∆ ) .7 For the sake of simplicity we consider explicit computations merely on R , but we stress that thisapproach can be applied, mutatis mutandis, to construct Poisson–Hopf algebra deformations ofLie–Hamilton systems defined on any manifold. Let us consider the local coordinates { x , . . . , x n } on an n -dimensional manifold M . Geometrically,every non-autonomous system of first-order differential equations on M of the formd x i d t = f i ( t , x , . . . , x n ) , i =
1, . . . , n , (3.1)where f i : R n + → R are arbitrary functions, amounts to a t -dependent vector field X : R × M → T M given by X t : R × M (cid:51) ( t , x , . . . , x n ) (cid:55)→ n ∑ i = f i ( t , x , . . . , x n ) ∂∂ x i ∈ T M . (3.2)This justifies to represent (3.2) and its related system of differential equations (3.1) by X t (cf. [47]).According to the Lie–Scheffers Theorem [43, 44, 101], see the theorem 1.2, a system X is a Lie systemif, and only if, X t ( x , . . . , x n ) : = X ( t , x , . . . , x n ) = (cid:96) ∑ i = b i ( t ) X i ( t , x , . . . , x n ) , (3.3)for some t -dependent functions b ( t ) , . . . , b (cid:96) ( t ) and vector fields X , . . . , X (cid:96) on M that span an (cid:96) -dimensional real Lie algebra V of vector fields, i.e. the Vessiot–Guldberg Lie algebra of X .A Lie system X is, furthermore, a LH one [9, 15, 29, 47, 49, 81] if it admits a Vessiot–Guldberg Liealgebra V of Hamiltonian vector fields relative to a Poisson structure. This amounts to the existence,around each generic point of M , of a symplectic form, ω , such that: L X i ω =
0, (3.4)for a basis X , . . . , X (cid:96) of V (cf. Lemma 4.1 in [9]). To avoid minor technical details and to highlightour main ideas, hereafter it will be assumed, unless otherwise stated, that the symplectic form andremaining structures are defined globally. More accurately, a local description around a genericpoint in M could easily be carried out.Each vector field X i admits a Hamiltonian function h i given by the rule: ι X i ω = d h i , (3.5)where ι X i ω stands for the contraction of the vector field X i with the symplectic form ω . Since ω isnon-degenerate, every function h induces a unique associated Hamiltonian vector field X h (Chapter1).8 Chapter 3. Poisson–Hopf Algebra Deformations of Lie Systems
The core in what follows is the fact that the space C ∞ ( H ∗ ω ) can be endowed with a Poisson–Hopfalgebra structure. We recall that an associative algebra A with a product m and a unit η is said to be a Hopf algebra over R [2, 51, 105] if there exist two homomorphisms called coproduct ( ∆ : A −→ A ⊗ A ) and counit ( (cid:101) : A −→ R ) , along with an antihomomorphism, the antipode γ : A −→ A , such thatthe following diagram (1.25) is commutative, section 1.2.If A is a commutative Poisson algebra and ∆ is a Poisson algebra morphism, then ( A , m , η , ∆ , (cid:101) , γ ) is a Poisson–Hopf algebra over R . We recall that the Poisson bracket on A ⊗ A reads { a ⊗ b , c ⊗ d } A ⊗ A = { a , c } ⊗ bd + ac ⊗ { b , d } , ∀ a , b , c , d ∈ A .In our particular case, C ∞ ( H ∗ ω ) becomes a Hopf algebra relative to its natural associative algebrawith unit provided that ∆ ( f )( x , x ) : = f ( x + x ) , m ( h ⊗ g )( x ) : = h ( x ) g ( x ) , (cid:101) ( f ) : = f ( ) , η ( )( x ) : = γ ( f )( x ) : = f ( − x ) ,for every x , x , x ∈ H ω and f , g , h ∈ C ∞ ( H ∗ ω ) . Therefore, the space C ∞ ( H ∗ ω ) becomes a Poisson–Hopf algebra by endowing it with the Poisson structure defined by the Kirillov–Kostant–Souriaubracket related to a Lie algebra structure on H ω . In this section we propose a systematic procedure to obtain deformations of LH systems by usingLH algebras and deformed Poisson–Hopf algebras that lead to appropriate extensions of the theoryof LH systems. Explicitly, the construction is based upon the following four steps:1. Consider a LH system X (3.3) on R n with respect to a symplectic form ω and admitting a LHalgebra H ω spanned by a basis of functions h , . . . , h (cid:96) ∈ C ∞ ( R n ) with structure constants c kij ,i.e. { h i , h j } ω = (cid:96) ∑ k = c kij h k , i , j =
1, . . . , (cid:96) .2. Introduce a Poisson–Hopf algebra deformation H z , ω of C ∞ ( H ∗ ω ) with deformation parameter z ∈ R (in a quantum group setting we would have q : = e z ) as the space of smooth functions F ( h z ,1 , . . . , h z , (cid:96) ) with fundamental Poisson bracket given by { h z , i , h z , j } ω = F z , ij ( h z ,1 , . . . , h z , (cid:96) ) , (3.6)where F z , ij are certain smooth functions also depending smoothly on the deformation param-eter z and such thatlim z → h z , i = h i , lim z → ∇ h z , i = ∇ h i , lim z → F z , ij ( h z ,1 , . . . , h z , (cid:96) ) = (cid:96) ∑ k = c kij h k , (3.7)where ∇ stands for the gradient relative to the Euclidean metric on R n . Hence,lim z → { h z , i , h z , j } ω = { h i , h j } ω . (3.8)3. Define the deformed vector fields X z , i by the rule ι X z , i ω : = d h z , i , (3.9)so that lim z → X z , i = X i . (3.10) .1. Formalism X (3.3) by X z : = (cid:96) ∑ i = b i ( t ) X z , i . (3.11)Now some remarks are in order. First, note that for a given LH algebra H ω there exist as manyPoisson–Hopf algebra deformations as non-equivalent Lie bialgebra structures δ on H ω [51], wherethe 1-cocycle δ essentially provides the first-order deformation in z of the coproduct map ∆ . Forthree-dimensional real Lie algebras the full classification of Lie bialgebra structures is known, andsome classification results are also known for certain higher-dimensional Lie algebras (see [25, 11]and references therein). Once a specific Lie bialgebra ( H ω , δ ) is chosen, the full Poisson–Hopf alge-bra deformation can be systematically obtained by making use of the Poisson version of the quan-tum duality principle for Hopf algebras, as we will explicitly see in the next section for an ( sl ( ) , δ ) Lie bialgebra.Second, the deformed vector fields X z , i (3.9) will not, in general, span a finite-dimensional Lie alge-bra, which implies that (3.11) is not a Lie system. In fact, the sequence of Lie algebra morphisms(1.11) and the properties of Hamiltonian vector fields [148] lead to [ X z , i , X z , j ] = [ ϕ ( h z , i ) , ϕ ( h z , j )] = ϕ ( { h z , i , h z , j } ω ) = ϕ ( F z , ij ( h z ,1 , . . . , h z , l )) = − (cid:96) ∑ k = ∂ F z , ij ∂ h z , k X z , k .In other words, (cid:2) X z , i , X z , j (cid:3) = (cid:96) ∑ k = G kz , ij ( x , y ) X z , k , (3.12)where the G kz , ij ( x , y ) are smooth functions relative to the coordinates x , y and the deformation pa-rameter z . Despite this, the relations (3.8) and the continuity of ϕ imply that [ X i , X j ] = ϕ ( { h i , h j } ) ω = ϕ (cid:18) lim z → { h z , i , h z , j } ω (cid:19) = lim z → ϕ { h z , i , h z , j } ω = lim z → [ X z , i , X z , j ] .Hence lim z → G kz , ij ( x , y ) = constantholds for all indices. Geometrically, the conditions (3.12) establish that the vector fields X z , i span aninvolutive smooth generalized distribution D z . In particular, the distribution D is spanned by theVessiot–Guldberg Lie algebra (cid:104) X , . . . , X l (cid:105) . This causes D to be integrable on the whole R n in thesense of Stefan–Sussman [148, 120, 152]. The integrability of D z , for z (cid:54) =
0, can only be ensured onopen connected subsets of R n where D z has constant rank [148].Third, although the vector fields X z , i depend smoothly on z , the distribution D z may change abruptly.For instance, consider the case given by the LH system X = ∂ x + ty ∂ x relative to the symplectic form ω = d x ∧ d y and admitting a LH algebra H ω = (cid:104) h : = y , h : = y /2 (cid:105) . Let us define h z ,1 : = y and h z ,2 : = y /2 + zx . Then X z = ∂ x + t ( y ∂ x − z ∂ y ) and dim D ( x , y ) =
1, but dim D z ( x , y ) = z (cid:54) =
0. Hence, the deformation of LH systems may change in an abrupt way the dynamical andgeometrical properties of the systems X z (cycles, periodic solutions, etc).Fourth, the deformation parameter z provides an additional degree of freedom that enables thecontrol or modification of the deformed system X z . In fact, as z can be taken small, perturbations ofthe initial Lie system X can be obtained from the deformed one X z in a natural way.And, finally, we stress that, by construction, the very same procedure can be applied to other2 n − dimensional manifolds different to R n , to higher dimensions as well as to multiparameterPoisson–Hopf algebra deformations of Lie algebras endowed with two or more deformation pa-rameters.0 Chapter 3. Poisson–Hopf Algebra Deformations of Lie Systems
Remark.
The coalgebra method employed in [15] to obtain superposition rules and constants of mo-tion for LH systems on a manifold M relies almost uniquely in the Poisson–Hopf algebra structurerelated to C ∞ ( g ∗ ) and a Poisson map D : C ∞ ( g ∗ ) → C ∞ ( M ) ,where we recall that g is a Lie algebra isomorphic to a LH algebra, H ω , of the LH system.Relevantly, quantum deformations allow us to repeat this scheme by substituting the Poisson alge-bra C ∞ ( g ∗ ) with a quantum deformation C ∞ ( g ∗ z ) , where z ∈ R , and obtaining an adequate Poissonmap D z : C ∞ ( g ∗ z ) → C ∞ ( M ) .The above procedure enables us to deform the LH system into a z -parametric family of Hamilto-nian systems whose dynamics is determined by a Steffan–Sussmann distribution and a family ofPoisson algebras. If z tends to zero, then the properties of the (classical) LH system are recoveredby a limiting process, hence enabling to construct new deformations exhibiting physically relevantproperties. The fact that H z , ω we are handling Poisson–Hopf algebra allows us to apply the coalgebra formal-ism established in [15] in order to obtain t -independent constants of the motion for X z .Let S ( H ω ) be the symmetric algebra of H ω , i.e. the associative unital algebra of polynomials on theelements of H ω . The Lie algebra structure on H ω can be extended to a Poisson algebra structure in S ( H ω ) by requiring [ v , · ] to be a derivation on the second entry for every v ∈ H ω . Then, S ( H ω ) canbe endowed with a Hopf algebra structure with a non-deformed (trivial) coproduct map ∆ definedby ∆ : S ( H ω ) → S ( H ω ) ⊗ S ( H ω ) , ∆ ( v i ) : = v i ⊗ + ⊗ v i , i =
1, . . . , (cid:96) , (3.13)which is a Poisson algebra homomorphism relative to the Poisson structure on S ( H ω ) and the oneinduced in S ( H ω ) ⊗ S ( H ω ) . Recall that every element of S ( H ω ) can be understood as a functionon H ∗ ω . Moreover, as S ( H ω ) is dense in the space C ∞ ( H ∗ ω ) of smooth functions on the dual H ∗ ω ofthe LH algebra H ω , the coproduct in S ( H ω ) can be extended in a unique way to ∆ : C ∞ ( H ∗ ω ) → C ∞ ( H ∗ ω ) ⊗ C ∞ ( H ∗ ω ) .Similarly, all structures on S ( H ω ) can be extended turning C ∞ ( H ∗ ω ) into a Poisson–Hopf algebra.Indeed, the resulting structure is the natural one in C ∞ ( H ∗ ω ) given in section 2.2.Let us assume now that C ∞ ( H ∗ ω ) has a Casimir invariant C = C ( v , . . . , v (cid:96) ) ,where v , . . . , v (cid:96) is a basis for H ω . The initial LH system allows us to define a Lie algebra morphism φ : H ω → C ∞ ( M ) , where M is a submanifold of R n where all functions h i : = φ ( v i ) , for i =
1, . . . , (cid:96) ,are well defined. Then, the Poisson algebra morphisms D : C ∞ ( H ∗ ω ) → C ∞ ( M ) , D ( ) : C ∞ ( H ∗ ω ) ⊗ C ∞ ( H ∗ ω ) → C ∞ ( M ) ⊗ C ∞ ( M ) , (3.14)defined respectively by D ( v i ) : = h i ( x , y ) , D ( ) ( ∆ ( v i )) : = h i ( x , y ) + h i ( x , y ) , i =
1, . . . , (cid:96) , (3.15)lead to the t -independent constants of motion F ( ) : = F and F ( ) for the Lie system X given in (3.3)where F : = D ( C ) , F ( ) : = D ( ) ( ∆ ( C )) . (3.16) .1. Formalism H z , ω with deformedcoproduct ∆ z and Casimir invariant C z = C z ( v , . . . , v (cid:96) ) , where { v , . . . , v (cid:96) } fulfill the same Poissonbrackets (3.6), and such that lim z → ∆ z = ∆ , lim z → C z = C .Following [15], the element C z turns out to be the cornerstone in the construction of the deformedconstants of the motion for the ‘generalized’ LH system X z .3 Part II
Applications to the Theory ofQuantum Poisson–Hopf Algebras sl ( ) -related Systems sl ( ) Once the general description of our approach has been introduced, we present in this section thegeneral properties of the Poisson analogue of the so-called non-standard quantum deformation ofthe simple real Lie algebra sl ( ) . This deformation will be applied in the sequel to get deformationsof the Milne–Pinney equation or Ermakov system and of some Riccati equations, since all thesesystems are known to be endowed with a LH algebra H ω isomorphic to sl ( ) [9, 15, 29].Let us consider the basis { J , J + , J − } for sl ( ) with Lie brackets and Casimir operator given by [ J , J ± ] = ± J ± , [ J + , J − ] = J , C = J + ( J + J − + J − J + ) . (4.1)Amongst the three possible quantum deformations of sl ( ) , we shall hereafter consider the non-standard (triangular or Jordanian) quantum deformation, U z ( sl ( )) (see [17, 116, 137] for furtherdetails). The Hopf algebra structure of U z ( sl ( )) has the following deformed coproduct and com-patible deformed commutation rules ∆ z ( J + ) = J + ⊗ + ⊗ J + , ∆ z ( J j ) = J j ⊗ e zJ + + e − zJ + ⊗ J j , j ∈ {− , 3 } , [ J , J + ] z = sinh ( zJ + ) z , [ J , J − ] z = − J − cosh ( zJ + ) − cosh ( zJ + ) J − , [ J + , J − ] z = J .The counit and antipode can be explicitly found in [17, 116], and the deformed Casimir reads [17] C z = J + sinh ( zJ + ) z J − + J − sinh ( zJ + ) z +
12 cosh ( zJ + ) .Let g be the Lie algebra of G . It is well known (see [51, 105]) that quantum algebras U z ( g ) are Hopfalgebra duals of quantum groups G z . On the other hand, quantum groups G z are just quantizationsof Poisson–Lie groups, which are Lie groups endowed with a multiplicative Poisson structure, i.e. aPoisson structure for which the Lie group multiplication is a Poisson map. In the case of U z ( sl ( )) ,such Poisson structure on SL ( ) is explicitly given by the Sklyanin bracket coming from the classical r -matrix r = zJ ∧ J + , (4.2)which is a solution of the (constant) classical Yang–Baxter equation.Moreover, the ‘quantum duality principle‘ [60, 136] states that quantum algebras can be thought ofas ‘quantum dual groups’ G ∗ z , which means that any quantum algebra can be obtained as the Hopfalgebra quantization of the dual Poisson–Lie group G ∗ . The usefulness of this approach to constructexplicitly the Poisson analogue of quantum algebras was developed in [25].In the case of U z ( sl ( )) , the Lie algebra g ∗ of the dual Lie group G ∗ is given by the dual of thecocommutator map δ that is obtained from the classical r -matrix as δ ( x ) = [ x ⊗ + ⊗ x , r ] , ∀ x ∈ g . (4.3)6 Chapter 4. Poisson–Hopf Algebra Deformations of sl ( ) -related Systems In our case, from (4.1) and (4.2) we explicitly obtain δ ( J ) = z J ∧ J + , δ ( J + ) = δ ( J − ) = z J − ∧ J + ,and the dual Lie algebra g ∗ reads [ j + , j ] = − z j , [ j + , j − ] = − z j − , [ j , j − ] =
0, (4.4)where { j , j + , j − } is the basis of g ∗ , and { J , J + , J − } can now be interpreted as local coordinateson the dual Lie group G ∗ . The dual Lie algebra (4.4) is the so-called ‘book’ Lie algebra, and thecomplete set of its Poisson–Lie structures was explicitly obtained in [11] (see also [10], where bookPoisson–Hopf algebras were used to construct integrable deformations of Lotka–Volterra systems).In particular, if we consider the coordinates on G ∗ given by v = J + , v = J , v = − J − ,the Poisson–Lie structure on the book group whose Hopf algebra quantization gives rise to thequantum algebra U z ( sl ( )) is given by the fundamental Poisson brackets [11] { v , v } z = − sinhc ( zv ) v , { v , v } z = − v , { v , v } z = − cosh ( zv ) v , (4.5)together with the coproduct map ∆ z ( v ) = v ⊗ + ⊗ v , ∆ z ( v k ) = v k ⊗ e zv + e − zv ⊗ v k , k =
2, 3, (4.6)which is nothing but the group law for the book Lie group G ∗ in the chosen coordinates (see [10, 11,25] for a detailed explanation). Therefore, (4.5) and (4.6) define a Poisson–Hopf algebra structureon C ∞ ( G ∗ ) , which can be thought of as a Poisson–Hopf algebra deformation of the Poisson algebra C ∞ ( sl ( ) ∗ ) , since we have identified the local coordinates on C ∞ ( G ∗ ) with the generators of theLie–Poisson algebra sl ( ) ∗ .Notice that we have introduced in (4.5) the hereafter called cardinal hyperbolic sinus function (seeAppendix B) defined by sinhc ( x ) : = sinh ( x ) x . (4.7)Summarizing, the Poisson–Hopf algebra given by (4.5) and (4.6), together with its Casimir function C z = sinhc ( zv ) v v − v , (4.8)will be the deformed Poisson–Hopf algebra that we will use in the sequel in order to constructdeformations of LH systems based on sl ( ) . Note that the usual Poisson–Hopf algebra C ∞ ( sl ( ) ∗ ) is smoothly recovered under the z → { v , v } = − v , { v , v } = − v , { v , v } = − v , (4.9)with undeformed coproduct (3.13) and Casimir C = v v − v . (4.10)We stress that this application of the ‘quantum duality principle’ would allow one to obtain thePoisson analogue of any quantum algebra U z ( g ) , which by following the method here presentedcould be further applied in order to construct the corresponding deformation of the LH systemsassociated to the Lie–Poisson algebra g . In particular, the Poisson versions of the other quantumalgebra deformations of sl ( ) can be obtained in the same manner with no technical obstructions(for instance, see [25] for the explicit construction of the ‘standard’ or Drinfel’d–Jimbo deformation). Some properties of this function along with its relationship with Lie systems are given in the Appendix A .2. Poisson–Hopf deformations of sl ( ) Lie–Hamilton systems Remark.
Since U z ( sl ( )) is a Poisson algebra, one can define a Lie algebra representation v ∈ sl ( ) (cid:55)→ [ v , · ] z ∈ End ( U z ( sl ( ))) , which makes U z ( sl ( )) into a sl ( ) -space. Similarly, C ∞ z ( sl ( ) ∗ ) is also a sl ( ) -space relative to the Lie algebra representation induced by the Poisson structure on C ∞ z ( sl ( ) ∗ ) ,i.e. ρ : v ∈ sl ( ) (cid:55)→ { v , ·} sl ( ) , z ∈ End ( C ∞ z ( sl ( ) ∗ )) .There exists a z -parametrized family of linear morphisms of the form φ z : P ∈ U z ( sl ( )) (cid:55)→ f P ∈ C ∞ z ( sl ( ) ∗ ) , ∀ z ∈ R ,satisfying that φ z ([ v , · ]) = { v , φ z ( · ) } sl ( ) ∗ , z for every v ∈ sl ( ) , i.e. φ z is a morphism of sl ( ) -spaces.There is a canonical way of constructing φ by setting φ ([ P ]) to be the unique symmetric polyno-mial in the equivalence class [ P ] . This construction is no longer available for U z ( sl ( )) . To define φ z is enough to use that every class of equivalence [ P ] in U z ( sl ( )) has a unique decomposition asa linear combination of the elements of every Poincaré–Birkhoff–Witt basis for U z ( sl ( )) . Then, φ z isthe linear morphism on U z ( sl ( )) that acts as the identity on the elements of the chosen Poincaré–Birkhoff–Witt basis.To illustrate the above point, let us recall that U z ( sl ( )) can be defined as the algebra generated bythe operators J − , J , K : = e zJ + , K − : = e − zJ + .Then, a Poincaré–Birkhoff–Witt basis is given by the polynomials J m − K p J l , where m , l ≥ p ∈ Z .In other words, every element in U z ( sl ( )) admits a unique representation as a polynomial in thisbasis. This allows us to define a morphism of sl ( ) -spaces: φ z : P ( J − , K , J ) ∈ U z ( sl ( )) (cid:55)→ f P ( J − , K , J ) ∈ C ∞ z ( sl ( ) ∗ ) , ∀ z ∈ R .Hence, any Casimir element of the Poisson–Hopf algebra U z ( sl ( )) gives rise to a Casimir of C ∞ z ( sl ( ) ∗ ) .For instance, C z (4.8) is the Poisson analog of − C z (4.2). sl ( ) Lie–Hamilton systems
This section concerns the analysis of Poisson–Hopf deformations of LH systems on a manifold M with a Vessiot–Guldberg algebra isomorphic to sl ( ) . Our geometric analysis will allow us tointroduce the notion of a Poisson–Hopf Lie system that, roughly speaking, is a family of non-autonomous Hamiltonian systems of first-order differential equations constructed as a deformationof a LH system by means of the representation of the deformation of a Poisson–Hopf algebra in aPoisson manifold.Let us endow a manifold M with a symplectic structure ω and consider a Hamiltonian Lie groupaction Φ : SL ( R ) × M → M . A basis of fundamental vector fields of Φ , let us say { X , X , X } ,enable us to define a Lie system X t = ∑ i = b i ( t ) X i ,for arbitrary t -dependent functions b ( t ) , b ( t ) , b ( t ) , and { X , X , X } spanning a Lie algebra iso-morphic to sl ( ) . As is well known, there are only three non-diffeomorphic classes of Lie algebras ofHamiltonian vector fields isomorphic to sl ( ) on the plane [9]. Since X , X , X admit Hamiltonianfunctions h , h , h , the t -dependent vector field X admits a t -dependent Hamiltonian function h = ∑ i = b i ( t ) h i .Due to the cohomological properties of sl ( ) (see e.g. [148]), the Hamiltonian functions h , h , h canalways be chosen so that the space (cid:104) h , h , h (cid:105) spans a Lie algebra isomorphic to sl ( ) with respectto {· , ·} ω .8 Chapter 4. Poisson–Hopf Algebra Deformations of sl ( ) -related Systems Let { v , v , v } be the basis for sl ( ) given in (4.4) and let M be a manifold where the functions h , h , h are smooth. Further, the Poisson–Hopf algebra structure of C ∞ ( sl ∗ ( )) is given by (4.9). Inthese conditions, there exists a Poisson algebra morphism D : C ∞ ( sl ∗ ( )) → C ∞ ( M ) satisfying D ( f ( v , v , v )) = f ( h , h , h ) , ∀ f ∈ C ∞ ( sl ∗ ( )) .Recall that the deformation C ∞ ( sl ∗ z ( )) of C ∞ ( sl ∗ ( )) is a Poisson–Hopf algebra with the new Pois-son structure induced by the relations (4.5). Let us define the submanifold O = : { θ ∈ sl ∗ ( ) : v ( θ ) (cid:54) = } of sl ∗ ( ) . Then, the Poisson structure on sl ∗ ( ) can be restricted to the space C ∞ ( O ) . Inturn, this enables us to expand the Poisson–Hopf algebra structure in C ∞ ( sl ∗ ( )) to C ∞ ( O ) . Withinthe latter space, the elements v z ,1 : = v , v z ,2 : = sinhc ( zv ) v , v z ,3 : = sinhc ( zv ) v v + c ( zv ) v , (4.11)are easily verified to satisfy the same commutation relations with respect to {· , ·} as the elements v , v , v in C ∞ ( sl ∗ z ( )) with respect to {· , ·} z (4.5), i.e. { v z ,1 , v z ,2 } = − sinhc ( zv z ,1 ) v z ,1 , { v z ,1 , v z ,3 } = − v z ,2 , { v z ,2 , v z ,3 } = − ch ( zv z ,1 ) v z ,3 . (4.12)In particular, from (4.11) with z = { v , v } = − v , { v , v } = { v , v } v v = − v , { v , v } = − v v { v , v } − c v { v , v } = − v v − c v = − v .The functions v z ,1 , v z ,2 , v z ,3 are not functionally independent, as they satisfy the constraintsinhc ( zv z ) v z v z − v z = c /4. (4.13)The existence of the functions v z ,1 , v z ,2 , v z ,3 and the relation (4.13) with the Casimir of the deformedPoisson–Hopf algebra is by no means casual. Let us explain why v z ,1 , v z ,2 , v z ,3 exist and how toobtain them easily.Around a generic point p ∈ sl ∗ ( ) , there always exists an open U p containing p where both Poissonstructures give a symplectic foliation by surfaces. Examples of symplectic leaves for {· , ·} and {· , ·} z are displayed in Fig. 4.1.The splitting theorem on Poisson manifolds [148] ensure that if U p is small enough, then there existtwo different coordinate systems { x , y , C } and { x z , y z , C z } where the Poisson bivectors related to {· , ·} and {· , ·} z read Λ = ∂ x ∧ ∂ y and Λ z = ∂ x z ∧ ∂ y z . Hence, C z and C are Casimir functions for Λ z and Λ , respectively. Moreover, x z = x z ( x , y , C ) , y z = y z ( x , y , C ) , C z = C z ( x , y , C ) . It follows fromthis that Φ : f ( x z , y z , C z ) ∈ C ∞ z ( U p ) (cid:55)→ f ( x , y , C ) ∈ C ∞ ( U p ) is a Poisson algebra morphism.If { v , v , v } are the standard coordinates on sl ∗ ( ) and the relations (4.5) are satisfied, then v i = ξ i ( x z , y z , C z ) holds for certain functions ξ , ξ , ξ : R → R . Hence, the ˆ v z , i = ξ i ( x , y , C ) close thesame commutation relations relative to {· , ·} as the v i do with respect to {· , ·} z . As C is a Casimirinvariant, the functions v z , i : = ξ i ( x , y , c ) , with a constant value of c , still close the same commutationrelations among themselves as the v i . Moreover, the functions v z , i become functionally dependent.Indeed, C z = C z ( v , v , v ) = C z ( ξ ( x z , y z , C z ) , ξ ( x z , y z , C z ) , ξ ( x z , y z , C z )) .Hence, c = C z ( ξ ( x z , y z , c ) , ξ ( x z , y z , c ) , ξ ( x z , y z , c )) and we conclude that c = C z ( v z ,1 , v z ,2 , v z ,3 ) . .2. Poisson–Hopf deformations of sl ( ) Lie–Hamilton systems F IGURE sl ∗ ( ) given by the surfaces withconstant value of the Casimir for the Poisson structure in sl ∗ ( ) (left) and its defor-mation (right). Such submanifolds are symplectic submanifolds where the Poissonbivectors Λ and Λ z admit a canonical form. The previous argument allows us to recover the functions (4.11) in an algorithmic way. Actually,the functions x z , y z , C z and x , y , C can be easily chosen to be x z : = v , y z : = − v sinhc ( zv ) v , C z : = sinhc ( zv ) v v − v ,as well as x = v , y = − v / v , C = v v − v .Therefore, ξ ( x z , y z , C z ) = x z , ξ ( x z , y z , C z ) = − y z sinhc ( zx z ) x z , ξ ( x z , y z , C z ) = C z + x z y z sinhc ( zx z ) sinhc ( zx z ) x z .Assuming that C z = c /4, replacing x z , y z by x = v , y = − v / v , respectively, and taking intoaccount that v z , i : = ξ i ( x , y , c ) , one retrieves (4.11).It is worth mentioning that due to the simple form of the Poisson bivectors in splitting form forthree-dimensional Lie algebras, this method can be easily applied to such a type of Lie algebras.Next, the above relations enable us to construct a Poisson algebra morphism D z : f ( v , v , v ) ∈ C ∞ ( sl ∗ z ( )) (cid:55)→ D ( f ( v z , v z , v z )) ∈ C ∞ ( M ) for every value of z allowing us to pass the structure of the Poisson–Hopf algebra C ∞ ( sl ∗ z ( )) to C ∞ ( M ) . As a consequence, D z ( C z ) satisfies the relations { D z ( C z ) , h z , i } ω = i =
1, 2, 3.Using the symplectic structure on M and the functions h z , i written in terms of { h , h , h } , one caneasily obtain the deformed vector fields X z , i in terms of the vector fields X i . Finally, as X z , t = ∑ i = b i ( t ) X z , i holds, it is straightforward to verify that the brackets X z , i D ( C z ) = { D ( C z ) , h z , i } = D ( C z ) is a t -independent constant of the motion for each of the deformedLH system X z , t .Consequently, deformations of LH-systems based on sl ( ) can be treated simultaneously, startingfrom their classical LH counterpart. The final result is summarized in the following statement.0 Chapter 4. Poisson–Hopf Algebra Deformations of sl ( ) -related Systems Theorem 4.1. If φ : sl ( ) → C ∞ ( M ) is a morphism of Lie algebras with respect to the Lie bracket in sl ( ) and a Poisson bracket in C ∞ ( M ) , then for each z ∈ R there exists a Poisson algebra morphism D z : C ∞ ( sl ∗ z ( )) → C ∞ ( M ) such that for a basis { v , v , v } satisfying the commutation relations (4.9) is givenby D z ( f ( v , v , v ))= f (cid:18) φ ( v ) , sinhc ( z φ ( v )) φ ( v ) , sinhc ( z φ ( v )) φ ( v ) φ ( v ) + c ( z φ ( v )) φ ( v ) (cid:19) . Provided that h i : = φ ( v i ) , the deformed Hamiltonian functions h z , i : = D z ( v i ) adopt the formh z ,1 = h , h z ,2 = sinhc ( zh ) h , h z ,3 = sinhc ( zh ) h h + c ( zh ) h , which satisfy the commutation relations (4.12).The Hamiltonian vector fields X z , i associated with h z , i through (3.9) turn out to be X z ,1 = X , X z ,2 = h h (cid:0) cosh ( zh ) − sinhc ( zh ) (cid:1) X + sinhc ( zh ) X , X z ,3 = (cid:34) h h (cid:0) ch ( zh ) − ( zh ) (cid:1) − c ch ( zh ) h sinhc ( zh ) (cid:35) X + h h sinhc ( zh ) X , and satisfy the following commutation relations coming from [ X z ,1 , X z ,2 ] = ch ( zh z ,1 ) X z ,1 , [ X z ,1 , X z ,3 ] = X z ,2 , [ X z ,2 , X z ,3 ] = ch ( zh z ,1 ) X z ,3 + z sinhc ( zh z ,1 ) h z ,1 h z ,3 X z ,1 .As a consequence, the deformed Poisson–Hopf system can be generically described in terms of theVessiot–Guldberg Lie algebra corresponding to the non-deformed LH system as follows: X z , t = ∑ i = b i ( t ) X z , i = (cid:20) b ( t ) + b ( t ) h h (cid:0) cosh ( zh ) − sinhc ( zh ) (cid:1)(cid:21) X + b ( t ) (cid:34) h h (cid:0) ch ( zh ) − ( zh ) (cid:1) − c ch ( zh ) h sinhc ( zh ) (cid:35) X + sinhc ( zh ) (cid:18) b ( t ) + b ( t ) h h (cid:19) X .This unified approach to nonequivalent deformations of LH systems possessing a common under-lying Lie algebra suggests the following definition. Definition 4.2.
Let ( C ∞ ( M ) , {· , ·} ) be a Poisson algebra. A Poisson–Hopf Lie system is pair consistingof a Poisson–Hopf algebra C ∞ ( g ∗ z ) and a z-parametrized family of Poisson algebra representations D z : C ∞ ( g ∗ z ) → C ∞ ( M ) with z ∈ R . Next, constants of the motion for X z , t can be deduced by applying the coalgebra approach intro-duced in [15] in the way briefly described in Section 3. In the deformed case, we consider thePoisson algebra morphisms D z : C ∞ ( sl ∗ z ( )) → C ∞ ( M ) , D ( ) z : C ∞ ( sl ∗ z ( )) ⊗ C ∞ ( sl ∗ z ( )) → C ∞ ( M ) ⊗ C ∞ ( M ) , .3. Deformations of sl ( ) Lie–Hamilton systems in R D z ( v i ) : = h z , i ( x ) ≡ h ( ) z , i , i =
1, 2, 3, D ( ) z ( ∆ z ( v )) = h z ,1 ( x ) + h z ,1 ( x ) ≡ h ( ) z ,1 , D ( ) z ( ∆ z ( v k )) = h z , k ( x ) e zh z ,1 ( x ) + e − zh z ,1 ( x ) h z , k ( x ) ≡ h ( ) z , k , k =
2, 3,where x s ( s =
1, 2 ) are global coordinates in M . We remark that, by construction, the functions h ( ) z , i satisfy the same Poisson brackets (4.12). Then t -independent constants of motion are given by (see(3.16)) F z ≡ F ( ) z : = D z ( C z ) , F ( ) z : = D ( ) z ( ∆ z ( C z )) ,where C z is the deformed Casimir (4.8). Explicilty, they read F z = sinhc (cid:16) zh ( ) z ,1 (cid:17) h ( ) z ,1 h ( ) z ,3 − (cid:16) h ( ) z ,2 (cid:17) = c F ( ) z = sinhc (cid:16) zh ( ) z ,1 (cid:17) h ( ) z ,1 h ( ) z ,3 − (cid:16) h ( ) z ,2 (cid:17) . sl ( ) Lie–Hamilton systems in R sl ( ) according to the local classification performed in [9], which wasbased on the previous results [73]. Consider the manifold M = R and the coordinates x = ( x , y ) .According to [9, 29], there are only thee classes of LH systems, denoted by P , I and I and theycorrespond to a positive, negative and zero value of the Casimir constant c , respectively. Recallthat these are non-diffeomorphic, so that there does not exist any local t -independent change ofvariables mapping one into another.Table 4.1 summarizes the three cases, covering vector fields, Hamiltonian functions, symplecticstructure and t -independent constants of motion. The particular LH systems which are diffeormor-phic within each class are also mentioned [29]. Notice that for all of them it is satisfy the followingcommutation relations for the vector fields and Hamiltonian functions (the latter with respect tocorresponding ω ): [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , { h , h } ω = − h , { h , h } ω = − h , { h , h } ω = − h .By applying the theorem 4.1 with the results of Table 4.1 we obtain the corresponding deformationswhich are displayed in Table 4.2. It is straightforward to verify that the classical limit z → Last chapter showed that deformations of a LH system with a fixed LH algebra H ω (cid:39) g can beobtained through a Poisson algebra C ∞ ( g ∗ ) , a given deformation and a certain Poisson morphism D : C ∞ ( g ∗ ) → C ∞ ( M ) . This section presents a simple method to obtain D from an arbitrary g ∗ ontoa symplectic manifold R n .2 Chapter 4. Poisson–Hopf Algebra Deformations of sl ( ) -related Systems T ABLE sl ( ) . For each class, it is displayed, in this order,a basis of vector fields X i , Hamiltonian functions h i , symplectic form ω , the constantsof motion F and F ( ) as well as the corresponding specific LH systems. • Class P with c = > X = ∂∂ x X = x ∂∂ x + y ∂∂ y X = ( x − y ) ∂∂ x + xy ∂∂ yh = − y h = − xy h = − x + y y ω = d x ∧ d yy F = F ( ) = ( x − x ) + ( y + y ) y y – Complex Riccati equation– Ermakov system, Milne–Pinney and Kummer–Schwarz equations with c > • Class I with c = − < X = ∂∂ x + ∂∂ y X = x ∂∂ x + y ∂∂ y X = x ∂∂ x + y ∂∂ yh = x − y h = x + y ( x − y ) h = xyx − y ω = d x ∧ d y ( x − y ) F = − F ( ) = − ( x − y )( x − y )( x − y )( x − y ) – Split-complex Riccati equation– Ermakov system, Milne–Pinney and Kummer–Schwarz equations with c <
0– Coupled Riccati equations • Class I with c = X = ∂∂ x X = x ∂∂ x + y ∂∂ y X = x ∂∂ x + xy ∂∂ yh = − y h = − x y h = − x y ω = d x ∧ d yy F = F ( ) = ( x − x ) y y – Dual-Study Riccati equation– Ermakov system, Milne–Pinney and Kummer–Schwarz equations with c =
0– Harmonic oscillator– Planar diffusion Riccati system
Theorem 4.3.
Let g be a Lie algebra whose Kostant–Kirillov–Souriau Poisson bracket admits a symplecticfoliation in g ∗ with a n-dimensional S ⊂ g ∗ . Then, there exists a LH algebra on the plane given by Φ : g → C ∞ (cid:0) R n (cid:1) relative to the canonical Poisson bracket on the plane.Proof. The Lie algebra g gives rise to a Poisson structure on g ∗ through the Kostant–Kirillov–Souriaubracket {· , ·} . This induces a symplectic foliation on g ∗ , whose leaves are symplectic manifoldsrelative to the restriction of the Poisson bracket. Such leaves are characterized by means of theCasimir functions of the Poisson bracket. By assumption, one of these leaves is 2 n -dimensional.In such a case, the Darboux Theorem warrants that the Poisson bracket on each leave is locallysymplectomorphic to the Poisson bracket of the canonical symplectic form on R n (cid:39) T ∗ R n . Inparticular, there exist some Darboux coordinates mapping the Poisson bracket on such a leaf intothe canonical symplectic bracket on T ∗ R n . The corresponding change of variables into the canonicalform in Darboux coordinates can be understood as a local diffeomorphism h : S k → R n mapping .3. Deformations of sl ( ) Lie–Hamilton systems in R Λ k on the leaf S k into the canonical Poisson bracket on T ∗ R n . Hence, h givesrise to a canonical Poisson algebra morphism φ h : C ∞ ( S k ) → C ∞ ( T ∗ R n ) .As usual, a basis { v , . . . , v r } of g can be considered as a coordinate system on g ∗ . In view of thedefinition of the Kostant–Kirillov–Souriau bracket, they span an r -dimensional Lie algebra. In fact,if [ v i , v j ] = ∑ rk = c kij v k for certain constants c kij , then { v i , v j } = ∑ rk = c kij v k . Since S k is a symplecticsubmanifold, there is a local immersion ι : S k (cid:44) → g ∗ which is a Poisson manifold morphism. Inconsequence, { ι ∗ v i , ι ∗ v j } = r ∑ k = c kij ι ∗ v k .Hence, the functions ι ∗ v i span a finite-dimensional Lie algebra of functions on S . Since S is 2 n -dimensional, there exists a local diffeomorphism φ : S → R n and Φ : v ∈ g (cid:55)→ φ ◦ ι ∗ v ∈ C ∞ (cid:0) R n (cid:1) is a Lie algebra morphism.Let us apply the above mechanism to explain the existence of three types of LH systems on theplane. We already know that the Lie algebra sl ( ) gives rise to a Poisson algebra in C ∞ ( g ∗ ) . In thestandard basis v , v , v with commutation relations (4.9), the Casimir is (4.10). It turns out that thesymplectic leaves of this Casimir are of three types:• A one-sheeted hyperboloid when v v − v = k < v v − v = v v − v = k > Λ = − v ∂∂ v ∧ ∂∂ v − v ∂∂ v ∧ ∂∂ v − v ∂∂ v ∧ ∂∂ v .Then, we have a changes of variables passing from the above form into Darboux coordinates¯ v = v , ¯ v = − v / v , C = v v − v .Then, v = v , v = − ¯ v ¯ v , v = ( C + ¯ v ¯ v ) / ¯ v .On a symplectic leaf, the value of C is constant, say C = c /4, and the restrictions of the previousfunctions to the leaf read ι ∗ v = v , ι ∗ v = − ¯ v ¯ v , ι ∗ v = c / ( v ) + ¯ v ¯ v .This can be viewed as a mapping Φ : sl ( ) → C ∞ ( R ) such that φ ( v ) = x , φ ( v ) = − xy , φ ( v ) = c / ( x ) + xy ,which is obviously a Lie algebra morphism relative to the standard Poisson bracket in the plane.It is simple to proof that when c is positive, negative or zero, one obtains three different types ofLie algebras of functions and their associated vector fields span the Lie algebras P , I and I asenunciated in [9]. Observe that since φ ( v ) φ ( v ) − φ ( v ) = c /4, there exists no change of variableson R mapping one set of variables into another for different values of c . Hence, Theorem 4.3ultimately explains the real origin of all the sl ( ) -LH systems on the plane.It is known that su ( ) admits a unique Casimir, up to a proportional constant, and the symplecticleaves induced in su ∗ ( ) are spheres. The application of the previous method originates a uniqueLie algebra representation, which gives rise to the unique LH system on the plane related to so ( ) .All the remaining LH systems on the plane can be generated in a similar fashion. The deformationsof such Lie algebras will generate all the possible deformations of LH systems on the plane.4 Chapter 4. Poisson–Hopf Algebra Deformations of sl ( ) -related Systems T ABLE sl ( ) -LH systems writ-ten in Table 1. The symplectic form ω is the same given in Table 1 and F ≡ F z . • Class P with c = > X z ,1 = ∂∂ x X z ,2 = x ch ( z / y ) ∂∂ x + y shc ( z / y ) ∂∂ y X z ,3 = (cid:32) x − y shc ( z / y ) (cid:33) ch ( z / y ) ∂∂ x + xy shc ( z / y ) ∂∂ y h z ,1 = − y h z ,2 = − xy shc ( z / y ) h z ,3 = − x shc ( z / y ) + y y shc ( z / y ) F ( ) z = ( x − x ) y y sinhc ( z / y ) sinhc ( z / y ) e z / y e − z / y + ( y + y ) y y sinhc ( z / y + z / y ) sinhc ( z / y ) sinhc ( z / y ) e z / y e − z / y • Class I with c = − < X z ,1 = ∂∂ x + ∂∂ y X z ,2 = x + y (cid:18) zx − y (cid:19) (cid:18) ∂∂ x + ∂∂ y (cid:19) + x − y (cid:18) zx − y (cid:19) (cid:18) ∂∂ x − ∂∂ y (cid:19) X z ,3 =
14 ch (cid:18) zx − y (cid:19) (cid:20) ( x + y ) + ( x − y ) shc − (cid:18) zx − y (cid:19)(cid:21) (cid:18) ∂∂ x + ∂∂ y (cid:19) + ( x − y ) shc (cid:18) zx − y (cid:19) (cid:18) ∂∂ x − ∂∂ y (cid:19) h z ,1 = x − y h z ,2 = ( x + y ) shc (cid:16) zx − y (cid:17) ( x − y ) h z ,3 = ( x + y ) shc (cid:16) zx − y (cid:17) − ( x − y ) ( x − y ) shc (cid:16) zx − y (cid:17) F ( ) z = ( x − x + y − y ) ( x − y )( x − y ) sinhc (cid:18) zx − y (cid:19) sinhc (cid:18) zx − y (cid:19) e − zx − y e zx − y − ( x + x − y − y ) sinhc (cid:16) zx − y + zx − y (cid:17) ( x − y )( x − y ) e zx − y ( x − y ) sinhc (cid:0) zx − y (cid:1) + e − zx − y ( x − y ) sinhc (cid:0) zx − y (cid:1) • Class I with c = X z ,1 = ∂∂ x X z ,2 = x ch (cid:16) z / y (cid:17) ∂∂ x + y (cid:16) z / y (cid:17) ∂∂ y X z ,3 = x ch (cid:16) z / y (cid:17) ∂∂ x + xy shc (cid:16) z / y (cid:17) ∂∂ y h z ,1 = − y h z ,2 = − x y shc (cid:16) z / y (cid:17) h z ,3 = − x y shc (cid:16) z / y (cid:17) F ( ) z = ( x − x ) y y sinhc (cid:16) z / y (cid:17) sinhc (cid:16) z / y (cid:17) e z / y e − z / y In this section we construct the non-standard deformation of the well-known Milne–Pinney (MP)equation [111, 123], which is known to be a LH system [9, 29]. Recall that the MP equation corre-sponds to the equation of motion of the isotropic oscillator with a time-dependent frequency and a‘centrifugal’ term. As we will show in the sequel, the main feature of this deformation is that thenew oscillator system has both a position-dependent mass and a time-dependent frequency.
The MP equation [111, 123] has the following expressiond x d t = − Ω ( t ) x + cx , (5.1)where Ω ( t ) is any t -dependent function and c ∈ R . By introducing a new variable y : = d x /d t , thesystem (5.1) becomes a first-order system of differential equations on T R , where R : = R \{ } , ofthe form d x d t = y , d y d t = − Ω ( t ) x + cx . (5.2)This system is indeed part of the one-dimensional Ermakov system [47, 62, 96, 98] and diffeomor-phic to the one-dimensional t -dependent frequency counterpart [9, 15, 29] of the Smorodinsky–Winternitz oscillator [68].The system (5.2) determines a Lie system with associated t -dependent vector field [29] X = X + Ω ( t ) X , (5.3)where X : = − x ∂∂ y , X : = (cid:18) y ∂∂ y − x ∂∂ x (cid:19) , X : = y ∂∂ x + cx ∂∂ y , (5.4)span a Vessiot–Guldberg Lie algebra V MP of vector fields isomorphic to sl ( ) (for any value of c )with commutation relations given by [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . (5.5)The vector fields of V MP are defined on R x (cid:54) = , where they span a regular distribution of order two.Furthermore, X is a LH system with respect to the symplectic form ω = d x ∧ d y and the vectorfields (5.4) admit Hamiltonian functions given by h = x , h = − xy , h = (cid:16) y + cx (cid:17) , (5.6)6 Chapter 5. Deformation of ODEs that fulfill the following commutation relations with respect to the Poisson bracket induced by ω : { h , h } ω = − h , { h , h } ω = − h , { h , h } ω = − h . (5.7)Then, the functions h , h , h span a LH algebra H MP ω (cid:39) sl ( ) of functions on R x (cid:54) = ; the t -dependentHamiltonian associated with the t -dependent vector field (5.3) reads h = h + Ω ( t ) h . (5.8)We recall that this Hamiltonian is a natural one, that is, it can be written in terms of a kinetic energy T and potential U by identifying the variable y as the conjugate momentum p of the coordinate x : h = T + U = p + Ω ( t ) x + c x . (5.9)Hence h determines the composition of a one-dimensional oscillator with a time-dependent fre-quency Ω ( t ) and unit mass with a Rosochatius or Winternitz potential; the latter is just a centrifugalbarrier whenever c > h and, obviously, when c vanishes, these reduce to the equations of motionof a harmonic oscillator with a time-dependent frequency.We stress that it has been already proved in [9, 29] that the MP equations (5.2) comprise the three different types of possible sl ( ) -LH systems according to the value of the constant c : class P for c >
0; class I for c <
0; and class I for c =
0. This means that any other LH system related toa Vessiot–Guldberg Lie algebra of Hamiltonian vector fields isomorphic to sl ( ) must be, up to a t -independent change of variables, of the form (5.2) for a positive, zero or negative value of c .This implies that the second-order Kummer–Schwarz equations [42, 102] and several types of Ric-cati equations [45, 61, 64, 106, 142, 143, 153] are comprised within H MP ω (depending on the sign of c ). The relationships amongst all of these systems are ensured by construction and these can beexplicitly obtained through either diffeomorphisms or changes of variables (see [9, 29] for details).The constants of motion for the MP equations can be obtained by applying the coalgebra formalismintroduced in [15] and briefly summarized in section 2.4. Explicitly, let us consider the Poisson–Hopf algebra C ∞ ( H MP ∗ ω ) with basis { v , v , v } , coproduct (3.13), fundamental Poisson brackets(4.9) and Casimir (4.10). The Poisson algebra morphisms (3.14) D : C ∞ ( H MP ∗ ω ) → C ∞ ( R x (cid:54) = ) , D ( ) : C ∞ ( H MP ∗ ω ) ⊗ C ∞ ( H MP ∗ ω ) → C ∞ ( R x (cid:54) = ) ⊗ C ∞ ( R x (cid:54) = ) ,defined by (3.15), where h i are the Hamiltonian functions (5.6), lead to the t -independent constantsof the motion F ( ) : = F and F ( ) given by (3.16), through the Casimir (4.10), for the Lie system X (5.2); namely [15] F = h ( x , y ) h ( x , y ) − h ( x , y ) = c F ( ) = (cid:0) [ h ( x , y ) + h ( x , y )] [ h ( x , y ) + h ( x , y )] (cid:1) − (cid:0) h ( x , y ) + h ( x , y ) (cid:1) = ( x y − x y ) + c ( x + x ) x x . (5.10)We observe that F ( ) is just a Ray–Reid invariant for generalized Ermakov systems [96, 129] andthat it is related to the one obtained in [8, 20] from a coalgebra approach applied to superintegrablesystems.By permutation of the indices corresponding to the variables of the non-trivial invariant F ( ) , wefind two other constants of the motion: F ( ) = S ( F ( ) ) , F ( ) = S ( F ( ) ) , (5.11) .1. Deformed Milne–Pinney equation and oscillator systems S ij is the permutation of variables ( x i , y i ) ↔ ( x j , y j ) . Since ∂ ( F ( ) , F ( ) ) / ∂ ( x , y ) (cid:54) =
0, bothconstants of motion are functionally independent (note that the pair ( F ( ) , F ( ) ) is functionally inde-pendent as well). From these two invariants, the corresponding superposition rule can be derivedin a straightforward manner. Its explicit expression can be found in [15]. In order to apply the non-standard deformation of sl ( ) described in chapter 4 to the MP equation,we need to find the deformed counterpart h z , i ( i =
1, 2, 3 ) of the Hamiltonian functions h i (5.6), sofulfilling the Poisson brackets (4.5), by keeping the canonical symplectic form ω .This problem can be rephrased as the one consistent in finding symplectic realizations of a givenPoisson algebra, which can be solved once a particular symplectic leave is fixed as a level set for theCasimir functions of the algebra, where the generators of the algebra can be expressed in terms ofthe corresponding Darboux coordinates. In the particular case of the U z ( sl ( )) algebra, the explicitsolution (modulo canonical transformations) was obtained in [18] where the algebra (4.5) was foundto be generated by the functions v ( q , p ) = q , v ( q , p ) = −
12 sinh zq zq qp , v ( q , p ) =
12 sinh zq zq p + zc sinh zq ,where ω = d q ∧ d p , and the Casimir function (4.8) reads C z = c /4. In practical terms, such asolution can easily be found by solving firstly the non-deformed case z → v i ( q , p ) functions under the constraint that the Casimir C z has to take a constantvalue. With this result at hand, the corresponding deformed vector fields X z , i can be computed byimposing the relationship (3.9) and the final result is summarized in the following statement. Proposition 5.1. (i) The Hamiltonian functions defined byh z ,1 : = x , h z ,2 : = −
12 sinhc ( zx ) xy , h z ,3 : = (cid:18) sinhc ( zx ) y + ( zx ) cx (cid:19) , (5.12) close the Poisson brackets (4.5) with respect to the symplectic form ω = d x ∧ d y on R x (cid:54) = , namely { h z ,1 , h z ,2 } ω = − sinhc ( zh z ,1 ) h z ,1 , { h z ,1 , h z ,3 } ω = − h z ,2 , { h z ,2 , h z ,3 } ω = − cosh ( zh z ,1 ) h z ,3 , (5.13) where sinhc ( x ) is defined in (4.7). Relations (5.13) define the deformed Poisson algebra C ∞ ( H MP ∗ z , ω ) .(ii) The vector fields X z , i corresponding to h z , i read X z ,1 = − x ∂∂ y , X z ,2 = (cid:18) cosh ( zx ) −
12 sinhc ( zx ) (cid:19) y ∂∂ y −
12 sinhc ( zx ) x ∂∂ x , X z ,3 = sinhc ( zx ) y ∂∂ x + (cid:34) cx cosh ( zx ) sinhc ( zx ) + sinhc ( zx ) − cosh ( zx ) x y (cid:35) ∂∂ y , which satisfy [ X z ,1 , X z ,2 ] = cosh ( zx ) X z ,1 , [ X z ,1 , X z ,3 ] = X z ,2 , [ X z ,2 , X z ,3 ] = cosh ( zx ) X z ,3 + z (cid:16) c + x y sinhc ( zx ) (cid:17) X z ,1 . (5.14)8 Chapter 5. Deformation of ODEs
Since lim z → sinhc ( zx ) = z → cosh ( zx ) =
1, it can directly be checked that all the clas-sical limits (3.7), (3.8) and (3.10) are fulfilled. As expected, the Lie derivative of ω with respect toeach X z , i vanishes.At this stage, it is important to realize that, albeit (5.13) are genuine Poisson brackets definingthe Poisson algebra C ∞ ( H MP ∗ z , ω ) , the commutators (5.14) show that X z , i do not span a new Vessiot–Guldberg Lie algebra; in fact, the commutators give rise to linear combinations of the vector fields X z , i with coefficients that are functions depending on the coordinates and the deformation parame-ter.Consequently, proposition 5.1 leads to a deformation of the initial Lie system (5.3) and of the LHone (5.8) defined by X z : = X z ,3 + Ω ( t ) X z ,1 , h z : = h z ,3 + Ω ( t ) h z ,1 . (5.15)Thus we obtain the following z -parametric system of differential equations that generalizes (5.2):d x d t = sinhc ( zx ) y ,d y d t = − Ω ( t ) x + cx cosh ( zx ) sinhc ( zx ) + sinhc ( zx ) − cosh ( zx ) x y . (5.16)From the first equation, we can write y = ( zx ) d x d t ,and by substituting this expression into the second equation in (5.16), we obtain a deformation ofthe MP equation (5.1) in the formd x d t + (cid:18) x − zx tanh ( zx ) (cid:19) (cid:18) d x d t (cid:19) = − Ω ( t ) x sinhc ( zx ) + c zx tanh ( zx ) .Note that this really is a deformation of the MP equation in the sense that the limit z → An essential feature of the formalism here presented is the fact that t -independent constants ofmotion for the deformed system X z (5.15) can be deduced by using the coalgebra structure of C ∞ ( H MP ∗ z , ω ) . Thus we start with the Poisson–Hopf algebra C ∞ ( H MP ∗ z , ω ) with deformed coproduct ∆ z given by (4.6) and, following section 2.4 [15], we consider the Poisson algebra morphisms D z : C ∞ ( H MP ∗ z , ω ) → C ∞ ( R x (cid:54) = ) , D ( ) z : C ∞ ( H MP ∗ z , ω ) ⊗ C ∞ ( H MP ∗ z , ω ) → C ∞ ( R x (cid:54) = ) ⊗ C ∞ ( R x (cid:54) = ) ,which are defined by D z ( v i ) = h z , i ( x , y ) : = h ( ) z , i , i =
1, 2, 3, D ( ) z ( ∆ z ( v )) = h z ,1 ( x , y ) + h z ,1 ( x , y ) : = h ( ) z ,1 , D ( ) z (cid:0) ∆ z ( v j ) (cid:1) = h z , j ( x , y ) e zh z ,1 ( x , y ) + e − zh z ,1 ( x , y ) h z , j ( x , y ) : = h ( ) z , j , j =
2, 3,where h z , i are the Hamiltonian functions (5.12), so fulfilling (5.13). Hence (see [18]) h ( ) z ,1 = ( x + x ) , h ( ) z ,2 = − (cid:16) sinhc ( zx ) x y e zx + e − zx sinhc ( zx ) x y (cid:17) , h ( ) z ,3 = (cid:32) sinhc ( zx ) y + cx sinhc ( zx ) (cid:33) e zx +
12 e − zx (cid:32) sinhc ( zx ) y + cx sinhc ( zx ) (cid:33) . .1. Deformed Milne–Pinney equation and oscillator systems h ( ) z , i fulfill the Poisson brackets (5.13). The t -independentconstants of motion are then obtained through F z = D z ( C z ) , F ( ) z = D ( ) z ( ∆ z ( C z )) ,where C z is the Casimir (4.8); these are F z = sinhc (cid:0) zh ( ) z ,1 (cid:1) h ( ) z ,1 h ( ) z ,3 − (cid:0) h ( ) z ,2 (cid:1) = c F ( ) z = sinhc (cid:0) zh ( ) z ,1 (cid:1) h ( ) z ,1 h ( ) z ,3 − (cid:0) h ( ) z ,2 (cid:1) (5.17) = (cid:34) sinhc ( zx ) sinhc ( zx ) ( x y − x y ) + c sinhc (cid:0) z ( x + x ) (cid:1) sinhc ( zx ) sinhc ( zx ) ( x + x ) x x (cid:35) e − zx e zx ,so providing the corresponding deformed Ray–Reid invariant, being (5.10) its non-deformed coun-terpart with z =
0. Notice that this invariant is related to the so-called ‘universal constant of themotion’ coming from U z ( sl ( )) and given in [20]. As in (5.11), other equivalent constants of motioncan be deduced from F ( ) z by permutation of the variables. If we set p : = y , the t -dependent Hamiltonian h z in (5.15) can be written, through (5.12), as: h z = T z + U z =
12 sinhc ( zx ) p + Ω ( t ) x + c x sinhc ( zx ) ,so deforming h given in (5.9). The corresponding Hamilton equations are just (5.16).It is worth mentioning that h z can be interpreted naturally within the framework of position-dependent mass oscillators (see [16, 55, 56, 70, 113, 126, 127, 128] and references therein). The aboveHamiltonian naturally suggests the definition of a position-dependent mass function in the form m z ( x ) : = ( zx ) = zx sinh ( zx ) , lim z → m z ( x ) =
1, lim x →± ∞ m z ( x ) =
0. (5.18)Then h z can be rewritten as h z = p m z ( x ) + m z ( x ) Ω ( t ) (cid:104) x sinhc ( zx ) (cid:105) + c m z ( x ) (cid:34) x sinhc ( zx ) (cid:35) .Thus the Hamiltonian h z can be regarded as a system corresponding to a particle with position-dependent mass m z ( x ) under a deformed oscillator potential U z ,osc ( x ) with time-dependent fre-quency Ω ( t ) and a deformed potential U z ,RW ( x ) given by U z ,osc ( x ) : = x sinhc ( zx ) = sinh ( zx ) z , (5.19) U z ,RW ( x ) : = x sinhc ( zx ) = (cid:18) zx sinh ( zx ) (cid:19) ,such that lim z → U z ,osc ( x ) = x , lim x →± ∞ U z ,osc ( x ) = + ∞ ,lim z → U z ,RW ( x ) = x , lim x →± ∞ U z ,RW ( x ) = Chapter 5. Deformation of ODEs
The Hamilton equations (5.16) can easily be expressed in terms of m z ( x ) as˙ x = ∂ h MP z ∂ p = pm z ( x ) ,˙ p = − ∂ h MP z ∂ x = − m z ( x ) Ω ( t ) x sinhc ( zx ) + cm z ( x ) cosh ( zx ) x sinhc ( zx ) + p m (cid:48) z ( x ) m z ( x ) ,and the constant of the motion (5.17) turns out to be F ( ) z = (cid:34) ( x p − x p ) m z ( x ) m z ( x ) + c m z ( x ) m z ( x ) sinhc (cid:0) z ( x + x ) (cid:1) ( x + x ) x x (cid:35) e − zx e zx . - - - x m z ( x ) m z = ( x ) = m z = ( x ) m z = ( x ) m z = ( x ) F IGURE z . - - - x U z ,osc ( x ) U z = ( x ) = x U z = ( x ) U z = ( x ) U z = ( x ) F IGURE z . In this section we consider the complex Riccati equation given byd z d t = b ( t ) + b ( t ) z + b ( t ) z , z ∈ C , (5.1) .2. Deformed Riccati equation b i ( t ) are arbitrary t -dependent real coefficients. We recall that (5.1) is related to certain planarRiccati equations [61, 153] and that several mathematical and physical applications can be foundin [118, 135].By writing z = u + iv , we find that (5.1) gives rise to a system of the type (3.1), namelyd u d t = b ( t ) + b ( t ) u + b ( t )( u − v ) , d v d t = b ( t ) v + b ( t ) uv . (5.2)Thus the associated t -dependent vector field reads X = b ( t ) X + b ( t ) X + b ( t ) X , (5.3)where X = ∂∂ u , X = u ∂∂ u + v ∂∂ v , X = ( u − v ) ∂∂ u + uv ∂∂ v , (5.4)span a Vessiot–Guldberg Lie algebra V CR (cid:39) sl ( ) with the same commutation relations (5.5). Ithas already be proven that the system X is a LH one belonging to the class P [9, 29] and that theirvector fields span a regular distribution on R v (cid:54) = . The symplectic form, coming from (3.4), and thecorresponding Hamiltonian functions (3.5) turn out to be ω = d u ∧ d vv , h = − v , h = − uv , h = − u + v v , (5.5)which fulfill the commutation rules (5.7) so defining a LH algebra H CR ω . A t -dependent Hamiltonianassociated with X reads h = b ( t ) h + b ( t ) h + b ( t ) h . (5.6)In this case, the constants of the motion (3.16) are found to be F = F ( ) = ( u − u ) + ( v + v ) v v . (5.7)As commented above, the Riccati system (5.2) is locally diffeomorphic to the MP equations (5.2)with c >
0, both belonging to the same class P [9]. Explicitly, the change of variables x = ± c (cid:112) | v | , y = ∓ c u (cid:112) | v | , u = − yx , | v | = c x , c >
0, (5.8)map, in this order, the vector fields (5.4) on R x (cid:54) = , the symplectic form ω = d x ∧ d y , Hamiltonianfunctions (5.6) and the constant of motion (5.10) onto the vector fields (5.4) on R v (cid:54) = , (5.5) and (5.7)(up to a multiplicative constant ± c ).To obtain the corresponding (non-standard) deformation of the complex Riccati system (5.2), thevery same change of variables (5.8) can be considered since, in our approach, the symplectic form(5.5) is kept non-deformed. Thus, by starting from proposition 5.1 and applying (5.8) (with c = Proposition 5.2. (i) The Hamiltonian functions given byh z ,1 = − v , h z ,2 = − sinhc ( z / v ) uv , h z ,3 = − sinhc ( z / v ) u + v sinhc ( z / v ) v , fulfill the commutation rules (5.13) with respect to the Poisson bracket induced by the symplectic form ω (5.5) defining the deformed Poisson algebra C ∞ ( H CR ∗ z , ω ) .(ii) The corresponding vector fields X z , i read X z ,1 = ∂∂ u , X z ,2 = u cosh ( z / v ) ∂∂ u + v sinhc ( z / v ) ∂∂ v , X z ,3 = (cid:32) u − v sinhc ( z / v ) (cid:33) cosh ( z / v ) ∂∂ u + uv sinhc ( z / v ) ∂∂ v ,2 Chapter 5. Deformation of ODEs which satisfy [ X z ,1 , X z ,2 ] = cosh ( z / v ) X z ,1 , [ X z ,1 , X z ,3 ] = X z ,2 , [ X z ,2 , X z ,3 ] = cosh ( z / v ) X z ,3 + z (cid:18) + u v sinhc ( z / v ) (cid:19) X z ,1 .Next the deformed counterpart of the Riccati Lie system (5.3) and of the LH one (5.6) is defined by X z : = b ( t ) X z ,1 + b ( t ) X z ,2 + b ( t ) X z ,3 , h z : = b ( t ) h z ,1 + b ( t ) h z ,2 + b ( t ) h z ,3 . (5.9)And the t -independent constants of motion turn out to be F z = F ( ) z = (cid:32) sinhc ( z / v ) sinhc ( z / v ) ( u − u ) v v + sinhc ( z / v + z / v ) sinhc ( z / v ) sinhc ( z / v ) ( v + v ) v v (cid:33) e z / v e − z / v .Therefore the deformation of the system (5.2), defined by X z (5.9), readsd u d t = b ( t ) + b ( t ) u cosh ( z / v ) + b ( t ) (cid:32) u − v sinhc ( z / v ) (cid:33) cosh ( z / v ) ,d v d t = b ( t ) v sinhc ( z / v ) + b ( t ) uv sinhc ( z / v ) . As a last application, let us consider two coupled Riccati equations given by [106]d u d t = a ( t ) + a ( t ) u + a ( t ) u , d v d t = a ( t ) + a ( t ) v + a ( t ) v , (5.10)constituting a particular case of the systems of Riccati equations studied in [15, 45].Clearly, the system (5.10) is a Lie system associated with a t -dependent vector field X = a ( t ) X + a ( t ) X + a ( t ) X , (5.11)where X = ∂∂ u + ∂∂ v , X = u ∂∂ u + v ∂∂ v , X = u ∂∂ u + v ∂∂ v , (5.12)close on the commutation rules (5.5), so spanning a Vessiot–Guldberg Lie algebra V (cid:39) sl ( ) . Fur-thermore, X is a LH system which belongs to the class I [9, 29] restricted to R u (cid:54) = v . The symplecticform and Hamiltonian functions for X , X , X read ω = d u ∧ d v ( u − v ) , h = u − v , h = (cid:18) u + vu − v (cid:19) , h = uvu − v . (5.13)The functions h , h , h satisfy the commutation rules (5.7), thus spanning a LH algebra H ω . Hence,the t -dependent Hamiltonian associated with X is given by h = a ( t ) h + a ( t ) h + a ( t ) h . (5.14)The constants of the motion (3.16) are now F = − F ( ) = − ( u − v )( u − v )( u − v )( u − v ) . (5.15) .2. Deformed Riccati equation c < x = ± ( | c | ) (cid:112) | u − v | , y = ∓ ( | c | ) ( u + v ) (cid:112) | u − v | , c < u = ± | c | x − yx , v = ∓ | c | x − yx , (5.16)which map the MP vector fields (5.4) with domain R x (cid:54) = , symplectic form ω = d x ∧ d y , Hamiltonianfunctions (5.6) and constant of motion (5.10) onto (5.12) with domain R u (cid:54) = v , (5.13) and (5.15) (up toa multiplicative constant ±| c | ), respectively.As in the previous section, the (non-standard) deformation of the coupled Riccati system (5.10) isobtained by starting again from proposition 5.1 and now applying the change of variables (5.16)with c = − Proposition 5.3. (i) The Hamiltonian functions given byh z ,1 = u − v , h z ,2 =
12 sinhc (cid:0) zu − v (cid:1)(cid:18) u + vu − v (cid:19) , (5.17) h z ,3 = sinhc (cid:0) zu − v (cid:1) ( u + v ) − ( u − v ) (cid:0) zu − v (cid:1) ( u − v ) , satisfy the commutation relations (5.13) with respect to the symplectic form ω (5.13) and define the deformedPoisson algebra C ∞ ( H ∗ z , ω ) .(ii) Their corresponding deformed vector fields turn out to be X z ,1 = ∂∂ u + ∂∂ v , X z ,2 = ( u + v ) cosh (cid:0) zu − v (cid:1) (cid:18) ∂∂ u + ∂∂ v (cid:19) + ( u − v ) sinhc (cid:0) zu − v (cid:1) (cid:18) ∂∂ u − ∂∂ v (cid:19) , X z ,3 = (cid:34) ( u + v ) + ( u − v ) sinhc (cid:0) zu − v (cid:1) (cid:35) cosh (cid:0) zu − v (cid:1) (cid:18) ∂∂ u + ∂∂ v (cid:19) + ( u − v ) sinhc (cid:0) zu − v (cid:1) (cid:18) ∂∂ u − ∂∂ v (cid:19) , which fulfill [ X z ,1 , X z ,2 ] = cosh (cid:0) zu − v (cid:1) X z ,1 , [ X z ,1 , X z ,3 ] = X z ,2 , [ X z ,2 , X z ,3 ] = cosh (cid:0) zu − v (cid:1) X z ,3 − z (cid:34) − (cid:18) u + vu − v (cid:19) sinhc (cid:0) zu − v (cid:1)(cid:35) X z ,1 .The deformed counterpart of the coupled Ricatti Lie system (5.11) and of the LH one (5.14) is definedby X z : = a ( t ) X z ,1 + a ( t ) X z ,2 + a ( t ) X z ,3 , h z : = a ( t ) h z ,1 + a ( t ) h z ,2 + a ( t ) h z ,3 . (5.18)And the t -independent constants of motion are F z = − F ( ) z = e − zu − v e zu − v ( u − v )( u − v ) (cid:104) sinhc (cid:0) zu − v (cid:1) sinhc (cid:0) zu − v (cid:1) ( u − u + v − v ) − e zu − v ( u − v ) sinhc (cid:0) zu − v (cid:1) + e − zu − v ( u − v ) sinhc (cid:0) zu − v (cid:1) sinhc (cid:0) zu − v + zu − v (cid:1) ( u + u − v − v ) .Therefore, the deformation of the system (5.10) is determined by X z (5.18). Note that the resultingsystem presents a strong interaction amongst the variables ( u , v ) through z , which goes far beyond4 Chapter 5. Deformation of ODEs the initial (naive) coupling corresponding to set the same t -dependent parameters a i ( t ) in both one-dimensional Riccati equations; namelyd u d t = a ( t ) + a ( t ) (cid:2) ( u + v ) cosh (cid:0) zu − v (cid:1) + ( u − v ) sinhc (cid:0) zu − v (cid:1)(cid:3) + a ( t ) (cid:34)(cid:32) ( u + v ) + ( u − v ) sinhc (cid:0) zu − v (cid:1) (cid:33) cosh (cid:0) zu − v (cid:1) + ( u − v ) sinhc (cid:0) zu − v (cid:1)(cid:35) ,d v d t = a ( t ) + a ( t ) (cid:2) ( u + v ) cosh (cid:0) zu − v (cid:1) − ( u − v ) sinhc (cid:0) zu − v (cid:1)(cid:3) + a ( t ) (cid:34)(cid:32) ( u + v ) + ( u − v ) sinhc (cid:0) zu − v (cid:1) (cid:33) cosh (cid:0) zu − v (cid:1) − ( u − v ) sinhc (cid:0) zu − v (cid:1)(cid:35) .5 h h h , which is the second of the relevant subalgebras of the two-photon Lie algebra h , correspondingto the highest dimensional Lie algebra of vector fields on the real plane that appears in the con-text of planar Lie–Hamilton systems admits two non-equivalent quantum deformations leadingto correspondingly non-equivalent Lie–Poisson systems. The Lie algebra h is amidst the notablenon-semisimple Lie algebras used in physics, where it provides a unified description of coherent,squeezed and intelligent states of light ([156] and references therein), also having applications in thetheory of integrable systems [19] and the description of damped harmonic oscillators [29], amongothers.The two-photon Lie algebra h , as considered in [156], is spanned by the six operators N = a + a − , A + = a + , A i = a i , B + = a + , B i = a i , M = I, (6.1)where a + and a − are the generators of the boson algebra [ a − , a + ] = I. Over this basis, the commu-tation relations are given by (see [23]): [ A ± , B ± ] = [ A − , A + ] = M , [ M , · ] = [ B − , B + ] = N + M , [ A ± , B ∓ ] = ∓ A ∓ , [ N , A ± ] = ± A ± , [ N , B ± ] = ± B ± . (6.2)It follows at once from these relations that the operators { A + , A − , M } span a subalgebra isomorphicto the Heisenberg-Weyl algebra h , which can be extended to the oscillator algebra h spanned bythese operators along with the number operator N .The two-photon Lie algebra admits two independent Casimir operators, one corresponding to thecentre generator M and a second one having degree four in the generators and given by C = (cid:16) MB + − A + (cid:17) (cid:16) MB − − A − (cid:17) − (cid:18) MN − A − A + + M (cid:19) . (6.3)For computational convenience, let us consider the change of basis D = − ( N + M ) . Now, if weunderstand the generators (cid:110) M , A − , − A + , D , B − , − B + (cid:111) of the two-photon algebra as functions on h ∗ in the natural way [19], we can consider the coordinates { v , · · · , v } on h . Taking into accountthe induced Kirillov–Konstant–Souriau Poisson structure on C ∞ ( h ∗ ) with respect to the canonicalsymplectic 2-form in the plane ω = d x ∧ d y , we are led to the brackets { v , v } = v , { v , v } = − v , { v , v } = { v , v } = − v , { v , v } = v , { v , v } = − v , { v , v } = { v , v } = v , { v , v } = − v , { v , v } = v . (6.4)The Casimir functions are found to v and C = (cid:16) v v − v v − v v v (cid:17) − v (cid:16) v + v v (cid:17) . (6.5)6 Chapter 6. Oscillator Systems from h We observe that the cubic invariant is a consequence of the Lie algebra h having the structure of asemi-direct product of su (
1, 1 ) and a Heisenberg algebra [34]. Defining V = v v + v v , V = v v − v , V = v v + v (6.6)it follows at once that v C = V + V V , showing its relation to the fourth-order Casimir operator(6.3).Using now the identity ι X i ω = d v i , the corresponding Hamiltonian vector fields are given by X = ∂∂ x , X = ∂∂ y , X = x ∂∂ x − y ∂∂ y , X = y ∂∂ x , X = x ∂∂ y , (6.7)with the nontrivial commutation relations [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = − X . (6.8) h Lie–Hamilton systems
As observed earlier, the subalgebra of h generated by { N , M , A + , A − } is isomorphic to the four-dimensional Poincaré algebra h . This Lie algebra possesses two Casimir operators, given respec-tively by M and C = MN − A + A − − A − A + . The quantum deformation of h (see [22]) is deter-mined by the coassociative coproduct ∆ ( A + ) = ⊗ A + + A + ⊗ ∆ ( M ) = ⊗ M + M ⊗ ∆ ( N ) = ⊗ N + N ⊗ e zA + , ∆ ( A − ) = ⊗ A − + A − ⊗ e zA + + zN ⊗ e zA + M , (6.9)from which the deformed commutation relations result as: [ N , A + ] = e zA + − z , [ N , A − ] = − A − , [ A − , A + ] = M e zA + . (6.10)Al already used, we consider the change of basis D = − N − M . By (6.10), the deformed Hamilto-nian functions for h are given by v z = v , v z = e zv v , v z = v , v z = − e zv z v , (6.11)corresponding to the classical Hamiltonian functions v i in the limit z → z → v j , z = v j , 0 ≤ j ≤ { v z , v z } z = v z e zv z , { v z , v z } z = − v z , { v , v } z = e zv z − z . (6.13)With respect to the canonical 2-form ω = d x ∧ d y , the Poisson structure of U ( h ) is given by theHamiltonian functions v j (0 ≤ j ≤
3) with Poisson brackets (6.13) and Casimir invariants v and C = v v − v v .The t -dependent vector field X t = ∑ j = f i ( t ) X j , z leads to the Poisson–Hopf system dxdt = e − xz f ( t ) + − e − xz z f ( t ) , dydt = yz e − xz f ( t ) + f ( t ) − y e − xz f ( t ) , (6.14) .3. Poisson–Hopf deformation of h Lie–Hamilton systems from h f j ( t ) (1 ≤ j ≤ f ( t ) =
0, the preceding system correspondsto the quantum deformation of the Lie–Hamilton system based on the Heisenberg Lie algebra h .We further observe that, under this latter assumption, the classical and deformed systems essen-tially have the same form, as the system consists in this case of a separable equation and a linearfirst-order inhomogeneous equation. In this sense, the Poisson–Hopf system associated to h doesnot lead to new types of systems.For the generic deformation based on the Poincaré algebra, the deformed noncentral invariant isgiven by C z = v z v z − v z e − zv z − z .The constants of the motion of the deformed system (6.14), computed by means of the coalgebra,are easily verified to take the form F z ( C z ) = e − zx z (cid:0) zx + e − zx − (cid:1) y , F ( ) z ( C z ) = ( y + y ) − e − z ( x + x ) ( + z ( x + x )) ( y e zx + y e zx ) z (6.15)with the expected classical limits (see Table 2 in [29])lim z → F z ( C z ) =
0, lim z → F ( ) z ( C z ) = ( x − x )( y − y ) . (6.16) h Lie–Hamilton systems from h h is anatural extension of the preceding quantum deformation of the Poincaré algebra h seen previously.It has been extensively studied, for which reason we omit the details (these can be found e.g. in[23, 19]) and merely indicate the coproduct and commutation relations. The coassociative coproductfor the quantum two-photon algebra U z ( h ) is given by: ∆ ( A + ) = ⊗ A + + A + ⊗ ∆ ( M ) = ⊗ M + M ⊗ ∆ ( N ) = ⊗ N + N ⊗ e zA + , ∆ ( B + ) = ⊗ B + + B + ⊗ e − zA + , ∆ ( A − ) = ⊗ A − + A − ⊗ e zA + + zN ⊗ e zA + M , (6.17) ∆ ( B − ) = ⊗ B − + B − ⊗ e zA + + zN ⊗ e zA + ( A − − zMN ) − zA − ⊗ e zA + N .The corresponding compatible deformed commutation relations are thus given by: [ N , A + ] = e zA + − z , [ N , A − ] = − A − , [ A − , A + ] = M e zA + , [ N , B + ] = B + , [ N , B − ] = − B − − zA − N , [ A + , B + ] = [ A − , B + ] = − e − zA + z , [ A − , B − ] = − zA − , [ · , M ] =
0, (6.18) [ A + , B − ] = − (cid:16) + e zA + (cid:17) A − + z e zA + MN , [ B − , B + ] = (cid:16) + e − zA + (cid:17) N + M − zA − B + .8 Chapter 6. Oscillator Systems from h It follows from (6.18) that the deformed Hamiltonian functions adopt the form have the form v z = v , v z = e zv v , v z = v , v z = − e zv z v , v z =
12 e zv v , v z = − (cid:18) e − zv − z (cid:19) , (6.19)which faithfully reproduce the Hamiltonian functions v i in the limit z → z → v j , z = v j , 0 ≤ j ≤
5. (6.20)The explicit Poisson brackets for the deformation are thus { v z , v z } z = v z e zv z , { v z , v z } z = − v z , { v z , v z } z = − zv z { v z , v z } z = e − zv z − z , { v , v } z = e zv z − z , { v z , v z } z = − v z ( + e zv z ) − ze zv z v z ( v z + v z − ) , { v z , v z } z = { v z , v z } z = v z − zv z ( v z + v z − ) , { v z , v z } z = − v z , { v z , v z } z = v + e − zv z + (cid:18) v z − (cid:19) ( e − zv z − ) − zv v , { v z , ·} =
0. (6.21)As follows from inspection of the coassociative coproduct (6.17) of the quantum two-photon alge-bra, the deformed generators of h give rise to a subalgebra of the quantum deformation of h , hencethe deformed Hamiltonian functions can be identified with the v j , z in equation (6.19) for 0 ≤ j ≤ X j , z given in(6.24) for these indices. We observe that, in this case, the Hamiltonian functions { v z , v z , v z } donot close as a subalgebra, as they involve the remaining generators, indicating that we are dealingwith a different quantum deformation of h as that considered earlier extending the deformationon the subalgebra sl ( R ) ⊕ R . A cumbersome but routine computation shows that the Casimirinvariants for (6.21) are v z for being a central element and the function C z = zv z v z v z ( v z + v z − ) − v z ( e zv z − ) z e − zv z + v z v z − v z v z − v z v z v z − ( e zv z − ) ( v z ( e zv z + ) + ( v z − ) ( − e zv z )) z e zv z . (6.22)It is straightforward to verify that in the limit we obtain the cubic invariant (6.5)lim z → C z = C . (6.23)The Hamiltonian vector fields X j , z associated to the functions v j , z are given by X z = X , X z = e z v z X + zv z X , X z = X , X z = − e z v z z X − v z X X z = v z X + zv z X , X z = e − z v z e − z v z − z X . (6.24)with non-vanishing commutators [ X z , X z ] = − e z v z X z , [ X z , X z ] = X z , [ X z , X z ] = zv z X z , [ X z , X z ] = e − z v z X z , [ X z , X z ] = − e z v z X z , [ X z , X z ] = X z , [ X z , X z ] = − ( + e z v z ) X z − zv z e z v z X z , [ X z , X z ] = X z , [ X z , X z ] = e − z v z v z (cid:0) e − z v z − (cid:1) X z − e − z v z X z . (6.25) .3. Poisson–Hopf deformation of h Lie–Hamilton systems from h z → X j , z = X j , lim z → (cid:2) X j , z , X k , z (cid:3) = (cid:2) X j , X k (cid:3) , 0 ≤ j ≤
5. (6.26)For generic functions f i ( t ) , the deformed Poisson–Hopf system determined by the t -dependentvector field X t = ∑ j = f i ( t ) X i , z is explicitly given by dxdt = e − xz f ( t ) + − e − xz z f ( t ) + y e − xz f ( t ) , dydt = yz e − xz f ( t ) + f ( t ) − y e − xz f ( t ) + y z − xz f ( t ) + e xz ( e xz − ) z f ( t ) . (6.27)The t -independent constants of the motion of this system are obtained by application of the coalge-bra formalism (see [26]), from which the following expressions are obtained: F ( C z ) = F ( ) ( C z ) = F ( ) ( C z ) = e − z ( x + x ) z (cid:16) zx − zx + zx − − zx + e z ( x + x ) − z ( x + x ) − (cid:17) × ( y e zx + y e zx ) − y z e − zx (cid:16) e zx − (cid:17) ( e zx − ) − y z (cid:16) e zx − (cid:17) × ( e zx − ) e − zx + y y z e − z ( x + x ) (cid:16) e z ( x + x ) + e z ( x + x ) − z ( x + x ) − z ( x + x ) − z ( x + x ) + z ( x + x ) − zx + zx − zx − zx + zx − zx + (cid:17) . (6.28)It can be routinely verified that lim z → F ( ) ( C z ) =
0, which is in agreement with the well knownfact that F ( ) ( C ) = F ( ) ( C z ) can be constructed so that in the limit we recover the firstnonvanishing t -independent constant of the motion F ( ) ( C ) = ( x ( y − y ) + x ( y − y ) + x ( y − y )) of the classical Lie–Hamilton system [29]. As a physically interesting application of the preceding deformation, let us consider a one-dimensionaldamped oscillator of the form dxdt = a ( t ) x + b ( t ) y + f ( t ) , dpdt = − c ( t ) x − a ( t ) p − g ( t ) , (6.29)where a ( t ) , b ( t ) , c ( t ) , f ( t ) and g ( t ) are arbitrary functions. The system (6.29) is actually a Lie–Hamilton system associated to the t -dependent vector field X t = f ( t ) X − g ( t ) X + a ( t ) X + b ( t ) X − c ( t ) X ,where X = ∂∂ x , X = ∂∂ p , X = x ∂∂ x − p ∂∂ p , X = p ∂∂ x , X = x ∂∂ p (6.30)0 Chapter 6. Oscillator Systems from h are Hamiltonian vector fields with respect to the symplectic form ω = d x ∧ d p . The Hamiltonianfunctions h i associated to the vector fields (6.30) are respectively h = p , h = − x , h = xp , h = p , h = − x , (6.31)showing that the resulting Vessiot–Guldberg Lie algebra of the Lie–Hamilton system is isomorphicto the two-photon Lie algebra h as given in Equation (6.7). The quantum deformation of (6.29)therefore corresponds to a system of type (6.27), for the choice of functions f ( t ) = f ( t ) , f ( t ) = − g ( t ) , f ( t ) = a ( t ) , f ( t ) = b ( t ) and f ( t ) = c ( t ) . As follows from (6.28), the deformed systempossesses a non-vanishing z -dependent constant of the motion F ( ) ( C z ) .We observe that the operators (cid:110) B + , B − , N + M (cid:111) span a simple non-compact Lie algebra isomor-phic to su (
1, 1 ) (cid:39) sl ( R ) . As both of these subalgebras admit quantum deformations, it is naturalto ask whether these can be extended to the whole two-photon algebra in a consistent way, such thatall Poisson–Hopf deformations of Lie–Hamilton systems in the plane can be described uniformlyin terms of subalgebras of the two photon algebra. In this sense, an extension of the sl ( R ) -relateddeformed systems remain to be described. Various preliminary results in this direction have beenobtained, with a complete description currently in progress. The invariants and nonlinear superposition rule for this type of Lie–Hamilton system have been analyzed in detail in[29], for which reason we skip their detailed expressions. If one wants to study the evolution of a SIS epidemic exposed to a constant heat source, like centrallyheated buildings; one can make use of quantum stochastic differential equations. It was shown thatthe stochastic SIS-epidemic model can be interpreted as a Hamiltonian system. Lie–Hamiltoniansystems admit a quantum deformation, so does the stochastic SIS-epidemic model, because it is aLie–Hamilton system.This chapter [63] proposes a quantum version of a stochastic SIS-epidemic model without usingstochastic calculus, but using the proper Hamiltonian approximation for the mean and the variance.Epidemic models try to predict the spread of an infectious disease afflicting a specific population,see [31, 109]. These models are rooted in the works of Bernoulli in the 18th century, when he pro-posed a mathematical model to defend the practice of inoculating against smallpox [82]. This wasthe start of germ theory.At the beginning of the 20th century, the emergence of compartmental models was starting to de-velop. Compartmental models are deterministic models in which the population is divided intocompartments, each representing a specific stage of the epidemic. For example, S represents thesusceptible individuals to the disease, I designates the infected individuals, whilst R stands for therecovered ones. The evolution of these variables in time is represented by a system of ordinarydifferential equations whose independent variable, the time, is denoted by t . Some of these firstmodels are the Kermack–McKendrick [90] and the Reed–Frost [1] epidemic models, both describ-ing the dynamics of healthy and infected individuals among other possibilities. There are severaltypes of compartmental models [110], as it can be the SIS model, in which after the infection theindividuals do not acquire immunity, the SIR model, in which after the infection the individualsacquire immunity, the SIRS model, for which immunity only lasts for a short period of time, theMSIR model, in which infants are born with immunity, etc. In this present work our focus is on theSIS model. The susceptible-infectious-susceptible (SIS) epidemic model assumes a population of size N andone single disease disseminating. The infectious period extends throughout the whole course of thedisease until the recovery of the patient with two possible states, either infected or susceptible. Thisimplies that there is no immunization in this model. In this approach the only relevant variableis the instantaneous density of infected individuals ρ = ρ ( τ ) depending on the time parameter τ , and taking values in [
0, 1 ] . The density of infected individuals decreases with rate γρ , where γ is the recovery rate, and the rate of growth of new infections is proportional to αρ ( − ρ ) , wherethe intensity of contagion is given by the transmission rate α . These two processes are modelledthrough the compartmental equation d ρ d τ = αρ ( − ρ ) − γρ . (7.1)2 Chapter 7. Applications to Biology
One can redefine the timescale as t : = ατ and introduce ρ : = − γ / α , so we can rewrite (7.1) as d ρ dt = ρ ( ρ − ρ ) . (7.2)Clearly, the equilibrium density is reached if ρ = ρ = ρ . Although compartmental equationshave proven their efficiency for centuries, they are still based on strong hypotheses. For example,the SIS model works more efficiently under the random mixing and large population assumptions.The first assumption is asking homogeneous mixing of the population, that is, individuals contactwith each other randomly and do not gather in smaller groups, as abstaining themselves fromcertain communities. This assumption is nevertheless rarely justified. The second assumption isthe rectangular and stationary age distribution, which means that everyone in the population livesto an age L , and for each age up to L , which is the oldest age, there is the same number of peoplein each subpopulation. This assumption seems feasible in developed countries where there existsvery low infant mortality, for example, and a long live expectancy. Nonetheless, it looks reasonableto implement probability at some point to permit random variation in one or more inputs overtime. Some recent experiments provide evidence that temporal fluctuations can drastically alter theprevalence of pathogens and spatial heterogeneity also introduces an extra layer of complexity as itcan delay the pathogen transmission. It is needless to point out that fluctuations should be considered in order to capture the spread ofinfectious diseases more closely. Nonetheless, the introduction of these fluctuations is not trivial[63]. One way to account for fluctuations is to consider stochastic variables. On the other hand, itseems that in the case of SIS models there exist improved differential equations for the mean andvariance of infected individuals.Recently, in [114], the model assumes the spreading of the disease as a Markov chain in discretetime in which at most one single recovery or transmission occurs in the duration of this infinitesimalinterval.As a result [91], the first two equations for instantaneous mean density of infected people (cid:104) ρ (cid:105) andthe variance σ = (cid:104) ρ (cid:105) − (cid:104) ρ (cid:105) are d (cid:104) ρ (cid:105) dt = (cid:104) ρ (cid:105) ( ρ − (cid:104) ρ (cid:105) ) − σ , d σ dt = σ ( ρ − (cid:104) ρ (cid:105) ) − ∆ − N (cid:104) ρ ( − ρ ) (cid:105) + γ N α (cid:104) ρ (cid:105) , (7.3)where ∆ = (cid:104) ρ (cid:105) − (cid:104) ρ (cid:105) . This system finds excellent agreement with empirical data [114]. Equa-tions (7.2) and (7.3) are equivalent when σ becomes irrelevant compared to (cid:104) ρ (cid:105) . Therefore, a gener-alization of compartmental equations only requires mean and variance, neglecting higher statisticalmoments. The skewness coefficient vanishes as a direct consequence of this assumption, so that ∆ : = σ (cid:104) ρ (cid:105) . For a big number of individuals ( N (cid:29) ) , the resulting equations are d ln (cid:104) ρ (cid:105) dt = ρ − (cid:104) ρ (cid:105) − σ (cid:104) ρ (cid:105) ,12 d ln σ dt = ρ − (cid:104) ρ (cid:105) . (7.4)The system right above can be obtained from a stochastic expansion as it is given in [150], as well. .2. The SIS model The investigation of the geometric and the algebraic foundations of a system permits to employseveral powerful techniques of geometry and algebra while performing the qualitative analysisof the system. This even results in an analytical/general solution of the system in our case. Forexample [97] and [115] for Lie symmetry approach to solve the classical SIS model. The Hamiltoniananalysis of a system plays an important role in the geometrical analysis of a given system.In [114], the SIS system (7.4) involving fluctuations has been recasted in Hamiltonian form in thefollowing way: the dependent variables are the mean (cid:104) ρ (cid:105) and the variance σ , and they both dependon time. Then, we define the dynamical variables q = (cid:104) ρ (cid:105) and p = σ , so the system (7.4) turnsout to be dqdt = q ρ − q − p , dpdt = − p ρ + pq . (7.5)We employ the abbreviation SISf for system (7.5) to differentiate it from the classical SIS model in(7.2). The letter “f" accounts for “fluctuations".We have computed the general solution to this system, finding a more general solution than the oneprovided by Nakamura and Martínez in [114]. Indeed, we have found obstructions in their modelsolution. We shall comment this in the last section gathering all our new results.The general solution for this system reads: q ( t ) = ρ e ρ t ( C ρ − ) e ρ t + C C ρ ( C ρ − ) e ρ t + C ρ ( C e ρ t + C ) , p ( t ) = C + C ρ − ρ − C + C e − ρ t . (7.6)In order to develop a geometric theory for this system of differential equations, we need to choosecertain particular solutions that we shall make use of. Here we present three different choices andtheir corresponding graphs according to the change of variables q = (cid:104) ρ (cid:105) and p = σ .4 Chapter 7. Applications to Biology F IGURE
Let us turn now to interpret these equations geometrically on a symplectic manifold. The symplectictwo-form ω = dq ∧ dp is a canonical skew-symmetric tensorial object in two-dimensions. For achosen (real-valued) Hamiltonian function h = h ( q , p ) , the dynamics is governed by a Hamiltonianvector field X h defined through the Hamilton equation ι X h ω = dh . (7.7)In terms of the coordinates ( q , p ) , the Hamilton equations (7.7) become dqdt = ∂ h ∂ p , dpdt = − ∂ h ∂ q . (7.8)It is possible to realize that the SISf system (7.5) is a Hamiltonian system since it fulfills the Hamiltonequations (7.7). To see this, consider the Hamiltonian function h = qp ( ρ − q ) + p . (7.9)and substitute it into (7.8). A direct calculation will lead us to (7.5). The skew-symmetry of thesymplectic two-form implies that the Hamiltonian function is constant all along the motion. In .2. The SIS model F IGURE holonomic Classical Mechanics, where the Hamiltonian is taken to be the total energy, this corre-sponds to the conservation of energy.
The model (7.5) can be generalized to a model represented by a time-dependent vector field X t = ρ ( t ) X + X , (7.10)where the constitutive vector fields are computed to be X = q ∂∂ q − p ∂∂ p , X = (cid:18) − q − p (cid:19) ∂∂ q + qp ∂∂ p . (7.11)The generalization comes from the fact that ρ ( t ) is no longer a constant, but it can evolve in time.First, for the vector fields in (7.11), a direct calculation shows that the Lie bracket [ X , X ] = X (7.12)6 Chapter 7. Applications to Biology F IGURE is closed within the Lie algebra. This implies that the SISf model (7.5) is a Lie system. The Vessiot-Guldberg algebra spanned by X , X is an imprimitive Lie algebra of type I according to theclassification presented in [12].If we copy the configuration space twice, we will have four degrees of freedom ( q , p , q , p ) andwe will archieve precisely two first-integrals as a consequence of the Fröbenius theorem. A first-integral for X t has to be a first-integral for X and X simultaneously. We define the diagonalprolongation (cid:101) X of the vector field X in the decomposition (7.12). Then we look for a first integral F such that (cid:101) X [ F ] vanishes identically. Notice that if F is a first-integral of the vector field (cid:101) X thenit is a first integral of (cid:101) X due to the commutation relation. For this reason, we start by integratingthe prolonged vector field (cid:101) X = q ∂∂ q + q ∂∂ q − p ∂∂ p − p ∂∂ p , (7.13) .2. The SIS model dq q = dq q = dp − p = dp − p . (7.14)Fix the dependent variable q and obtain a new set of dependent variables, say ( K , K , K ) , whichare computed to be K = q q , K = q p , K = q p . (7.15)Now, this induces the following basis in the tangent space ∂∂ K = q ∂∂ q − q p q ∂∂ p − q p q ∂∂ p , ∂∂ K = q ∂∂ p , ∂∂ K = q ∂∂ p . (7.16)provided that q is not zero. Introducing the coordinate changes exhibited in (7.15) into the diagonalprojection (cid:101) X of the vector field X , we arrive at the following expression (cid:101) X = (cid:32) K − (cid:32) + K (cid:33)(cid:33) ∂∂ K + (cid:32)(cid:32) K + K (cid:33) K − (cid:32) + K (cid:33) K (cid:33) ∂∂ K + (cid:32) K K − (cid:32) + K (cid:33) K (cid:33) ∂∂ K .To integrate the system once more, we use the method of characteristics again and obtain d ln | K | − K = d ln | K | K + K K − − K = d ln | K | K − (cid:18) + K (cid:19) . (7.17)We obtain two first integrals by integrating in pairs ( K , K ) and ( K , K ) , where K is fixed. Aftersome cumbersome calculations we obtain K = K (cid:0) k K + k k K + k − (cid:1) ( K + )( K − ) k ( k K + k ) , K = K (cid:18) k K + k K + k − k (cid:19) ( K + )( K − ) . (7.18)By substituting back the coordinate transformation (7.15) into the solution (7.18), we arrive at thefollowing implicit equations q = − q (cid:16) k k ± (cid:113) k p q + k k p q − k p q − k p q + k (cid:17) k ( − p q + k ) , p = q k + q q k k + q k − q k ( q k + q k q − q k q − k q ) . (7.19)Let us notice that the equations (7.19) depend on a particular solution ( q , p ) and two constants ofintegration ( k , k ) which are related to initial conditions.Let us show now the graphs and values of the initial conditions for which the solution reminds usof sigmoid behavior, which is the expected growth of ρ ( t ) . As particular solution for ( q , p ) , wehave made use of particular solution 2 given in Figure 7.2 through its corresponding values of q , p through the change of variables q = (cid:104) ρ (cid:105) and p = σ .Since the solution (7.19) is quite complicated, one may look for a solution of a linearized model. Wefirst employ the following change of coordinates { u = ln | K | , v = ln | K | , w = ln | K |} . (7.20)8 Chapter 7. Applications to Biology F IGURE
In terms of these new variables, the system (7.17) reads du − e − u = dve − v + e v − w − − e − u = dw e − v − ( + e − u ) . (7.21)One can solve the system above by introducing a linear approximation1 − e − u (cid:39) u , e − v + e v − w − − e − u (cid:39) u − w ,2 e − v − ( + e − u ) (cid:39) u − v , (7.22)after which (7.21) reads du u = dv u − w = dw u − v . (7.23)We can solve now v and w in terms of u and obtain v ( u ) = k u −√ + k u √ + u , w ( u ) = √ (cid:16) k u −√ − k u √ (cid:17) . (7.24)We need to isolate the constants of integration k and k . Hence, the two first integrals read now k = u √ (cid:0) √ v − √ u + w (cid:1) /2 √ k = u − √ (cid:0) √ v − √ u − w (cid:1) /2 √
2. (7.25)Now, if we substitute the coordinate changes in (7.20) and in (7.15), we arrive at the followinggeneral solution q = q exp (cid:16) − ln ( q p ) + k + k (cid:17) , p = q exp (cid:16) √ ( k − k ) ln ( q p ) + k + k (cid:17) . (7.26)which can be written as q = q (cid:16) q p (cid:17) − + k + k , p = q (cid:16) q p (cid:17) √ k − k + k + k . (7.27) .3. Lie–Hamilton analysis of the SISf model ( q , p ) and two constants of integration ( k , k ) , as in (7.19).Let us show now the graphs and values of the initial conditions for which the solution reminds usof sigmoid behavior, which is the expected growth of ρ ( t ) . As particular solution for ( q , p ) , wehave made use of particular solution 2 given in Figure 7.2 through its corresponding values of q , p through the change of variables q = < ρ > and p = σ . In this section, we shall show that the SISf model (7.5) is a Lie-Hamilton system [101]. Among thedeveloped methods for Lie–Hamilton systems, we consider a very important recent method for theobtainance of solutions as superposition principles through the Poisson coalgebra method [12].We have already proven in (7.12) that (7.5) defines a Lie system. In order to see if it is a Lie–Hamilton system, we first need to check whether the vector fields in (7.12) are Hamiltonian vectorfields. Consider now the canonical symplectic form ω = dq ∧ dp . It is easy to check that the vectorfields X and X in (7.12) are Hamiltonian with respect to the Hamiltonian functions h = − qp , h = − q p + p , (7.28)respectively. It is easy to see that the Poisson bracket of these two functions reads { h , h } = h .It means that the Hamiltonian functions form a finite dimensional Lie algebra, denoted in the lit-erature as I r = A (cid:39) R (cid:110) R , and it is isomorphic to the one defined by vector fields X , X . TheHamiltonian function for the total system is h = ρ ( t ) h + h = − q p + p − ρ ( t ) qp (7.29)and it is exactly the Hamiltonian function (7.9) proposed in [114].Lie-Hamilton systems can also be integrated in terms of a superposition rule. We need to find aCasimir function for the Poisson algebra, but unfortunately, there exists no nontrivial Casimir inthis particular case. It is interesting to see how a symmetry of the Lie algebra { X , X } commuteswith the Lie bracket, i.e. the vector field Z = − p ( C p q + C pq + C )( pq − )( pq + ) ∂∂ p + C p q + C p q − C pq − C p ( pq − ) ( pq + ) ∂∂ q (7.30)fulfills [ X , Z ] = [ X , Z ] =
0. Notice too that Z is a conformal vector field, that is, L Z ω = − ( C /2 ) ω . (7.31)Since it is a Hamiltonian system, one would expect that a first integral for Z , let us say f , wouldPoisson commute with the Poisson algebra { h , h } , since Z = − ˆ Λ ( d f ) . Nonetheless, this is notthe case unless f = constant. This implies that the Casimir is a constant, hence trivial and thecoalgebra method can not be directly applied. However, there is a way in which we can circumventthis problem by considering an inclusion of the algebra I r = A as a Lie subalgebra of a Lie algebra toanother class admitting a Lie–Hamiltonian algebra with a non-trivial Casimir. In this case, we willconsider the algebra, denoted by I (cid:39) iso (
1, 1 ) , due to the simple form of its Casimir. If we obtainthe superposition rule for I , we simultaneously obtain the superposition for I r = A as a byproduct.0 Chapter 7. Applications to Biology iso (
1, 1 ) The Lie–Hamilton algebra iso (
1, 1 ) has the commutation relations { h , h } = h , { h , h } = − h , { h , h } = h , { h , ·} =
0, (7.32)with respect to ω = dx ∧ dy in the basis { h = y , h = − x , h = xy , h = } . The Casimir associatedto this Lie–Hamilton algebra is C = h h + h h . Let us apply the coalgebra method to this case.Mapping the representation without coproduct, the first iteration is trivial, i.e., F =
0. We could usethe second-order coproduct and third-order coproduct ∆ ( ) and ∆ ( ) , or the second-order coproduct ∆ ( ) together with the permuting sub-indices property. We need three constants of motion, thiswould be equivalent to integrating the diagonal prolongation (cid:101) X on ( R ) . Using the coalgebramethod and sub-index permutation, one obtains F ( ) = ( x − x )( y − y ) = k , F ( ) = ( x − x )( y − y ) = k , F ( ) = ( x − x )( y − y ) = k . (7.33)From them, we can choose two functionally independent constants of motion. Our choice is F ( ) = k , F ( ) = k . The introduction of k simplifies the final result, with expression x ( x , y , x , y , k , k , k ) = ( x + x ) + k − k ± B ( y − y ) , y ( x , y , x , y , k , k , k ) = ( y + y ) + k − k ∓ B ( x − x ) , (7.34)where B = (cid:113) k + k + k − ( k k + k k + k k ) . (7.35)In the case that matters to us, I r = A , the third constant k is a function k = k ( x , y , x , y ) and B ≥
0. Notice though that this superposition rule is expressed in the basis (7.32), therefore, weneed the change of coordinates between the present iso (
1, 1 ) and our problem (7.28). See that thecommutation relation { h , h } = − h in (7.32) coincides with the commutation relation { h , h } = h of our pandemic system (7.28). So, by comparison, we see there is a change of coordinates x = − qp , y = q − qp . (7.36)This way, introducing this change (7.36) in (7.34), the superposition principle for our Hamiltonianpandemic system reads q = A × C − p = − C × D − (7.37)where A : = (cid:32) q + q + ( k − k ± B )( p − p ) (cid:33) (cid:18) p + p + ( k − k ∓ B )( q − q ) (cid:19) , C : = (cid:32) q + q + ( k − k ± B )( p − p ) (cid:33) − D : = q + q + ( k − k ± B )( p − p ) (cid:18) p + p + ( k − k ∓ B )( q − q ) (cid:19) .Here, ( q , p ) and ( q , p ) are two particular solutions and k , k , k are constants of integration. .4. A deformed SISf model < ρ > = q ( t ) and σ = p using the two particular solutions inFigure 7.2 and Figure 7.3 provided in the introduction. Notice that we have renamed c = ( k − k ± B ) and k = ( k − k ∓ B ) . For the SISf model (7.5), we start with the Vessiot–Guldberg algebra (7.12) labelled as I r = A . To obtaina deformation of a Lie algebra I r = A , we need to rely on a bigger Lie algebra, in this case, we makeuse of sl ( ) . To this end, consider the vector fields X and X in (7.11), and let X be a vector fieldgiven by X : = p q ( − p q + c + ) + c ( p q − ) ∂∂ q − p q ( c + )( p q − ) ∂∂ p , (7.38)where c ∈ R . Then, { X , X , X } span a Vessiot–Lie algebra V isomorphic to sl ( ) that satisfies thefollowing commutation relations [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X . (7.39)This vector field X admits a Hamiltonian function, say h , with respect to the canonical symplecticform on R , so that we have the family h = − qp , h = p − q p , h = p q + c − p q . (7.40)Hence, { h , h , h } span a Lie–Hamilton algebra H ω ; isomorphic to sl ( ) where the commutationrelations with respect to the Poisson bracket induced by the canonical symplectic form ω on R aregiven by { h , h } ω = h , { h , h } ω = − h , { h , h } ω = h . (7.41) Step 1.
Applying the non-standard deformation of sl ( ) in [13] we arrive at the Hamiltonian func-tions h z ;1 = − shc ( zh z ;2 ) qp , h z ;2 = p − q p , h z ;3 = − p (cid:0) shc ( zh z ;2 ) q p + c (cid:1) shc ( zh z ;2 )( q p − ) , (7.42)Accordingly, the Poisson brackets are computed to be { h z ;1 , h z ;2 } ω = shc ( zh z ;2 ) h z ;2 , { h z ;2 , h z ;3 } ω = h z ;1 , { h z ;1 , h z ;3 } ω = − cosh ( zh z ;2 ) h z ;3 , (7.43) Step 2.
The vector fields X z ;1 and X z ;2 associated to the Hamiltonian functions h z ;1 and h z ;2 exhibitedin (7.42) are X z ,1 = cosh (cid:16) z ( p − q p ) (cid:17) ( p q − ) (cid:20) ( − p q ) ∂∂ q + ( p q − p q ) ∂∂ p (cid:21) + shc (cid:16) z ( p − q p ) (cid:17) ( p q − ) (cid:20) q ∂∂ q − p ( p q + ) ∂∂ p (cid:21) , X z ,2 = (cid:18) − q − p (cid:19) ∂∂ q + qp ∂∂ p . (7.44)2 Chapter 7. Applications to Biology
We do not write explicitly the expression of the vector field X z ;3 because it does not play a relevantrole in our system. The deformed vector fields keep the commutation relations [ X z ;1 , X z ;2 ] = cosh (cid:18) z (cid:18) p − q p (cid:19)(cid:19) X z ;2 . (7.45) Step 3.
The total Hamiltonian function for the deformed model is h z = ρ ( t ) h z ;1 + h z ;2 = − ρ ( t ) shc ( zh z ;2 ) qp + p − q p . (7.46)so that the deformed dynamics is computed to be dqdt = cosh (cid:16) z ( p − q p ) (cid:17) ( p q − ) ( − p q ) + shc (cid:16) z ( p − q p ) (cid:17) ( p q − ) q ρ ( t ) − q − p , dpdt = cosh (cid:16) z ( p − q p ) (cid:17) ( p q − ) ( p q − p q ) − p shc (cid:16) z ( p − q p ) (cid:17) ( p q − ) ( p q + ) ρ ( t ) − qp . (7.47)This system describes a family of z-parametric differential equations that generalizes the SISf model(7.5), where the demographic interaction and both rates allow a more realistic representation of theepidemic evolution. According to the kind of deformation, this may be called a quantum familySISf model. Note that the SISf model can be recovered in the limit when z tends to zero. For the present case, the constants of motion are computed to be F ( ) = c F ( ) = (cid:16) h ( ) + h ( ) (cid:17) (cid:16) h ( ) + h ( ) (cid:17) − (cid:16) h ( ) + h ( ) (cid:17) , (7.48)after the quantization, the latter one becomes F ( ) z = shc (cid:16) zh ( ) z ;2 (cid:17) h ( ) z ;2 h ( ) z ;3 − (cid:16) h ( ) z ;1 (cid:17) , (7.49)where h ( ) z ; j : = D ( ) z ( ∆ z ( v j )) . This coproduct ∆ z can be described as a follows ∆ z ( v ) = v ⊗ + ⊗ v , ∆ z ( v j ) = v j ⊗ e zv + e − zv ⊗ v j , j =
1, 3.More explictly, using the expressions given in (5.12), we have h ( ) z ; j = h z ; j ( q , p ) e zh z ;2 ( q , p ) + h z ; j ( q , p ) e − zh z ;2 ( q , p ) , j =
1, 3 h ( ) z ;2 = h z ;2 ( q , p ) + h z ;2 ( q , p ) . (7.50)So, to retrieve another constant of motion we can apply the trick of permuting indices. Then, herewe have a second constant of motion, writing it implicitly, F ( ) z , ( ) = shc (cid:16) zh ( ) z ;2 ( ) (cid:17) h ( ) z ;2 ( ) h ( ) z ;3 ( ) − (cid:16) h ( ) z ;1 ( ) (cid:17) , (7.51)where the sub-index ( ) means that the variables ( q , p ) are interchanged with ( q , p ) when theyappear in the deformed Hamiltonian functions h z ; j and h ( ) z ; j ( ) = h z ; j ( q , p ) e zh z ;2 ( q , p ) + h z ; j ( q , p ) e − zh z ;2 ( q , p ) , j =
1, 3 h ( ) z ;2 ( ) = h z ;2 ( q , p ) + h z ;2 ( q , p ) . .4. A deformed SISf model h z ;2 ( q , p ) = − shc ( zh z ;2 ) q p , h z ;2 = p − q p , h z ;3 = − p (cid:0) shc ( zh z ;2 ) q p + c (cid:1) shc ( zh z ;2 )( q p − ) , h z ;2 ( q , p ) = − shc ( zh z ;2 ) q p , h z ;2 = p − q p , h z ;3 = − p (cid:0) shc ( zh z ;2 ) q p + c (cid:1) shc ( zh z ;2 )( q p − ) , (7.52)whilst in (7.51) h z ;2 ( q , p ) = − shc ( zh z ;2 ) q p , h z ;2 = p − q p , h z ;3 = − p (cid:0) shc ( zh z ;2 ) q p + c (cid:1) shc ( zh z ;2 )( q p − ) , h z ;2 ( q , p ) = − shc ( zh z ;2 ) q p , h z ;2 = p − q p , h z ;3 = − p (cid:0) shc ( zh z ;2 ) q p + c (cid:1) shc ( zh z ;2 )( q p − ) . (7.53)If we set these two first integrals equal to a constant, F ( ) z , ( ) = k and F ( ) z = k , with k , k ∈ R , one is able to retrieve a superposition principle for q = q ( q , q , p , p , k , k ) and p = p ( q , q , p , p , k , k ) . Notice that here ( q , p ) and ( q , p ) are two pairs of particular solutionsand k , k are two constants over the plane to be related to initial conditions [63].4 Chapter 7. Applications to Biology F IGURE .4. A deformed SISf model F IGURE In this work, the notion of Poisson–Hopf deformation of LH systems has been proposed. Thisframework differs radically from other approaches to the LH systems theory [15, 41, 47, 49, 154], asour resulting deformations do not formally correspond to LH systems, but to an extended notion ofthem that requires a (non-trivial) Hopf structure and is related with the non-deformed LH systemby means of a limiting process in which the deformation parameter z vanishes. Moreover, theintroduction of Poisson–Hopf structures allows for the generalization of the type of systems underinspection, since the finite-dimensional Vessiot–Guldberg Lie algebra is replaced by an involutivedistribution in the Stefan–Sussman sense (Chapter 3).In Chapter 4, the Poisson analogue of the non-standard quantum deformation of sl ( ) has beenstudied, establishing explicitly the constants of the motion for the quantum deformed systems. Thethree non-equivalent LH systems in the plane based on the Lie algebra sl ( ) have been describedin unified form, which provides a nice geometrical interpretation of both these systems and theircorresponding quantum deformations. Chapter 5 is devoted to the analysis of specific systems ofdifferential equations and their deformed counterpart. We first consider the Milne–Pinney equa-tion, the deformations of which provide us with new oscillator systems with the particularity thatthe mass of the particle is dependent on the position, and where the constants of the motion are ex-plicitly obtained. In particular, the Schrödinger problem for position-dependent mass Hamiltoniansis directly connected with the quantum dynamics of charge carriers in semiconductor heterostruc-tures and nanostructures (see, for instance, [27, 79, 151]). In this context, it is worthy to be observedthat the standard or Drinfel’d–Jimbo deformation of sl ( ) would not lead to an oscillator with aposition-dependent mass as, in that case, the deformation function would be sinhc ( zqp ) instead ofsinhc ( zq ) ; this can clearly be seen in the corresponding symplectic realization given in [14]. Thisfact explains that we have chosen the non-standard deformation of sl ( ) due to its physical ap-plications. In spite of this, the Drinfel’d–Jimbo deformation could provide additional deformationfor the Milne–Pinney equation, leading to systems that are non-equivalent to those studied here.In any case, these examples suggest an alternative approach to dynamical systems with a noncon-stant mass, for which the classical tools are of limited applicability. The second type of LH systemsthat has been studied corresponds to the complex and coupled Riccati equations, which have beenextensively studied in the literature. For them, the deformed versions of the corresponding LH sys-tems and their constants of the motion have been obtained. The main results of Chapters 3-5 havebeen published in [13] and [14]. In Chapter 6 we focus on oscillator systems obtained as a deforma-tion of LH systems based on the oscillator algebra h , seen as a subalgebra of the 2-photon algebra h . In particular, these deformations can be obtained as the restriction of the corresponding de-formed h Poisson–Hopf systems. An illustrative example of this type of deformations is given bya generalization of the damped oscillator. It remains to explicitly determine a superposition rule forsuch systems, a problem that it is currently in progress. In Chapter 7 the affine Lie algebra b , seenas a subalgebra of sl ( ) , is used to obtain quantum deformed systems applicable in the context ofepidemiological models. This approach constitutes a novelty, as the techniques usually employedfor this type of models are essentially of stochastic nature. The results of this chapter have recentlybeen submitted for publication.There is a plethora of problems and applications that emerge from the Poisson–Hopf algebra de-formation formalism. Although the results have been principally focused on the two-dimensionalcase, for which an explicit classification of LH algebras exists [9, 15], the results are valid for arbi-trary manifolds and higher-dimensional Vessiot–Guldberg Lie algebras. A systematic analysis ofthe known systems would certainly lead to a richer spectrum of properties for the deformed sys-tems that deserve further investigation. In particular, the dynamical properties of specific systems8 Chapter 8. Conclusions of differential equations can be studied with these techniques, and it is expected that some new andintriguing features will emerge from this analysis.As a byproduct, and related to the current COVID-19 pandemic, one may wonder whether a de-scription of the pandemic could be related to a SISf-pandemic model. The SISf model is a very firstapproximation for a trivial infection process, in which there are only two possible states for an indi-vidual in the population: they are either infected or susceptible to the infection. Hence, this modeldoes not provide the possibility of acquiring immunity at any point. It seems that COVID-19 pro-vides some certain types of immunity, but only a thirty percent of the infected individuals, hence,a SIR model that considers “R" for recovered individuals (not susceptible anymore, i.e., immune) isnot a proper model for the current situation. It would be interesting to have a model contemplat-ing immune and nonimmunized individuals simultaneously. Currently we are still in search of astochastic Hamiltonian model that includes potential immunity and nonimmunity.
One of the most important questions to be addressed is whether the Poisson–Hopf algebra approachcan provide an effective procedure to derive a deformed analogue of superposition principles fordeformed LH systems. It would also be interesting to know whether such a description is simulta-neously applicable to the various non-equivalent deformations, like an extrapolation of the notionof Lie algebra contraction to Lie systems. Another open problem worthy to be considered is thepossibility of getting a unified description of such systems in terms of a certain amount of fixed ‘el-ementary’ systems, thus implying a first rough systematization of LH-related systems from a moregeneral perspective than that of finite-dimensional Lie algebras. Some possible future work in thisdirection can be summarized as follows:• In the classification of LH systems on the plane, the so-called 2-photon algebra plays a centralrole, as it is the highest dimensional algebra that can appear with the properties of a LH alge-bra. The study of their quantum deformations is, therefore, a fundamental question to com-plete the analysis of the deformations of the LH systems on the plane. It should be noted thatthere are essentially two different possibilities for these deformations, depending on the struc-ture of two prominent subalgebras, the algebra h and sl ( ) , which gives rise to systems anddeformations with different properties. The first case, based on the oscillator algebra h , haspartially been considered in Chapter 6. However, there still remains the problem of obtainingan effective superposition rule, the implementation of which is currently being studied. Theanalysis must be completed identifying particular classes of systems of differential equationsthat can be deformed by this procedure, and that can be interpreted as perturbations of theinitial system. The second case, based on the extension of the results for sl ( ) to the 2-photonalgebra, is structurally quite different due to the corresponding quantum deformation. It isexpected that new systems with diverging properties will emerge from this analysis. From thepoint of view of applications, these systems have a multitude of interesting properties, suchas new systems of the Lotka–Volterra type or oscillator systems with variable frequencies ormasses dependent on one or more parameters, but whose dynamics can be characterized bythe existence of a procedure for the systematic and explicit construction of the constants ofmotion and the superposition rules. The complete analysis of the Poisson–Hopf deformationsof the LH systems based on the 2-photon algebra is currently in progress, to be sent soon forpublication.• On the other hand, it should be noted that, currently, there is no classification of LH systemsfor dimension n ≥
3. A problem of interest that arises in this context is to analyze the possi-bility of generating new LH systems, both classical and deformed, by means of the extensionof the systems on the plane, in combination with the projections of the realizations of vectorfields. In this context, it is known that projections of Lie algebra realizations associated witha linear representation give rise to non-linear realizations. Analyzing the question from theperspective of functional algebras (Hamiltonians), it is conceivable that there exist compatiblesymplectic structures that give rise to LH systems in higher dimensions, as well as a depen-dence of the symplectic forms of the representation. Criteria of this type can be combined .1. Future work b as a subalgebra of sl ( ) , it is natural to con-struct the corresponding model but considering b as a subalgebra of the oscillator algebra h .Again, the quite distinct quantum deformation leads to systems with different properties, andboth approaches should be compared in detail, analyzing the numerical solutions deducedfrom both approaches. The first steps in this direction are also currently under scrutiny.• We would also like to extend our study to more complicated compartmental models, althoughat a first glance we have not been able to identify more Lie systems, at least in their currentform. We suspect that the Hamiltonian description of these compartmental models couldnonetheless behave as a Lie system, as it has happened in our presented case. This shall bepart of our future endeavors. Moreover, one could inspect in more meticulous detail how thesolutions of the quantum-deformed system (7.47) recover the nondeformed solutions whenthe introduced parameter tends to zero. We need to further study how this precisely modelsa heat bath, and if this new integrable system could correspond to other models apart frominfectious models. We would like to figure out whether it is possible to modelize subatomicdynamics with the resulting deformed Hamiltonian (7.46). There exists a stochastic theory ofLie systems developed in [95] that could be another starting point to deal with compartmentalsystems. In the present work we were lucky to find a theory with fluctuations that happenedto match a stochastic expansion, but this is rather more of an exception than a rule. Indeed,it seems that the most feasible way to propose stochastic models is using the stochastic Lietheory instead of expecting a glimpse of luck with fluctuations. As stated, finding particularsolutions is by no means trivial. The analytic search is a very intricate task. We think that inorder to fit particular solutions in the superposition principle, one may need to compute theseparticular solutions numerically. Some specific numerical methods for particular solutions ofLie systems can be devised in [124].• Finally, starting from the Chebyshev equation, it has been shown that the point of Noethersymmetries of this equation can be expressed for arbitrary n in terms of the Chebyshev poly-nomials T n ( x ) , U n ( x ) of first and second kind, respectively. Moreover, it has been observedthat the generic realization of the Lie point symmetry algebra sl ( R ) can be enlarged to moregeneral linear homogeneous second-order ODEs, the solutions of which are expressible interms of trigonometric or hyperbolic functions. In particular, the commutators of the genericpoint symmetries show that various of the algebraic relations of the general solutions actuallyarise as a consequence of the symmetry. The same conclusions hold for the structure of thefive-dimensional subalgebra of Noether symmetries. The realization of the symmetry gener-ators has been shown to remain valid for differential equations of hypergeometric type, en-abling us to obtain realizations of sl ( R ) in terms of hypergeometric functions in general andvarious orthogonal polynomials in particular, such as the Chebyshev or Jacobi polynomials.Another remarkable fact emerges from this analysis; namely, that the forcing terms are al-ways independent on the “velocities" y (cid:48) , y (cid:48) . This is again a consequence of the chosen genericrealization, and the question whether other generic realizations in terms of the general solu-tion of the ODE (or system) enable to determine forcing terms that explicitly depend on thederivatives, and even lead to autonomous differential equations (systems), arises naturally.In this context, it would be desirable to obtain a realization of sl ( R ) that not only enablesto describe generically the point and Noether symmetries of the Jacobi polynomials, but alsoapplies to the differential equations associated to the remaining families of orthogonal poly-nomials, specifically the Laguerre and Hermite polynomials. This would allow to constructfurther non-linear equations and systems possessing a subalgebra of Noether symmetries, thegenerators of which are given in terms of these orthogonal polynomials.1 Part III
Appendix A Lie algebras: Elementary properties
A.1 Lie algebras
In this Appendix, we recall the main structural properties of Lie algebras used in this work. Detailscan be found in [141, 155].Given a Lie algebra g of dimension n over a basis { X , · · · , X n } with commutators (cid:2) X i , X j (cid:3) = C kij X k , (A.1)the structure constants (cid:110) C kij (cid:111) correspond to the coefficients of a skew-symmetric 2-covariant 1-contravariant tensor µ defined on the linear space underlying g and satisfying the Jacobi identity. If (cid:8) ω , · · · , ω n (cid:9) denotes the dual basis of 1-forms to { X , · · · , X n } , the Maurer-Cartan equations of g are defined as d ω k = − C kij ω i ∧ ω j , 1 ≤ i , j , k ≤ n . (A.2)In particular, the Jacobi identity is satisfied if and only of the 2-forms d ω k are closed [132]. Definition A.1.
The adjoint representation ad : g → End ( g ) of a Lie algebra g is given byad ( X ) ( Y ) = [ X , Y ] , X , Y ∈ g . (A.3) Definition A.2.
The Killing form κ of g is the bilinear symmetric form κ : g × g → R defined by κ ( X , Y ) = Tr ( ad ( X ) · ad ( Y )) , X , Y ∈ g . (A.4)In particular, it follows that a Lie algebra g is semisimple if and only if κ is non-degenerate, i.e.,det () (cid:54) =
0. For real Lie algebras, the signature of the Killing form further determines the isomor-phism class [155].
Proposition A.3 (Levi decomposition) . Any Lie algebra g admits a decomposition g = s −→⊕ r , (A.5) where r is a maximal solvable ideal of g and s (cid:39) g / r is the maximal semisimple subalgebra. The semisimple algebra s of (A.5) is usually called the Levi subalgebra of g .The Lie algebras sl ( R ) and sl ( R ) are semisimple by the preceding criterion. Moreover, they aresimple Lie algebras [139].5 B The hyperbolic sinc function
B.1 The hyperbolic sinc function
The hyperbolic counterpart of the well-known sinc function is defined bysinhc ( x ) : = sinh ( x ) x = (cid:40) sinh ( x ) x , for x (cid:54) = x = x = ( x ) = ∞ ∑ n = x n ( n + ) ! .And its derivative is given bydd x sinhc ( x ) = cosh ( x ) x − sinh ( x ) x = cosh ( x ) − sinhc ( x ) x .Hence the behaviour of sinhc ( x ) and its derivative remind that of the hyperbolic cosine and sinefunctions, respectively. We represent them in figure B.1.A novel relationship of the sinhc function (and also of the sinc one) with Lie systems can be estab-lished by considering the following second-order ordinary differential equation t d x d t + x d t − η t x =
0, (B.1)where η is a non-zero real parameter. Its general solution can be written as x ( t ) = A sinhc ( η t ) + B cosh ( η t ) t , A , B ∈ R .Notice that if we set η = i λ with λ ∈ R ∗ we recover the known result for the sinc function: t d x d t + x d t + λ t x = x ( t ) = A sinc ( λ t ) + B cos ( λ t ) t . (B.2)Next the differential equation (B.1) can be written as a system of two first-order differential equa-tions by setting y = d x /d t , namelyd x d t = y , d y d t = − t y + η x .These equations determine a Lie system with associated t -dependent vector field X = − t X + X + η X , (B.3)where X = y ∂∂ y , X = y ∂∂ x , X = x ∂∂ y , X = x ∂∂ x + y ∂∂ y ,fulfill the commutation relations [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X − X , [ X , · ] = Appendix B. The hyperbolic sinc function - - x - - shc ( x ) ch ( x ) d dx shc ( x ) sh ( x ) F IGURE
B.1: The hyperbolic sinc function versus the hyperbolic cosine function andthe derivative of the former versus the hyperbolic sine function.
Hence, these vector fields span a Vessiot–Guldberg Lie algebra V isomorphic to gl ( ) with domain R x (cid:54) = . In fact, V is diffeomorphic to the class I (cid:39) gl ( ) of the classification given in [9]. Thediffemorphism can be explictly performed by means of the change of variables u = y / x and v = x , leading to the vector fields of class I with domain R v (cid:54) = given in [9] X = u ∂∂ u , X = − u ∂∂ u − uv ∂∂ v , X = ∂∂ u , X = − v ∂∂ v .Therefore X (B.3) is a Lie system but not a LH one since there does not exist any compatible sym-plectic form satisfying (3.4) for class I as shown in [9].Finally, we point out that the very same result follows by starting from the differential equation(B.2) associated with the sinc function.7 C Orthogonal systems andsymmetries of ODEs
C.1 Point symmetries of ordinary differential equations
Symmetries of differential equations can be formulated by various different approaches, from theclassical one by means of vector fields and their k th -order prolongations to their reformulation interms of differential forms (see e.g. [30, 83, 84, 100, 152] and references therein). From the computa-tional point of view, a rather convenient approach to determine symmetries of differential equationsis based on the reformulation of the symmetry condition in terms of differential operators [84]. Asin the following we adopt this method for the computation of symmetries, we briefly review themain facts of the procedure (see e.g. [141] for details). It is well known that a scalar second-orderordinary differential equation y (cid:48)(cid:48) = ω (cid:0) x , y , y (cid:48) (cid:1) (C.1)can be formulated in equivalent form in terms of the partial differential equation (PDE) A f = (cid:18) ∂∂ x + y (cid:48) ∂∂ y + ω (cid:0) x , y , y (cid:48) (cid:1) ∂∂ y (cid:48) (cid:19) f =
0. (C.2)A vector field X = ξ ( x , y ) ∂∂ x + η ( x , y ) ∂∂ y ∈ X (cid:0) R (cid:1) is called a (Lie) point symmetry generator ofthe equation (C.1) if the prolonged vector field˙ X = X + (cid:18) d η dx − d ξ dx y (cid:48) (cid:19) ∂∂ y (cid:48) (C.3)satisfies the commutator (cid:2) ˙ X , A (cid:3) = − d ξ dx A . (C.4)We observe in particular that the condition on the prolongation of the symmetry generator X is au-tomatically given by the commutator. If we expand the latter, it follows that the only non-vanishingcomponent is that related to the basic vector field ∂∂ y (cid:48) . From this we extract the equation definingthe symmetry condition (cid:0) y (cid:48) (cid:1) ∂ ξ∂ y + (cid:0) y (cid:48) (cid:1) (cid:18) ∂ ξ∂ x ∂ y − ∂ξ∂ y ∂ω∂ y (cid:48) − ∂ η∂ y (cid:19) + ξ ∂ω dx + η ∂ω∂ y + ∂η∂ x ∂ω∂ y (cid:48) − ∂ η∂ x + ω (cid:18) ∂ξ∂ x − ∂η∂ y (cid:19) + y (cid:48) (cid:18) ω ∂ξ∂ x + ∂ω∂ y (cid:48) (cid:18) ∂η∂ y − ∂ξ∂ x (cid:19) + ∂ ξ∂ x − ∂ η∂ x ∂ y (cid:19) =
0. (C.5)As the components ξ and η of the symmetry generator X do not depend on y (cid:48) , the latter equationcan be separated into a system of partial differential equations. In particular, for a second-orderlinear homogeneous differential equation y (cid:48)(cid:48) + g ( x ) y (cid:48) + g ( x ) y =
0, (C.6)8
Appendix C. Orthogonal systems and symmetries of ODEs we have ω = − g ( x ) y (cid:48) − g y ( x ) , and the preceding symmetry condition separates into the follow-ing four partial differential equations (PDEs in short): ∂ ξ∂ y = (cid:18) ∂ ξ∂ x ∂ y − g ( x ) ∂ξ∂ y − ∂ η∂ x ∂ y (cid:19) = ∂ ξ∂ x − g ( x ) ∂ξ∂ x − y g ( x ) ∂ξ∂ y − ∂ η∂ x ∂ y + dg ( x ) dx ξ =
0; (C.7) g ( x ) (cid:18) ∂η∂ y − ∂ξ∂ x − η (cid:19) − g ( x ) ∂η∂ x − ∂ η∂ x − y dg ( x ) dx ξ = sl ( R ) [30, 84, 100]. For linear ho-mogeneous ordinary differential equations, the vector field X = y ∂∂ y is always a point symmetry.Moreover, if y ( x ) = λ T ( x ) + λ U ( x ) ; λ , λ ∈ R (C.8)denotes the general solution of (C.6), two additional independent symmetries of the equation canbe chosen as X = T ( x ) ∂∂ y , X = U ( x ) ∂∂ y . (C.9)These three symmetries satisfy the commutators [ X , X i ] = − X i , i =
2, 3; [ X , X ] =
0, (C.10)therefore span a solvable Lie algebra of type A [104, 139]. It is well known that a scalar second-order ODE admits a Lie algebra of point symmetries of dimension n =
0, 1, 2, 3, 8 [100, 103]. Fromthe various studies concerning the structure of linearizable differential equations (see e.g. [6, 40,103, 104] and references therein), it follows that invariance with respect to the solvable algebra A implies that the symmetry algebra of the ODE (C.6) is maximal, hence isomorphic to sl ( R ) . Bymeans of a local transformation, the ODE can be reduced to the free particle equation z (cid:48)(cid:48) = W { T ( x ) , U ( x ) } as W = det (cid:18) T ( x ) U ( x ) ddt T ( x ) ddt U ( x ) (cid:19) . (C.11) C.1.1 Symmetries of the Chebyshev equation
Chebyshev polynomials possibly constitute the simplest case of orthogonal polynomials, and pos-sess various interesting structural properties that have found extensive application in numericalanalysis [130]. In contrast to the other classical orthogonal polynomials, the Laguerre, Legendreand Hermite polynomials, which appear in a wide variety of physical problems and therefore areof considerable importance in the description of natural phenomena [89], Chebyshev polynomialsappear only marginally (e.g. in connection with the Lissajous figures [71], although they have foundextensive application in approximation theory and numerical methods [130].The Chebyshev polynomials of first and second kind are defined by T n ( x ) = cos ( n arccos x ) = (cid:104)(cid:16) x + i (cid:112) − x (cid:17) n + (cid:16) x − i (cid:112) − x (cid:17) n (cid:105) , (C.12) U n ( x ) = sin ( n arccos x ) = i (cid:104)(cid:16) x + i (cid:112) − x (cid:17) n − (cid:16) x − i (cid:112) − x (cid:17) n (cid:105) . (C.13)An alternative formulation of Chebyshev polynomials is given by means of the functions V n ( x ) = cos ( n arcsin x ) , W n ( x ) = sin ( n arcsin x ) (C.14) .1. Point symmetries of ordinary differential equations V m + ( x ) = ( − ) m U m + ( x ) , V m ( x ) = ( − ) m T m ( x ) , W m + ( x ) = ( − ) m T m + ( x ) , W m ( x ) = ( − ) m U m ( x ) . (C.15)For the weight function ϕ ( x ) = (cid:16) √ − x (cid:17) − and the closed interval [ −
1, 1 ] , the Chebyshev poly-nomials satisfy the orthogonality relations (cid:90) − T n ( x ) T m ( x ) dx √ − x = m (cid:54) = n π m = n (cid:54) = π m = n = (cid:90) − U n ( x ) U m ( x ) dx √ − x = m (cid:54) = n π m = n (cid:54) = m = n = n ≥ T n ( x ) and U n ( x ) are respectively given by a n th -order differential operator T n ( x ) = ( − ) n n n ! ( n ) ! (cid:112) − x d n d x n (cid:104) − x (cid:105) n − , (C.18) U n ( x ) = ( − ) n − n n ! n ( n ) ! d n − d x n − (cid:104) − x (cid:105) n − . (C.19)From the latter identities we can easily deduce the relations ddx T n ( x ) = n √ − x U n ( x ) ; ddx U n ( x ) = − n √ − x T n ( x ) , (C.20)from which we get ddx T n ( x ) ddx U n ( x ) + n T n ( x ) U n ( x ) − x = n . Using these relations, it is straightforward to verify that the polynomials T n ( x ) and U n ( x ) are independent solutions of the linear homogeneous second-order ODE (cid:16) − x (cid:17) y (cid:48)(cid:48) − x y (cid:48) + n y =
0. (C.22)As observed in the previously, three of the symmetry generators are immediate: X = y ∂∂ y , X = T n ( x ) ∂∂ y , X = U n ( x ) ∂∂ y . (C.23)We conclude that the Chebyshev equation (C.22) has maximal symmetry L (cid:39) sl ( R ) , and henceconstitutes a linearizable equation. Discarding the case n = n =
1, we find that the symmetries (C.23), togetherwith the following vector fields, form a basis of L : X = (cid:112) − x y (cid:18) x ∂∂ x + y ∂∂ y (cid:19) ; X = (cid:16) x − (cid:17) y ∂∂ x + y x ∂∂ y ; X = x (cid:112) − x (cid:18) x ∂∂ x + y ∂∂ y (cid:19) ; X = x (cid:16) x − (cid:17) ∂∂ x + (cid:16) x + (cid:17) y ∂∂ y ; X = (cid:112) − x (cid:18)(cid:16) x − (cid:17) ∂∂ x + yx ∂∂ y (cid:19) . (C.24)As T ( x ) = x and U ( x ) = √ − x , it follows at once that the symmetry generators (C.24) can allbe expressed in terms of the Chebyshev polynomials for n =
1, using appropriately the relations(C.18)-(C.19) and those derived from them. It is therefore natural to ask whether this realization forthe Lie algebra sl ( R ) can be modified in order to describe the point symmetries of the Chebyshevequation for arbitrary n . The answer, which is in the affirmative, will be proven to remain valid fordifferential equations having solutions of trigonometric and hyperbolic types. This specific realization differs from that considered in [67]. Appendix C. Orthogonal systems and symmetries of ODEs
C.2 Functional realization of sl ( R ) As follows from equation (C.12), the Chebyshev polynomials constitute a particular case of trigono-metric functions of the type y ( x ) = λ sin H ( x ) + λ cos H ( x ) , (C.25)with H ( x ) being an arbitrary differentiable function and λ , λ ∈ R . Functions of the form (C.25)can be shown to be solutions to the linear second-order homogeneous equation dy dx − d Hdx dHdx dydx + (cid:18) dHdx (cid:19) y =
0. (C.26)In analogy with the previous example, it is reasonable to ask whether for this equation, that also ex-hibits maximal sl ( R ) -symmetry, the symmetry generators can be described generically in termsof the fundamental solutions T ( x ) = sin H ( x ) and U ( x ) = cos H ( x ) . Making the substitution g ( x ) = (cid:16) dHdx (cid:17) , the ODE (C.26) transforms onto dy dx − g (cid:48) ( x ) g ( x ) dydx + g ( x ) y =
0. (C.27)Skipping the assumption that g ( x ) is obtained from the derivative of H ( x ) , we can formulate thesymmetry problem for the more general ODE (C.27). Without loss of generality, we can supposethat the general solution of this equation is given by y ( x ) = λ T ( x ) + λ U ( x ) , where T ( x ) and U ( x ) are two independent solutions. Proposition C.1.
For arbitrary functions g ( x ) (cid:54) = , the vector fieldsX = − T (cid:48) ( x ) g ( x ) y ∂∂ x + T ( x ) y ∂∂ y ; X = − U (cid:48) ( x ) g ( x ) y ∂∂ x + U ( x ) y ∂∂ y (C.28) are point symmetries of (C.27).Proof. Let T ( x ) be a solution of the ODE (C.27). Denoting dTdx = T (cid:48) ( x ) , the prolongation ˙ X of thevector field X is explicitly given by˙ X = − T (cid:48) ( x ) g ( x ) y ∂∂ x + T ( x ) y ∂∂ y + (cid:18) y T (cid:48) ( x ) + yy (cid:48) T ( x ) g ( x ) − T (cid:48) ( x ) g (cid:48) ( x ) g ( x ) + y (cid:48) T (cid:48) ( x ) g ( x ) (cid:19) ∂∂ y (cid:48) .(C.29)We further define the quantity R = (cid:16) d Tdx − g (cid:48) ( x ) g ( x ) dTdx + g ( x ) T ( x ) (cid:17) , which reduces to zero as T ( x ) solves the equation. If we now evaluate the commutator (C.4), after some simplification we obtainthe following expression for the symmetry condition: (cid:18) − y (cid:48) g ( x ) + yy (cid:48) g (cid:48) ( x ) g ( x ) + y (cid:19) R − yy (cid:48) g ( x ) d Rdx =
0, (C.30)showing that X is a point symmetry. Permuting T ( x ) and U ( x ) , the same argument shows that X is also a symmetry of the ODE.Clearly the vector fields { X , · · · , X } are independent, as their component in ∂∂ y depends on differ-ent powers of y and U ( x ) , T ( x ) are independent solutions of the ODE. In order to complete a basis .2. Functional realization of sl ( R ) X = [ X , X ] = − T ( x ) U (cid:48) ( x ) g ( x ) ∂∂ x + T ( x ) U ( x ) g ( x )+ T (cid:48) ( x ) U (cid:48) ( x ) g ( x ) y ∂∂ y , X = [ X , X ] = − T ( x ) T (cid:48) ( x ) g ( x ) ∂∂ x + T ( x ) g ( x )+( T (cid:48) ( x )) g ( x ) y ∂∂ y , (C.31) X = [ X , X ] = − T (cid:48) ( x ) U ( x ) g ( x ) ∂∂ x + T ( x ) U ( x ) g ( x )+ T (cid:48) ( x ) U (cid:48) ( x ) g ( x ) y ∂∂ y We observe that, according to the ODE (C.27), the solutions T ( x ) and U ( x ) are functionally relatedthrough g ( x ) , i.e., g ( x ) = T (cid:48) ( x ) C − T ( x ) = U (cid:48) ( x ) C − U ( x ) (C.32)for some constants C , C . As T ( x ) and U ( x ) do not themselves reduce to constants, a routinebut cumbersome computation shows that { X , X , X } are linearly independent vector fields. As aconsequence, { X , · · · , X } are linearly independent and can be taken as a basis of the symmetry al-gebra L of the differential equation (C.27). We will see that this realization describes the symmetriesof the ODE for arbitrary choices of g ( x ) , always yielding the same commutators.We now proceed to compute the structure constants of sl ( R ) over the preceding basis. Up to now,the only known commutators are the following: [ X , X ] = − X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = [ X , X ] = [ X , X ] = [ X , X ] = [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . (C.33)We observe in particular that X acts diagonally on the remaining generators, hence it must belongto the Cartan subalgebra h of s l ( R ) . Further, as h is an Abelian subalgebra of sl ( R ) , thebrackets (C.33) imply that a second generator of h must be a linear combination of X , X and X .However, as the functions g ( x ) , U ( x ) and T ( x ) are unknown, we ignore the coefficients of theremaining commutators. One possibility to circumvent this difficulty is to forget provisionally that { X , · · · , X } arise as symmetries of an ODE and focus only on the algebraic problem. We supposethat { X , · · · , X } are independent generators of an eight-dimensional Lie algebra, and that thecommutators (C.33) hold. Using the Jacobi condition, we can derive a parameterized expression forthe remaining brackets, or alternatively we can compute the Maurer–Cartan equations associatedto this basis [132]. Proceeding like this, it follows that { X , · · · , X } define a Lie algebra if thefollowing conditions hold: [ X , X ] = a a X , [ X , X ] = a X , [ X , X ] = a a X + a X , [ X , X ] = α X + a X + a X + a X , [ X , X ] = a ( a X − a X ) , [ X , X ] = a ( a X + X ) , [ X , X ] = a a X , [ X , X ] = [ X , X ] = − a a X , [ X , X ] = − a ( X + a X ) , [ X , X ] = a ( a X − a X ) , [ X , X ] = − a ( X + a X ) , [ X , X ] = − X , [ X , X ] = − a a X , [ X , X ] = a ( X − a X ) , [ X , X ] = a ( α X + a X + a X + a X ) , [ X , X ] = a ( X − a X ) , (C.34)where a , a , a ∈ R are arbitrary constants and α = − a (cid:0) a + a (cid:1) . As the Lie algebra must beisomorphic to s l ( R ) , its Killing form κ must be non-degenerate [132]. A routine computationshows that κ has the following determinant:det κ = − a (cid:16) a + a (cid:17) . (C.35)Thus, if κ is non-degenerate, then we must always have a (cid:54) = a (cid:54) = − a . As a (cid:54) =
0, a changeof scale always allows us to suppose that a = For the basic definitions on Lie algebras, see [139, 155]. Appendix C. Orthogonal systems and symmetries of ODEs
We now return to the interpretation of { X , · · · , X } as point symmetries of (C.27) in the realiza-tion (C.28)-(C.31). In order to satisfy the commutators (C.34), the functions T ( x ) , U ( x ) and theirderivatives T (cid:48) ( x ) , U (cid:48) ( x ) will have to satisfy certain supplementary constraints that depend on thespecific values of a and a . One one hand, such constraints will enable us to deduce relationsbetween T ( x ) , U ( x ) and further g ( x ) , in order to appear as the solutions of the differential equa-tion (C.27). On the other hand, we will derive the admissible values of a and a for which thecommutators are compatible with the generators being realized as vector fields. Developing formally the commutator of X and X , imposition of the identity [ X , X ] + a X = a T (cid:48) ( x ) + T ( x ) U (cid:48) ( x ) − U (cid:48) ( x ) − T ( x ) U ( x ) T (cid:48) ( x ) =
0. (C.36)It is not difficult to see that this equation admits an integrating factor, enabling us to rewrite (C.36)as dd x (cid:18) ( U ( x ) − a T ( x )) (cid:16) T ( x ) − (cid:17) − + β (cid:19) = β . As a consequence, we obtain that U ( x ) = a T ( x ) − β (cid:16) T ( x ) − (cid:17) , (C.38)where β (cid:54) = W (cid:54) =
0. Now the commutator [ X , X ] + X = ( T (cid:48) ( x )) + ( T ( x ) − ) g ( x ) = [ X , X ] we obtain, after some simplification, the numerical rela-tion β − a − a =
0. (C.40)With these conditions, the only remaining commutator that still imposes a constraint is [ X , X ] .Developing the latter leads to the condition a (cid:16) T ( x ) − (cid:17) (cid:18) a T ( x ) − (cid:113) a + a (cid:113) T ( x ) − (cid:19) =
0. (C.41)As the solution T ( x ) = cons. ∈ R is excluded by the previous conditions (as otherwise the Wron-skian is W = a =
0. In particular, we have that the admissible functionsfor which the realization (C.28)-(C.31) are point symmetries of the ODE (C.27) are given by g ( x ) = ( T (cid:48) ( x )) (cid:0) − T ( x ) (cid:1) − , (C.42) U ( x ) = √ a √ T −
1. (C.43)As a consequence, the squares of the functions U ( x ) and T ( x ) are related by U ( x ) − a T ( x ) + a =
0. (C.44)We observe in particular that, in addition, the identity g ( x ) T ( x ) U ( x ) + T (cid:48) ( x ) U (cid:48) ( x ) = g ( x ) with the solution of the ODE is satisfied.In view of this result, we will essentially obtain two types of functions (trigonometric and hyper-bolic) for which the symmetries are given by (C.28)-(C.31), depending on the sign of the parameter a . From (C.34), we obtain that the commutator table for the point symmetry algebra L and thegiven realization is the following: In other words, the values for which the vector fields define a realization of the Lie algebra sl ( R ) . Details can be foundin [73]. As the Lie bracket is skew-symmetric, we only display the commutators (cid:2) X i , X j (cid:3) for i < j . .2. Functional realization of sl ( R ) [ · , · ] X X X X X X X X X − X − X X X X X X X X X a X + a X X − a X X X − X a X X − X − X X X − a X + a X X X X C.2.1 Noether symmetries
The differential equation (C.27) can be seen as the equation of motion of a particle in one dimension.As such systems are always integrable and conservative (see e.g. [80, 121, 131]), it follows that thereexists a Lagrangian L ( x , y , y (cid:48) ) such that (C.27) arises as the Lagrange equation of second kind dd x (cid:18) ∂ L ∂ y (cid:48) (cid:19) − ∂ L ∂ y =
0. (C.47)In particular, as the ODE (C.27) can be reduced to the free particle equation z (cid:48)(cid:48) ( s ) = L must contain a five-dimensional subalgebra L NS corresponding to Noether symmetries [140].Recall that a point symmetry X = ξ ( x , y ) ∂∂ x + η ( x , y ) ∂∂ y is called a Noether symmetry if there existsa function V ( x , y ) such that the condition˙ X ( L ) + A ( ξ ) L − A ( V ) = ψ = ξ ( x , y ) (cid:20) y (cid:48) ∂ L ∂ y (cid:48) − L (cid:21) − η ( x , y ) ∂ L ∂ y (cid:48) + V ( x , y ) (C.49)will be a constant of the motion of the system [71, 131].For the Lagrangian defined as L (cid:0) x , y , y (cid:48) (cid:1) = (cid:112) g ( x ) (cid:16) ( y (cid:48) ) − g ( x ) y (cid:17) , (C.50)the equation of motion is equivalent to the differential equation (C.27), so that, without loss ofgenerality, we can suppose that L is the Lagrangian of the system.Evaluating the symmetry condition (C.48) for L and separating the resulting expression into powersof y (cid:48) , we obtain the system of PDEs for the components of a Noether symmetry: ∂ξ∂ y = ξ ( x , y ) g (cid:48) ( x ) − g ( x ) ∂η∂ y + g ( x ) ∂ξ∂ x =
0, (C.51)2 g ( x ) ∂ξ∂ y + − g ( x ) ∂η∂ x + g ( x ) ∂ V ∂ y =
0, (C.52) ξ ( x , y ) g ( x ) g (cid:48) ( x ) y + g ( x ) y η ( x , y ) + g ( x ) y ∂ξ∂ x + g ( x ) ∂ V ∂ x =
0. (C.53)4
Appendix C. Orthogonal systems and symmetries of ODEs
The first condition implies that ξ ( x , y ) = ϕ ( x ) . Inserting this into the second equation furthershows that η ( x , y ) satisfies the equation ∂η∂ y = (cid:18) ϕ (cid:48) ( x ) g (cid:48) ( x ) g ( x ) + ϕ (cid:48) ( x ) (cid:19) (C.54)with solution η ( x , y ) = (cid:16) ϕ (cid:48) ( x ) g (cid:48) ( x ) g ( x ) + ϕ (cid:48) ( x ) (cid:17) y + θ ( x ) . Therefore, the generic form of a Noethersymmetry is given by X = ϕ ( x ) ∂∂ x + (cid:18) (cid:18) ϕ ( x ) g (cid:48) ( x ) g ( x ) + ϕ (cid:48) ( x ) (cid:19) y + θ ( x ) (cid:19) ∂∂ y . (C.55)Reordering the terms and simplifying, the third equation can be brought to the form4 g ( x ) ∂ V ∂ y + y (cid:104) ϕ ( x ) (cid:16) g (cid:48) ( x ) − g ( x ) g (cid:48)(cid:48) ( x ) (cid:17) − g ( x ) (cid:0) g ( x ) ϕ (cid:48)(cid:48) ( x ) + g (cid:48) ( x ) ϕ (cid:48) ( x ) (cid:1)(cid:105) − g ( x ) θ (cid:48) ( x ) =
0, (C.56)from which the expression for V ( x , y ) is obtained as V ( x , y ) = − (cid:104) ϕ ( x ) (cid:16) g (cid:48) ( x ) − g ( x ) g (cid:48)(cid:48) ( x ) (cid:17) − g ( x ) ( g ( x ) ϕ (cid:48)(cid:48) ( x ) + g (cid:48) ( x ) ϕ (cid:48) ( x )) (cid:105) g ( x ) y + h ( x )+ g ( x ) θ (cid:48) ( x ) g ( x ) y . (C.57)Inserting ξ ( x , y ) , η ( x , y ) and V ( x , y ) into the equation (C.53) and simplifying the resulting expres-sion, we finally obtain the conditions to be satisfied by ϕ ( x ) and θ ( x ) in order to define a Noethersymmetry: θ ( x ) g ( x ) − g (cid:48) ( x ) g ( x ) θ (cid:48) ( x ) + θ (cid:48)(cid:48) ( x ) =
0, (C.58)4 g ( x ) ϕ (cid:48)(cid:48)(cid:48) ( x ) + ϕ (cid:48) ( x ) (cid:16) g ( x ) g (cid:48)(cid:48) ( x ) + g ( x ) − g ( x ) g (cid:48) ( x ) (cid:17) + ϕ ( x ) (cid:16) g ( x ) g (cid:48)(cid:48)(cid:48) ( x ) + g (cid:48) ( x ) − g ( x ) g (cid:48) ( x ) g (cid:48)(cid:48) ( x ) + g ( x ) g (cid:48) ( x ) (cid:17) =
0. (C.59)Now, as any Noether symmetry of (C.27) must be a linear combination of { X , · · · , X } , we con-clude from (C.28) and (C.31) that X = k X + k X + k X + k X + k X + k X , (C.60)as only these symmetries are at most linear in the variable y , being thus compatible with the form(C.55). Now X and X are clearly Noether symmetries if we take either θ ( x ) = T ( x ) or θ ( x ) = U ( x ) . As a consequence, the three remaining Noether symmetries must be a linear combinationof X , X , X and X . Instead of inspecting the preceding equation (C.59) for any arbitrary linearcombinations, we check whether X or X as given in (C.31) satisfy the condition. Take for instance X . Using the constraint (C.45) and the fact that T ( x ) , U ( x ) are independent solutions of the differentialequation (C.27), the expansion of condition (C.48) applied to X reduces to˙ X ( L ) + A ( ξ ) L − A ( V ) = y (cid:48) (cid:32) U ( x ) T (cid:48) ( x ) + T ( x ) U (cid:48) ( x ) (cid:112) g ( x ) y − ∂ V ∂ y (cid:33) − y (cid:112) g ( x ) (cid:0) T (cid:48) ( x ) U (cid:48) ( x ) − g ( x ) T ( x ) U ( x ) (cid:1) − ∂ V ∂ x . (C.61) Again, permuting T ( x ) and U ( x ) , the analysis is extensible to the vector field X . .2. Functional realization of sl ( R ) y (cid:48) we obtain the auxiliary function V ( x , y ) = U ( x ) T (cid:48) ( x ) + T ( x ) U (cid:48) ( x ) (cid:112) g ( x ) y + h ( x ) . (C.62)Inserting this into the last term of (C.61) and manipulating algebraically the expression we obtainthat y ( g ( x ) T ( x ) U ( x ) − T (cid:48) ( x ) U (cid:48) ( x )) (cid:112) g ( x ) − ∂ V ∂ x = − y [ U ( x ) Λ + T ( x ) Λ + Λ ] (cid:112) g ( x ) + h (cid:48) ( x ) , (C.63)where the Λ i (1 ≤ i ≤
3) are defined as Λ = (cid:18) T (cid:48)(cid:48) ( x ) − g (cid:48) ( x ) g ( x ) T (cid:48) ( x ) + g ( x ) T ( x ) (cid:19) , Λ = (cid:18) U (cid:48)(cid:48) ( x ) − g (cid:48) ( x ) g ( x ) U (cid:48) ( x ) + g ( x ) U ( x ) (cid:19) , Λ = (cid:0) T ( x ) U ( x ) g ( x ) + T (cid:48) ( x ) U (cid:48) ( x ) (cid:1) .As T ( x ) and U ( x ) are solutions of (C.27), it is immediate that Λ = Λ =
0, while Λ = h ( x ) = α ∈ R , the point symmetry X is alsoa Noether symmetry. Changing T ( x ) for U ( x ) further shows that X also constitutes a Noethersymmetry of the equation. Using that Noether symmetries are preserved by commutators [140], itfollows that [ X , X ] = − a X + a X is also a Noether symmetry. Proposition C.2.
Any Noether symmetry X of the ODE (C.27) has the formX = λ X + λ X + λ X + λ X + λ [ X , X ] (C.64) for some scalars λ , · · · , λ ∈ R . In particular, the Lie algebra L NS of Noether symmetries admits thefollowing Levi decomposition L NS = sl ( R ) −→⊕ V R , (C.65) where the Levi subalgebra s = sl ( R ) is generated by X , X and Y = [ X , X ] . The symmetries X , X transform according to the 2-dimensional irreducible representation V of s . The first part follows from the previous computations. Now, using Table (C.46), the commutatorsof the Noether symmetries are the following: [ · , · ] Y X X X X Y − a X a X − a X a X X Y a X X − X X X sl ( R ) and generated by X , X and Y . Thesymmetries X and X are easily seen to form a maximal solvable ideal of L NS , hence the Levidecomposition of the Lie algebra is given by (C.65).From (C.49) it is immediate to verify that the constants of the motion associated to the symmetries X and X are ψ = y (cid:48) T ( x ) − y T (cid:48) ( x ) (cid:112) g ( x ) , ψ = y (cid:48) U ( x ) − y U (cid:48) ( x ) (cid:112) g ( x ) . (C.67)For the symmetries in the Levi subalgebra sl ( R ) , the constants of the motion are quadratic in y (cid:48) and easily seen to be functionally dependent on ψ and ψ , just as it is expected from the freeparticle equation z (cid:48)(cid:48) = Appendix C. Orthogonal systems and symmetries of ODEs
C.3 Orthogonal functions as solutions to the ODE (C.27)
Among the many interesting questions arising in the theory of special functions, considerable at-tention has been devoted to the problem of obtaining and characterizing orthogonal polynomialsby means of differential equations [4, 32, 66, 78, 87, 99]. In this context, it is well known that theso-called classical orthogonal polynomials constitute essentially the only class to be determined bya second-order differential equation of Sturm-Liouville type [4, 78].An important structural result in the theory of classical orthogonal polynomials states that the Ro-drigues formula F n ( x ) = p ( x ) D n (cid:2) p ( x ) Q ( x ) n (cid:3) , (C.68)for quadratic polynomials Q ( x ) = ( b − x ) ( x − a ) , where p ( x ) is a weight function in the finiteinterval ( a , b ) , provides a polynomial F n ( x ) of degree n in x for any n ≥ p ( x ) = ( b − x ) α ( x − a ) β ; α > − β > −
1. (C.69)The F n ( x ) correspond to the class of Jacobi polynomials. For polynomials Q ( x ) of degrees d ≤ F n ( x ) for any n ≥
0. For the case of a weight function (C.69), the ODEhas the following form: ( x − a ) ( b − x ) F (cid:48)(cid:48) n ( x ) + ( a ( + α ) + b ( + β ) − ( + α + β ) x ) F (cid:48) n ( x )+ n ( n + + α + β ) F n ( x ) =
0. (C.70)As seen before, the Chebychev polynomials T n ( x ) and U n ( x ) arise from (C.68) for a = − b = α = β = − . In this context, the question arises whether, besides the Chebyshev casepreviously considered, there are other possible values of α , β , a and b such that the ODE (C.70) hasthe form (C.27). Starting from the function g ( x ) = n ( n + + α + β )( x − a ) ( b − x ) ,the constraint − g (cid:48) ( x ) g ( x ) = ( a ( + α ) + b ( + β ) − ( + α + β ) x )( x − a ) ( b − x ) (C.71)leads, after comparison of the differential equations (C.27) and (C.70), to the equation a + b − x = ( a ( + α ) + b ( + β ) − ( + α + β ) x ) . (C.72)It follows at once that this identity holds only if α = β = − , without any conditions on a and b .As a consequence, the orthogonal polynomials of the form P n ( x ) = ( b − x ) ( x − a ) D n (cid:104) ( b − x ) n − ( x − a ) n − (cid:105) , n ≥ ( x − a ) ( b − x ) F (cid:48)(cid:48) n ( x ) + (cid:18) a + b − x (cid:19) F (cid:48) n ( x ) + n F n ( x ) = g ( x ) given by g ( x ) = n ( x − a ) ( b − x ) . (C.75)The ODE (C.74) is clearly of hypergeometric type (see [3, 52, 89]), hence the orthogonal functionsobtained will always be expressible in terms of hypergeometric functions. For the general ODE for the classical orthogonal polynomials, see e.g. [147]. This case is generally seen as a special case of ultraspherical functions. See [3] for details. .3. Orthogonal functions as solutions to the ODE (C.27) C.3.1 Solutions of trigonometric type
If we consider the values b = − a , the ODE (C.74) admits the general solution of trigonometric type F n ( x ) = C cos (cid:18) n arctan x √ a − x (cid:19) + C sin (cid:18) n arctan x √ a − x (cid:19) . (C.76)With H ( x ) = n arctan (cid:18) x (cid:0) a − x (cid:1) − (cid:19) , taking T ( x ) = sin H ( x ) and U ( x ) = cos H ( x ) , the con-straints (C.43) and (C.43) are satisfied for a = −
1, while the relations (C.44) and (C.45) follow atonce. In this case, the symmetry generators of the differential equation are given by X = y ∂∂ y ; X = sin H ( x ) ∂∂ y ; X = cos H ( x ) ∂∂ y ; X = sin H ( x ) H (cid:48) ( x ) y ∂∂ x + y cos H ( x ) ∂∂ y ; X = − cos H ( x ) H (cid:48) ( x ) y ∂∂ x + y sin H ( x ) ∂∂ y ; X = sin H ( x ) H (cid:48) ( x ) ∂∂ x + y ( H ( x )) ∂∂ y ; X = − sin ( H ( x )) H (cid:48) ( x ) ∂∂ x − y (cid:16) + sin H ( x ) (cid:17) ∂∂ y ; X = − cos H ( x ) H (cid:48) ( x ) y ∂∂ x + y ( H ( x )) ∂∂ y .For this basis of generators, the commutator table of sl ( R ) is explicitly given by [ · , · ] X X X X X X X X X − X − X X X X X X X X X X − X X X X X − X − X X − X − X X X X − X X X X a =
1, the preceding solution (C.76) can be simplified by means of the trigono-metric identity arcsin x = arctan x √ − x ,and thus we recover the classical Chebyshev polynomials T n ( x ) and U n ( x ) . The basis of symmetriesof (C.22) is explicitly given by X = y ∂∂ y ; X = T n ( x ) ∂∂ y ; X = U n ( x ) ∂∂ y ; X = √ − x T n ( x ) n y ∂∂ x + y U n ( x ) ∂∂ y ; X = − √ − x U (cid:48) n ( x ) n y ∂∂ x + y T n ( x ) ∂∂ y ; X = √ − x T n ( x ) n ∂∂ x + yT n ( x ) U n ( x ) ∂∂ y ; X = − √ − x T n ( x ) U n ( x ) n ∂∂ x + y (cid:16) + T n ( x ) (cid:17) ∂∂ y ; X = − √ − x U n ( x ) n ∂∂ x + yT n ( x ) U n ( x ) ∂∂ y . (C.78)For n =
1, we recover exactly the vector fields in (C.24), showing that the generic realization de-scribes naturally the basis of symmetries of the Chebyshev equation (C.22) for arbitrary values of n .For a (cid:54) =
1, the orthogonal polynomials deduced from the Rodrigues formula are still deeply relatedto the Chebyshev case, and by means of a new scaled variable z = x a − , it can be shown that thefunctions T n (cid:16) xa (cid:17) , U n (cid:16) xa (cid:17) , n ≥ Appendix C. Orthogonal systems and symmetries of ODEs solve the differential equation (C.74). This case hence does not add essentially new variants.For a (cid:54) = b + a (cid:54) =
0, the general solution of (C.74) can be written as F n ( x ) = C cos (cid:32) n arctan a + b + x (cid:112) ( a − x ) ( x − b ) (cid:33) + C sin (cid:32) n arctan a + b + x (cid:112) ( a − x ) ( x − b ) (cid:33) . (C.80)In this case, the polynomials P n ( ) have nonzero terms in any even and odd order, hence they willbe expressible in terms of linear combinations of Jacobi polynomials [3, 145]. As an example, weenumerate the first five polynomials that arise from this choice of the parameters: n P n ( x ) (( a + b ) − x ) (cid:0)(cid:0) a + ab + b (cid:1) − ( a + b ) x + x (cid:1) (cid:0)(cid:0) a + a b + ab + b (cid:1) − ( a + b ) ( a + b ) x + ( a + b ) x − x (cid:1) (cid:0)(cid:0) a + a b + a b + ab + b (cid:1) − ( a + b ) (cid:0) a + ab + b (cid:1) x (cid:1) + (cid:0) (cid:0) a + ab + b (cid:1) x − ( a + b ) x + x (cid:1) ( a + b ) (cid:0) a + a b + a b + ab + b (cid:1) − (cid:0) a + ab + b (cid:1) × (cid:0) a + ab + b (cid:1) x + ( a + b ) (cid:0) a + ab + b (cid:1) x − (cid:0) a + ab + b (cid:1) x + ( a + b ) x − x The orthogonality relation for these polynomials is given by the formula (cid:90) ba P n ( x ) P m ( x ) ( b − x ) − ( x − a ) − dx = δ mn ∏ n − l = ( l + ) n + . (C.81)Let us finally observe that for the case a = b =
1, the differential equation x ( − x ) F (cid:48)(cid:48) n ( x ) + (cid:18) − x (cid:19) F (cid:48) n ( x ) + n F n ( x ) = F n ( x ) = C cos (cid:18) n arctan √ x √ b − x (cid:19) + C sin (cid:18) n arctan √ x √ b − x (cid:19) . (C.83)The Jacobi polynomials F n (cid:16) , x (cid:17) are obviously a solution to this equation, hence considering thetransformation x = − z , the orthogonal polynomials obtained from the Rodrigues formula canbe easily related to the Chebyshev polynomials T n ( − z ) . C.3.2 Solutions of hyperbolic type
Functions of hyperbolic type can be obtained as solutions of the ODE (C.27) choosing the auxiliaryfunction g ( x ) = − (cid:16) dHdx (cid:17) . In this case, we write the general solution as y ( x ) = λ sinh H ( x ) + λ cosh H ( x ) , (C.84) The general solution can also be expressed in terms of hyperbolic functions. .3. Orthogonal functions as solutions to the ODE (C.27) a = U ( x ) = cosh H ( x ) , T ( x ) = sinh H ( x ) . The basis of point symmetries is explicitly given by X = y ∂∂ y ; X = cosh H ( x ) ∂∂ y ; X = sinh H ( x ) ∂∂ y ; X = cosh H ( x ) H (cid:48) ( x ) y ∂∂ x + y sinh H ( x ) ∂∂ y ; X = sinh H ( x ) H (cid:48) ( x ) y ∂∂ x + y cosh H ( x ) ∂∂ y ; X = cosh H ( x ) H (cid:48) ( x ) ∂∂ x + y H ( x ) ∂∂ y ; X = sinh ( H ( x )) H (cid:48) ( x ) ∂∂ x + y (cid:16) + cosh H ( x ) (cid:17) ∂∂ y ; X = sinh H ( x ) H (cid:48) ( x ) y ∂∂ x + y ( H ( x )) ∂∂ y .We will see that for suitable choices of H ( x ) , the hyperbolic functions of (C.84) define orthonormalsystems of functions in the interval [ −
1, 1 ] .We start for example from the function g ( x ) = n (cid:16) − x (cid:17) (C.85)From (C.27) we have the differential equation y (cid:48)(cid:48) + x − x y (cid:48) + n (cid:16) − x (cid:17) y =
0. (C.86)It is not difficult to justify that no n th -order polynomials satisfy this equation for n >
0. The generalsolution can be written in terms of hyperbolic functions as y ( x ) = C sinh [ F n ( x )] + C cosh [ F n ( x )] , (C.87)where F n ( x ) is defined as F n ( x ) = n (cid:16) x (cid:112) x − − ln (cid:16) x + (cid:112) x − (cid:17)(cid:17) . (C.88)Making the substitution u = (cid:16) x √ x − − ln (cid:16) x + √ x − (cid:17)(cid:17) , it is straightforward to verify that (cid:90) − cosh ( F n ( x )) cosh ( F m ( x )) (cid:112) − x dx = i2 (cid:90) − i π cosh (cid:16) n v (cid:17) cosh (cid:16) m v (cid:17) dv (C.89)holds. The latter integral can be easily solved, and for n (cid:54) = m we obtain thati2 (cid:90) − i π cosh (cid:16) n v (cid:17) cosh (cid:16) m v (cid:17) dv = i2 (cid:34) sinh (cid:0) m − n v (cid:1) m − n + sinh (cid:0) m + n v (cid:1) m + n (cid:35) − i π . = (cid:34) sin (cid:0) m − n π (cid:1) m − n + sin (cid:0) m + n π (cid:1) m + n (cid:35) . (C.90)Now observe that if n and m have different parity, i.e., n = p and m = q +
1, then12 (cid:34) sin (cid:0) π (cid:1) q + − p + sin (cid:0) π (cid:1) q + p + (cid:35) = − q + ( − q − + p ) ( q + p + ) (cid:54) =
0, (C.91)whereas for the case of n , m having the same parity, the integral (C.89) vanishes identically. Ananalogous result is obtained if we compute the integrals for the hyperbolic sine functions.As a consequence, two possibilities are given to construct an orthogonal system of functions with g ( x ) of type (C.85): Observe that interchanging the role of T ( x ) and U ( x ) always changes the sign of a . Appendix C. Orthogonal systems and symmetries of ODEs
1. For g ( x ) = n (cid:0) − x (cid:1) , the fundamental solutions P n ( x ) = cosh (cid:104) n (cid:16) x (cid:112) x − − ln (cid:16) x + (cid:112) x − (cid:17)(cid:17)(cid:105) , (C.92) Q n ( x ) = sinh (cid:104) n (cid:16) x (cid:112) x − − ln (cid:16) x + (cid:112) x − (cid:17)(cid:17)(cid:105) (C.93)of the ODE y (cid:48)(cid:48) + x − x y (cid:48) + n (cid:16) − x (cid:17) y = w ( x ) = √ − x in the interval [ −
1, 1 ] .2. For g ( x ) = ( n + ) (cid:0) − x (cid:1) , the fundamental solutions P n ( x ) = cosh (cid:20) n + (cid:16) x (cid:112) x − − ln (cid:16) x + (cid:112) x − (cid:17)(cid:17)(cid:21) , (C.94) Q n ( x ) = sinh (cid:20) n + (cid:16) x (cid:112) x − − ln (cid:16) x + (cid:112) x − (cid:17)(cid:17)(cid:21) (C.95)of the ODE y (cid:48)(cid:48) + x − x y (cid:48) + ( n + ) (cid:16) − x (cid:17) y = w ( x ) = √ − x in the interval [ −
1, 1 ] .Considering different choices of the function g ( x ) , other orthogonal systems of functions can for-mally be obtained as solutions of the ODE (C.27). C.4 Non-linear deformations
We have seen previously that the linear homogeneous ODE (C.27) admits five Noether symmetriesif the equation is derived, via the Helmholtz condition, from the Lagrangian (C.50). The corre-sponding constants of the motion are obtained from (C.49), where the two constants linear in thevelocity y (cid:48) generate the remaining invariants. This fact, to a certain extent, is a direct consequenceof the maximal symmetry of the equation, implying in particular that it is linearizable [121]. In thissituation, we can ask how to modify the ODE (C.27) by addition of a “forcing" term such that themaximal symmetry is broken, but such that the resulting equation preserves a given subalgebra ofNoether symmetries. In order to avoid those variational symmetries with a constant of the motionlinear in y (cid:48) , this subalgebra should be chosen as the Levi subalgebra L NS . For our specific purposes,the problem can be formulated in the following terms: Does there exist a (non-linear) ODE and aLagrangian L (cid:48) such that the vector fields X and X of (C.31) are Noether symmetries?Consider an arbitrary function G ( x , y , y (cid:48) ) and define the extended Lagrangian L = (cid:112) g ( x ) (cid:16)(cid:0) y (cid:48) (cid:1) − g ( x ) y (cid:17) − G (cid:0) x , y , y (cid:48) (cid:1) . (C.96)The equation of motion associated to (C.96) has the form y (cid:48)(cid:48) − g (cid:48) ( x ) g ( x ) y (cid:48) + g ( x ) y − (cid:113) g ( x ) (cid:18) ddx (cid:18) ∂ G ∂ y (cid:48) (cid:19) − ∂ G ∂ y (cid:19) =
0, (C.97)which can be interpreted as a “deformation” of the ODE (C.27) by the forcing term G ( x , y , y (cid:48) ) . Wenow require that X and X from (C.31) are Noether symmetries of L , imposing additionally thatthe symmetry condition (C.48) is satisfied for the same function V ( x , y ) valid for the Lagrangian .4. Non-linear deformations sl ( R ) -subalgebra of Noether symmetries.We observe again that, as the vector fields X and X are obtained by interchanging the role of T ( x ) and U ( x ) , it suffices to compute the symmetry condition for only one of these symmetries. Wemake the computations for X . The Noether symmetry condition˙ X ( L ) + A ( ξ ) L − A ( V ) (C.98)with V ( x , y ) as given in (C.62) leads, after some simplification using the constraint (C.45), to thefollowing partial differential equation for G ( x , y , y (cid:48) ) : (cid:18) T ( x ) U ( x ) + T ( x ) U (cid:48) ( x ) g ( x ) (cid:19) G (cid:0) x , y , y (cid:48) (cid:1) + y T ( x ) U ( x ) ∂ G ∂ y − T ( x ) U (cid:48) ( x ) g ( x ) ∂ G ∂ x + (cid:32) y (cid:0) T ( x ) U (cid:48) ( x ) + T (cid:48) ( x ) U ( x ) (cid:1) − y (cid:48) T ( x ) (cid:32) U ( x ) + g (cid:48) ( x ) U (cid:48) ( x ) g ( x ) (cid:33)(cid:33) ∂ G ∂ y (cid:48) =
0. (C.99)Albeit complicated in form, this PDE can be solved. As we are interested in those solutions beingvalid for both the symmetries X and X , suppose that the Noether condition (C.48) is also satisfiedfor X . The corresponding PDE for G ( x , y , y (cid:48) ) is obtained from (C.99) replacing T ( x ) by U ( x ) .Taking the difference of these two equations leads to the auxiliary equation − W g ( x ) (cid:18) g (cid:48) ( x ) y (cid:48) ∂ G ∂ y (cid:48) + g ( x ) ∂ G ∂ x − g (cid:48) ( x ) G (cid:0) x , y , y (cid:48) (cid:1)(cid:19) =
0. (C.100)The general solution is easily found to be G (cid:0) x , y , y (cid:48) (cid:1) = ϕ (cid:32) y , y (cid:48) (cid:112) g ( x ) (cid:33) (cid:113) g ( x ) , (C.101)with ϕ an arbitrary function of its arguments. Inserting the latter into (C.99) and solving for ϕ showsthat the only functions G ( x , y , y (cid:48) ) for which X and X are Noether symmetries of the Lagrangian L in (C.96) are G (cid:0) x , y , y (cid:48) (cid:1) = α (cid:112) g ( x ) y , α ∈ R . (C.102)The perturbed equation of motion preserving the sl ( R ) -subalgebra of Noether symmetries istherefore y (cid:48)(cid:48) − g (cid:48) ( x ) g ( x ) y (cid:48) + g ( x ) y − α g ( x ) y =
0. (C.103)We observe that if we only require invariance by either X or X , the possibilities are wider. How-ever, in these cases the forcing term will always contain explicitly the function T ( x ) and U ( x ) , inaddition to g ( x ) . For this reason we leave this case aside. Proposition C.3.
For arbitrary functions g ( x ) (cid:54) = , the nonlinear ODE (C.103) possesses a Lie algebraof point symmetries isomorphic to sl ( R ) . Moreover, any point symmetry is a Noether symmetry for theLagrangian L = (cid:112) g ( x ) (cid:16)(cid:0) y (cid:48) (cid:1) − g ( x ) y (cid:17) − α (cid:112) g ( x ) y . (C.104) Recall that W is the Wronskian of (C.27). Appendix C. Orthogonal systems and symmetries of ODEs
Proof.
For the equation of motion associated to the Lagrangian L we have the auxiliary function ω (cid:0) x , y , y (cid:48) (cid:1) = g (cid:48) ( x ) g ( x ) y (cid:48) − g ( x ) y + α g ( x ) y . (C.105)The symmetry condition for point symmetries obtained from (C.5) is given by the system ∂ ξ∂ y = g (cid:48) ( x ) g ( x ) ∂ξ∂ y + ∂ ξ∂ x ∂ y − ∂ η∂ y =
0; (C.106) ∂ ξ∂ x + g (cid:48) ( x ) g ( x ) ∂ξ∂ x + g ( x ) (cid:18) α y − y (cid:19) ∂ξ∂ y + g (cid:48) ( x ) g ( x ) (cid:18) − g (cid:48) ( x ) g ( x ) (cid:19) ξ ( x , y ) − ∂ η∂ x ∂ y =
0; (C.107) g ( x ) (cid:18) y − α y (cid:19) (cid:18) ∂η∂ y − ∂ξ∂ x − g (cid:48) ( x ) ξ ( x , y ) (cid:19) − (cid:18) + α y (cid:19) g ( x ) η ( x , y ) − ∂ η∂ x + g (cid:48) ( x ) g ( x ) ∂η∂ x =
0. (C.108)From the first two equations we immediately obtain that ξ ( x , y ) = F ( x ) y + F ( x ) ; η ( x , y ) = y (cid:18) g (cid:48) ( x ) F ( x ) g ( x ) + F (cid:48) ( x ) (cid:19) + F ( x ) y + F ( x ) .(C.109)Inserting these functions into the equations (C.107) and (C.108) and separating with respect to thepowers of y , the free terms provide us with the two constraints F ( x ) = F ( x ) =
0. This simplifi-cation further leads to the relation F ( x ) g (cid:48) ( x ) + F (cid:48) ( x ) g ( x ) − g ( x ) F ( x ) =
0, (C.110)enabling us to write the components ξ and η of a point symmetry X as ξ ( x , y ) = F ( x ) , (C.111) η ( x , y ) = y (cid:18) F (cid:48) ( x ) + g (cid:48) ( x ) g ( x ) F ( x ) (cid:19) . (C.112)In order to satisfy the equation (C.108), the function F ( x ) must be a solution to the differentialequation d F dx + g (cid:48)(cid:48) ( x ) g ( x ) + g ( x ) − ( g (cid:48) ( x )) g ( x ) dF dx + g (cid:48)(cid:48)(cid:48) ( x ) g ( x ) − g (cid:48) ( x ) g (cid:48)(cid:48) ( x ) g ( x ) + g (cid:48) ( x ) g ( x ) + ( g (cid:48) ( x )) g ( x ) =
0. (C.113)This shows that the dimension of the symmetry algebra is at most 3. Now, since the Noether sym-metries X , X and [ X , X ] are also point symmetries of the equation [141], they automaticallysatisfy the ODE (C.113). From the commutator of the Noether symmetries it follows at once that L (cid:39) L NS (cid:39) sl ( R ) .This result implies in particular that the perturbed or deformed differential equation (C.103) isnot linearizable, and hence genuinely non-linear. It is worthy to be mentioned that differentialequations invariant under sl ( R ) have been analyzed by various authors in connection with non-linear equations, as well as in the context of systems admitting constants of the motion of a specifictype [33, 57, 74, 77, 122].We now determine the constants of the motion of the Lagrangian (C.104) using formula (C.49): forthe symmetry X , the appropriate function V ( x , y ) is given by (C.62) for h ( x ) =
0, resulting in the .4. Non-linear deformations ψ = − ( U (cid:48) ( x ) T (cid:48) ( x ) + U ( x ) T ( x ) g ( x )) g ( x ) (cid:112) g ( x ) yy (cid:48) + U ( x ) T (cid:48) ( x ) (cid:112) g ( x ) y + α T ( x ) U (cid:48) ( x ) (cid:112) g ( x ) y − T ( x ) U (cid:48) ( x ) g ( x ) (cid:0) y (cid:48) (cid:1) . (C.114)For X , the function V ( x , y ) is the same, and the constant of the motion ψ is obtained by simplypermuting U ( x ) and T ( x ) . In particular, taking the difference ψ − ψ we obtain the simplifiedinvariant ψ − ψ = W (cid:112) g ( x ) (cid:32) ( y (cid:48) ) g ( x ) + y − α y (cid:33) . (C.115)Now, for [ X , X ] = − a ( X − X ) the function V ( x , y ) satisfying condition (C.48) is V ( x , y ) = y T ( x ) T (cid:48) ( x ) √ g ( x ) and the constant of the motion ψ = − T ( x ) T (cid:48) ( x ) (cid:16) ( y (cid:48) ) + g ( x ) (cid:16) y + α y (cid:17)(cid:17) + y y (cid:48) (cid:16) g ( x ) T ( x ) − g ( x ) + (cid:16) T (cid:48) ( x ) (cid:17)(cid:17) g ( x ) (cid:112) g ( x ) .(C.116)Clearly the constants ψ = ψ − ψ and ψ are independent, and both quadratic in y (cid:48) . In contrastto the non-deformed ODE (C.27), for α (cid:54) = g ( x ) the reduced equation has an evenmore intricate form, and does not greatly simplify the integration of the non-linear equation. Withthe exception of g ( x ) =
1, that is easily seen to lead to the classical Pinney equation [123], forarbitrary non-constant g ( x ) it is more practical to use numerical methods for the resolution.As an example to illustrate this fact, we consider g ( x ) = x and the deformed equation y (cid:48)(cid:48) − y (cid:48) x + x y − α xy =
0. (C.117)For α = y ( x ) = C sin (cid:18) x √ x (cid:19) + C cos (cid:18) x √ x (cid:19) and Wronskian W = −√ x . For α (cid:54) = ψ = − √ x (cid:32) ( y (cid:48) ) x + y − α y (cid:33) , ψ = −√ x (cid:16) ( y (cid:48) ) y − xy − α x (cid:17) sin (cid:16) x √ x (cid:17) + xy y (cid:48) cos (cid:16) x √ x (cid:17) x √ x .In spite of their apparent simplicity, the fact that both constants of the motion depend explicitly onthe independent variable x makes their use rather complicated, so that we skip this step.In the following Figure we compare the solutions of (C.117) for the values α = α = α = Appendix C. Orthogonal systems and symmetries of ODEs F IGURE
C.1: y ( x ) for the non-linear ODE (C.117) and the linearizable ODE. C.4.1 Non-linear systems in N = dimensions Just as the scalar ODE (C.27) has been perturbed using a subalgebra of Noether symmetries, wecan consider the problem of deforming systems of differential equations along the same lines. Werecall that, in different contexts, variations of this ansatz have already been considered in the lit-erature (see [76] and references therein), although usuallly related to the time-dependent oscillatorequations.We start from the decoupled system in N = y (cid:48)(cid:48) − g (cid:48) ( x ) g ( x ) y (cid:48) + g ( x ) y = y (cid:48)(cid:48) − g (cid:48) ( x ) g ( x ) y (cid:48) + g ( x ) y =
0. (C.118)Clearly this system is linearizable and reducible to the free particle system (cid:8) z (cid:48)(cid:48) = z (cid:48)(cid:48) = (cid:9) , hencethe Lie algebra of point symmetries is isomorphic to the rank three simple Lie algebra sl ( R ) ofdimension 15 [5, 141]. The system (C.118) also arises as the equations of motion associated to theLagrangian L = (cid:112) g ( x ) (cid:16)(cid:0) y (cid:48) (cid:1) + (cid:0) y (cid:48) (cid:1) − g ( x ) (cid:16) y + y (cid:17)(cid:17) . (C.119)As for the scalar case, a point symmetry X = ξ ( x , y , y ) ∂∂ x + η ( x , y , y ) ∂∂ y + η ( x , y , y ) ∂∂ y is a Noether symmetry of (C.118) if the constraint (C.48) is satisfied for some function V ( x , y , y ) .The constant of the motion associated to X is then given by ψ = ξ (cid:20) y (cid:48) ∂ L ∂ y (cid:48) + y (cid:48) ∂ L ∂ y (cid:48) − L (cid:21) − η ∂ L ∂ y (cid:48) + η ∂ L ∂ y (cid:48) + V ( x , y , y ) . (C.120)As (C.118) has maximal symmetry, we know that the subalgebra of Noether symmetries must havedimension 8 [72]. Now, instead of solving the symmetry condition (C.48), we use the results ob-tained for the scalar case (C.27) to determine the Noether symmetries. We first consider the case ofNoether symmetries of the form Y = η ( x , y , y ) ∂∂ y + η ( x , y , y ) ∂∂ y . (C.121) .4. Non-linear deformations (cid:113) g ( x ) (cid:18)(cid:0) y (cid:48) (cid:1) ∂η ∂ y + (cid:0) y (cid:48) (cid:1) ∂η ∂ y (cid:19) + y (cid:48) y (cid:48) (cid:112) g ( x ) (cid:18) ∂η ∂ y + ∂η ∂ y (cid:19) + (cid:112) g ( x ) (cid:18) ∂η ∂ x + ∂η ∂ x (cid:19) − (cid:18) ∂ V ∂ y + ∂ V ∂ y (cid:19) − (cid:18) ∂ V ∂ x + (cid:113) g ( x ) ( y η + y η ) (cid:19) =
0. (C.122)From the quadratic powers in y (cid:48) and y (cid:48) it follows at once that η ( x , y , y ) = k y + f ( x ) , η ( x , y , y ) = − k y + f ( x ) . (C.123)Inserting these expressions into (C.122) we further obtain V ( x , y , y ) = ( y f ( x ) + y f ( x )) ,and the symmetry condition reduces to y (cid:16) f (cid:48)(cid:48) ( x ) − g (cid:48) ( x ) g ( x ) f (cid:48) ( x ) + g ( x ) f ( x ) (cid:17) + y (cid:16) f (cid:48)(cid:48) ( x ) − g (cid:48) ( x ) g ( x ) f (cid:48) ( x ) + g ( x ) f ( x ) (cid:17)(cid:112) g ( x ) =
0. (C.124)It follows at once that the functions f ( x ) and f ( x ) must be solutions to the ODE (C.27).This proves that, for the special type (C.121) of vector fields we obtain five independent Noethersymmetries Y = U ( x ) ∂∂ y , Y = U ( x ) ∂∂ y , Y = T ( x ) ∂∂ y , Y = T ( x ) ∂∂ y , Y = y ∂∂ y − y ∂∂ y . (C.125)In order to obtain the three remaining symmetries, we apply the results of the preceding sections.A routine computation shows that the vector fields Z = − T ( x ) U (cid:48) ( x ) g ( x ) ∂∂ x + ( T (cid:48) ( x ) U (cid:48) ( x ) + g ( x ) T ( x ) U ( x )) g ( x ) (cid:18) y ∂∂ y + y ∂∂ y (cid:19) , (C.126) Z = − U ( x ) T (cid:48) ( x ) g ( x ) ∂∂ x + ( T (cid:48) ( x ) U (cid:48) ( x ) + g ( x ) T ( x ) U ( x )) g ( x ) (cid:18) y ∂∂ y + y ∂∂ y (cid:19) (C.127)are Noether symmetries of (C.118) for the function V ( x , y , y ) = U ( x ) T (cid:48) ( x ) + T ( x ) U (cid:48) ( x ) (cid:112) g ( x ) (cid:16) y + y (cid:17) . (C.128)The vector fields Z , Z and Z = [ Z , Z ] are independent, and thus { Z , Z , Z , Y , · · · , Y } form abasis of the Lie algebra L NS of Noether symmetries of the system (C.118). In particular, { Z , Z , Z } generate a copy of sl ( R ) isomorphic to the Levi subalgebra of L NS . We further observe that Y corresponds to the infinitesimal generator of a rotation in the plane [73].Now we analyze the existence of Lagrangians L = (cid:112) g ( x ) (cid:16)(cid:0) y (cid:48) (cid:1) + (cid:0) y (cid:48) (cid:1) − g ( x ) (cid:16) y + y (cid:17) − G (cid:0) x , y , y , y (cid:48) , y (cid:48) (cid:1)(cid:17) (C.129)such that the vector fields Z and Z are Noether symmetries. The procedure is completely analo-gous to that of scalar ODEs previously considered, for which reason we omit the detailed compu-tations. Imposing that (C.48) is satisfied for Z leads, after some heavy algebraic manipulation, tothe PDE:2 T ( x ) (cid:16) U (cid:48) ( x ) g (cid:48) ( x ) + U ( x ) g ( x ) (cid:17) G (cid:0) x , y , y , y (cid:48) , y (cid:48) (cid:1) − T ( x ) U ( x ) g ( x ) U (cid:48) ( x ) ∂ G ∂ x + (cid:16) g ( x ) y (cid:0) T ( x ) U (cid:48) ( x ) + T (cid:48) ( x ) U ( x ) (cid:1) − T ( x ) (cid:16) U (cid:48) ( x ) g (cid:48) ( x ) + g ( x ) U ( x ) (cid:17) y (cid:48) (cid:17) ∂ G ∂ y + (cid:16) g ( x ) y (cid:0) T ( x ) U (cid:48) ( x ) + T (cid:48) ( x ) U ( x ) (cid:1) − T ( x ) (cid:16) U (cid:48) ( x ) g (cid:48) ( x ) + g ( x ) U ( x ) (cid:17) y (cid:48) (cid:17) ∂ G ∂ y T ( x ) U ( x ) g ( x ) (cid:18) y ∂ G ∂ y + y ∂ G ∂ y − U (cid:48) ( x ) g ( x ) ∂ G ∂ x (cid:19) =
0. (C.130) Using that T ( x ) and U ( x ) are solutions to the ODE (C.27), as well as the constraint (C.45). Appendix C. Orthogonal systems and symmetries of ODEs
For the vector field Z , the corresponding PDE satisfied by G is obtained from (C.130) permuting T ( x ) and U ( x ) . In this form, however, the equations are of little use, as all intervening functionsare unknown. We can transform the PDEs using the constraints satisfied by T ( t ) , U ( t ) and g ( t ) in order to obtain an equivalent pair of differential equations. Skipping the routine computations,such a set is given by the equations2 g ( x ) ∂ G ∂ t + y (cid:48) g (cid:48) ( x ) ∂ G ∂ y (cid:48) + y (cid:48) g (cid:48) ( x ) ∂ G ∂ y (cid:48) − g (cid:48) ( x ) G (cid:0) x , y , y , y (cid:48) , y (cid:48) (cid:1) =
0, (C.131) − T ( x ) U ( x ) (cid:18) y ∂ G ∂ y + y ∂ G ∂ y − G (cid:19) + (cid:0) T ( x ) U ( x ) y (cid:48) − y A (cid:1) ∂ G ∂ y (cid:48) + (cid:0) T ( x ) U ( x ) y (cid:48) − y A (cid:1) ∂ G ∂ y (cid:48) =
0, (C.132)where A = T ( x ) U (cid:48) ( x ) + T (cid:48) ( x ) U ( x ) . The first of these equations has the general solution G (cid:0) x , y , y , y (cid:48) , y (cid:48) (cid:1) = g ( x ) Φ (cid:32) y , y , y (cid:48) (cid:112) g ( x ) , y (cid:48) (cid:112) g ( x ) (cid:33) . (C.133)Inserting the latter into equation (C.132) and analyzing the terms depending on y (cid:48) , y (cid:48) , it is notdifficult to verify that the condition ∂ Φ ∂ y (cid:48) = ∂ Φ ∂ y (cid:48) = y ∂ Φ ∂ y + y ∂ Φ ∂ y − Φ ( y , y ) = Φ ( y , y ) = ϕ (cid:16) y y − (cid:17) y − . This shows that the most general function G satisfyingthe system (C.131)-(C.132) is given by G (cid:0) x , y , y , y (cid:48) , y (cid:48) (cid:1) = g ( x ) y ϕ (cid:18) y y (cid:19) . (C.136) Proposition C.4.
For any non-zero function ϕ (cid:16) y y (cid:17) , the non-linear systemy (cid:48)(cid:48) − g (cid:48) ( x ) g ( x ) y (cid:48) + g ( x ) y + g ( x ) y ϕ (cid:18) y y (cid:19) + g ( x ) y y ϕ (cid:48) (cid:18) y y (cid:19) =
0, (C.137) y (cid:48)(cid:48) − g (cid:48) ( x ) g ( x ) y (cid:48) + g ( x ) y − g ( x ) y ϕ (cid:48) (cid:18) y y (cid:19) = possesses exactly three Noether symmetries. The proof is essentially the same as that of Proposition 3 for the scalar case. The system correspondsto the equations of motion associated to the Lagrangian L = (cid:112) g ( x ) (cid:32)(cid:0) y (cid:48) (cid:1) + (cid:0) y (cid:48) (cid:1) − g ( x ) (cid:16) y + y (cid:17) − g ( x ) y ϕ (cid:18) y y (cid:19)(cid:33) (C.139)Successive reduction of the Noether symmetry condition (C.48) shows that any such symmetry hasthe components ξ ( x , y , y ) = φ ( x ) , η ( x , y , y ) = (cid:18) φ ( x ) g (cid:48) ( x ) g ( x ) + φ (cid:48) ( x ) (cid:19) y , η ( x , y , y ) = (cid:18) φ ( x ) g (cid:48) ( x ) g ( x ) + φ (cid:48) ( x ) (cid:19) y , .4. Non-linear deformations φ ( x ) satisfies the third-order equation (C.113). As the solutions to the lattercorrespond exactly to the symmetries Z , Z and Z preserved by the deformation, we conclude thatthe dimension of the Noether symmetry algebra is three, hence it must be isomorphic to sl ( R ) .As a consequence, the system (C.137)-(C.138) cannot be linearizable.As happened in the scalar case, the integration of the deformed system is quite cumbersome in spiteof the two constants of the motion, as these contain explicitly the independent variable x and theirexpression differs from being an easy one.As an example to illustrate one of these deformed systems, we consider the auxiliary functions g ( x ) = − x and ϕ ( y y ) = α y y . We are thus deforming the uncoupled system consisting of twoChebyshev equations. For the chosen forcing term, the deformed system equals y (cid:48)(cid:48) − x ( − x ) y (cid:48) + ( − x ) y − α y √ x − y =
0, (C.140) y (cid:48)(cid:48) − x ( − x ) y (cid:48) + ( − x ) y + α y √ x − y =
0. (C.141)Solving numerically the system for α = y ( ) = − y (cid:48) ( ) = y ( ) = y (cid:48) ( ) = − the solutions y ( x ) and y ( x ) give rise to the following graphical repre-sentation: F IGURE
C.2: Solutions y ( x ) (dashed) and y ( x ) for the system (C.140)-(C.141). Both solutions are approximatively oscillations, however with varying frequency. The resultingtrajectory in the plane { y , y } has a relatively complicated structure, as shows the following plot. For α = y ( t ) = U ( t ) , y = T ( t ) . Appendix C. Orthogonal systems and symmetries of ODEs F IGURE
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Rev. Mod. Phys. IBLIOGRAPHY
CV abreviado
A continuación presentamos una relación de los trabajos publicados, enviados a publicacióny en proceso de ejecución relacionados con la temática de esta memoria. Asimismo se indicanlas conferencias y comunicaciones presentadas con relación a esta temática.
Publicaciones
Campoamor-Stursberg, R. and Fernández-Saiz, E. (2015). Realizations of sl(3;R) in Terms ofChebyshev Polynomials and Orthogonal Systems of Functions. Symmetry Breaking and Vari-ational Symmetries.
In book: Ordinary and Partial Differential Equations . Chapter: 3. Nova SciencePublishers.Ballesteros, A.; Campoamor-Stursberg, R.: Fernández-Saiz, E.; Herranz, F. J. and de Lucas,J. (2018). Poisson–Hopf algebra deformations of Lie–Hamilton systems.
Journal of Physics A:Mathematical and Theoretical , (6), 065202.Ballesteros, A.; Campoamor-Stursberg, R.; Fernández-Saiz, E.; Herranz, F. J. and de Lucas,J. (2018). A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems basedon sl(2). Dobrev, V. (ed.). Quantum Theory and Symmetries with Lie Theory and Its Applications inPhysics
Volume 1, 2018. pp. 347–366.Esen, O.; Fernández-Saiz, E.; Sardón, C. and Zaj ˛ac, M. (2020). Geometry and solutions ofan epidemic SIS model permitting fluctuations and quantization, submitted arXiv preprintarXiv:2008.02484 [q-bio.PE] .Ballesteros, A.; Campoamor-Stursberg, R.: Fernández-Saiz, E.; Herranz, F. J. and de Lucas, J.A refinement of Poisson–Hopf deformations of Lie–Hamilton systems: The setup of explicitdeformed superposition rules and applications to the oscillator algebra.
In progress .Campoamor-Stursberg, R.: Fernández-Saiz, E.; Herranz, F. J. and Sardon, C. An alternativeepidemic SIS model based on the Poincaré algebra h . In progress . Conferencias y comunicaciones
Fernández–Saiz, E. (2016). Realizations of sl ( R ) in terms of Chebyshev polynomials. Postersession in , in Madrid.Fernández–Saiz, E. (2016). Lie symmetries and differential equations: Harrison-Estabrookmethod. Seminar in BurgosBallesteros, A.; Campoamor-Stursberg, R.: Fernández-Saiz, E.; Herranz, F. J. and de Lucas, J.(2017). Quantum algebras in Lie–Hamilton systems. Poster session in , in La Laguna.Ballesteros, A.; Campoamor-Stursberg, R.: Fernández-Saiz, E.; Herranz, F. J. and de Lucas,J. (2017). Quantum algebras in Lie–Hamilton systems: oscilator system. Poster session in VIberoamerican Meeting on Geometry, Mechanics and Control , in La Laguna (Tenerife).18
BIBLIOGRAPHY
Campoamor-Stursberg, R. and Fernández–Saiz, E. (2017). Realizations of sl(3,R): Chebyshevpolynomials. Poster session in
A Workshop to honour Prof. Orlando Ragnisco in his 70th anniver-sary , in Burgos.Ballesteros, A.; Campoamor-Stursberg, R.: Fernández-Saiz, E.; Herranz, F. J. and de Lucas, J.(2017). Poisson–Hopf algebra deformation of Lie systems. Talk in
X. International SymposiumQuantum Theory and Symmetries & 12-th edition of the International Workshop "Lie Theory and ItsApplications in Physics" , in Varna.Fernández–Saiz, E. (2018). Poisson-Hopf algebra deformations of Lie systems. Scientific Divul-gation Seminar in MadridBallesteros, A.; Campoamor-Stursberg, R.: Fernández-Saiz, E.; Herranz, F. J. and de Lucas, J.(2018). Poisson–Hopf algebra deformation of Lie systems. Talk in , in Madrid.Fernández–Saiz, E. (2018). Sistemas de Lie y sus aplicaciones. Seminar (Red de Doctorandosen Matemáticas UCM), in Madrid.Fernández–Saiz, E. (2018). Estructuras de Poisson: un billete de ida y vuelta a la mecánicacuántica. Scientific Divulgation Seminar (
3ª Jornada PhDay Complutense - Ciencias Matemáticas ),in Madrid.
Participación en organización de congresos
Fernández–Saiz, E. (2016) in the Organizing Committee of
X Workshop of Young Researchers inMathematics , in Madrid.Fernández–Saiz, E. (2017) in the Organizing Committee of