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Lie's reduction method and differential Galois theory in the complex analytic context
1. | Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C., Colombia |
2. | Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain |
References:
[1] |
C. Athorne and T. Hartl, Solvable structures and hidden symmetries,, J. Phys. A, 27 (1995), 3463.
|
[2] |
C. Athorne, Symmetries of linear ordinary differential equations,, J. Phys. A, 30 (1997), 4639.
doi: 10.1088/0305-4470/30/13/015. |
[3] |
C. Athorne, On the Lie symmetry algebra of general ordinary differential equation,, J. Phys. A, 31 (1998), 6605.
doi: 10.1088/0305-4470/31/31/008. |
[4] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58,, Contemp. Math., 509 (2010).
|
[5] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem,, J. Lie Theory, 20 (2010), 483. Google Scholar |
[6] |
R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'', Lectures at the R.G.I. in Park City, (1991). Google Scholar |
[7] |
A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'', Lecture Notes in Mathematics, 1226 (1986).
|
[8] |
J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'', Napoli Series on Physics and Astrophysics, (2000).
|
[9] |
J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations,, Rep. Math. Phys., 60 (2007), 237.
|
[10] |
J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle,, Acta Appl. Math., 66 (2001), 67.
doi: 10.1023/A:1010743114995. |
[11] |
J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008).
|
[12] |
J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory,, Acta Appl. Math., 70 (2002), 43.
doi: 10.1023/A:1013913930134. |
[13] |
G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières,, (French) [Singular foliations of codimension one, 56 (2006), 735.
|
[14] |
G. Casale, The Galois groupoid of Picard-Painlevé VI equation,, Algebraic, analytic and geometric aspects of complex differential equations and their deformations, (2007), 15.
|
[15] |
G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité,, Comment. Math. Helv., 83 (2008), 471.
doi: 10.4171/CMH/133. |
[16] |
G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'', (French) [Lessons on the general theory of surfaces. I, (1914).
|
[17] |
W. Fulton and J. Harris, "Representation Theory. A First Course,'', Graduate Texts in Mathematics, 129 (1991).
|
[18] |
A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales,, Compt. Rend. Acad. Sci. Paris, (1893), 964. Google Scholar |
[19] |
M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action,, J. Math. Phys., 40 (1999), 3104.
doi: 10.1063/1.532749. |
[20] |
J. E. Humphreys, "Linear Algebraic Groups,'', Graduate Texts in Mathematics, (1975).
|
[21] |
I. Kaplansky, "An Introduction to Differential Algebra,'', Actualités Sci. Ind., (1251).
|
[22] |
E. R. Kolchin, Galois theory of differential fields,, Amer. J. Math., 75 (1953), 753.
doi: 10.2307/2372550. |
[23] |
E. R. Kolchin, "Differential Algebra and Algebraic Groups,'', Pure and Applied Mathematics, 54 (1973).
|
[24] |
A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures,, J. Dynam. Control Systems, 1 (1995), 91.
doi: 10.1007/BF02254657. |
[25] |
S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten,, Math. Ann. Bd., 25 (1885), 71.
doi: 10.1007/BF01446421. |
[26] |
S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'', Lepziger Berichte, (1893). Google Scholar |
[27] |
S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'', (German) Bearbeitet und herausgegeben von Georg Scheffers, (1893).
|
[28] |
S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales,, Compt. Rend. Acad. Sci. Paris, (1893), 1233. Google Scholar |
[29] |
S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'', Reprinted by Chelsea books, (1967). Google Scholar |
[30] |
J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités,, J. Math. Pures Appl., 4 (1839), 423. Google Scholar |
[31] |
B. Malgrange, Le groupoïde de Galois d'un feuilletage,, (French) [The Galois groupoid of a foliation], 38 (2001), 465.
|
[32] |
B. Malgrange, On nonlinear differential Galois theory,, Chinese Ann. Math. Ser. B, 23 (2002), 219.
doi: 10.1142/S0252959902000213. |
[33] |
B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'', (French) [Lie Pseudogroups and Differential Galois Theory], (2010). Google Scholar |
[34] |
P. Malliavin, "Géométrie Différentielle Intrinsèque,'', (French) [Intrinsic Differential Geometry], (1972).
|
[35] |
M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases,, Phys. Rev. A (3), 52 (1995), 936.
doi: 10.1103/PhysRevA.52.936. |
[36] |
J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II,, Methods Appl. Anal., 8 (2001), 33.
|
[37] |
J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113.
|
[38] |
J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'', Progress in Mathematics, 179 (1999).
|
[39] |
J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations,, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845.
|
[40] |
K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants,, Nagoya Math. J., 113 (1989), 173.
|
[41] |
K. Nishioka, General solutions depending algebraically on arbitrary constants,, Nagoya Math. J., 113 (1989), 1.
|
[42] |
K. Nishioka, Lie extensions,, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82.
doi: 10.3792/pjaa.73.82. |
[43] |
K. Nomizu, "Lie Groups and Differential Geometry,'', The Mathematical Society of Japan, (1956).
|
[44] |
J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation,, (French) [Differential Galois theory and resummation], (1990), 117.
|
[45] |
M. Rosenlicht, A remark on quotient spaces,, An. Acad. Brasil. Ci., 35 (1963), 487.
|
[46] |
C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory],, Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16 (2001).
|
[47] |
J.-P. Serre, Géométrie algébrique et géométrie analytique, (French), 6 (): 1955.
|
[48] |
J.-P. Serre, Espaces fibrés algebriques (French),, Séminaire Claude Chevalley, 3 (1958), 1. Google Scholar |
[49] |
Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'', Translated from the Japanese by the author, 82 (1990).
|
[50] |
S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles,, J. Math. Phys., 25 (1984), 3155.
doi: 10.1063/1.526085. |
[51] |
M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory,, IEEE Trans. Automat. Control, 30 (1985), 266.
doi: 10.1109/TAC.1985.1103934. |
[52] |
H. Umemura, On the irreducibility of the first differential equation of Painlevé,, in, (1988), 771.
|
[53] |
H. Umemura, Galois theory of algebraic and differential equations,, Nagoya Math. J., 144 (1996), 1.
|
[54] |
H. Umemura, Differential Galois theory of infinite dimension,, Nagoya Math. J., 144 (1996), 59.
|
[55] |
H. Umemura, Sur l'équivalence des théories de Galois différentielles générales,, (French) [On the equivalence of general differential Galois theories], 346 (2008), 1155.
|
[56] |
M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328 (2003).
|
[57] |
E. Vessiot, Sur l'intégration des equations différentielles linéaires, (French), 9 (1892), 197.
|
[58] |
E. Vessiot, Sur une classe d'équations différentielles,, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53.
|
[59] |
E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires,, Compt. Rend. Acad. Sci. Paris, (1893), 1112. Google Scholar |
[60] |
E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales,, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894).
|
[61] |
E. Vessiot, Sur la théorie de Galois et ses diverses généralisations, (French), 21 (1904), 9.
|
[62] |
E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif, (French), 57 (1940), 1.
|
show all references
References:
[1] |
C. Athorne and T. Hartl, Solvable structures and hidden symmetries,, J. Phys. A, 27 (1995), 3463.
|
[2] |
C. Athorne, Symmetries of linear ordinary differential equations,, J. Phys. A, 30 (1997), 4639.
doi: 10.1088/0305-4470/30/13/015. |
[3] |
C. Athorne, On the Lie symmetry algebra of general ordinary differential equation,, J. Phys. A, 31 (1998), 6605.
doi: 10.1088/0305-4470/31/31/008. |
[4] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58,, Contemp. Math., 509 (2010).
|
[5] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem,, J. Lie Theory, 20 (2010), 483. Google Scholar |
[6] |
R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'', Lectures at the R.G.I. in Park City, (1991). Google Scholar |
[7] |
A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'', Lecture Notes in Mathematics, 1226 (1986).
|
[8] |
J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'', Napoli Series on Physics and Astrophysics, (2000).
|
[9] |
J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations,, Rep. Math. Phys., 60 (2007), 237.
|
[10] |
J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle,, Acta Appl. Math., 66 (2001), 67.
doi: 10.1023/A:1010743114995. |
[11] |
J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008).
|
[12] |
J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory,, Acta Appl. Math., 70 (2002), 43.
doi: 10.1023/A:1013913930134. |
[13] |
G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières,, (French) [Singular foliations of codimension one, 56 (2006), 735.
|
[14] |
G. Casale, The Galois groupoid of Picard-Painlevé VI equation,, Algebraic, analytic and geometric aspects of complex differential equations and their deformations, (2007), 15.
|
[15] |
G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité,, Comment. Math. Helv., 83 (2008), 471.
doi: 10.4171/CMH/133. |
[16] |
G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'', (French) [Lessons on the general theory of surfaces. I, (1914).
|
[17] |
W. Fulton and J. Harris, "Representation Theory. A First Course,'', Graduate Texts in Mathematics, 129 (1991).
|
[18] |
A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales,, Compt. Rend. Acad. Sci. Paris, (1893), 964. Google Scholar |
[19] |
M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action,, J. Math. Phys., 40 (1999), 3104.
doi: 10.1063/1.532749. |
[20] |
J. E. Humphreys, "Linear Algebraic Groups,'', Graduate Texts in Mathematics, (1975).
|
[21] |
I. Kaplansky, "An Introduction to Differential Algebra,'', Actualités Sci. Ind., (1251).
|
[22] |
E. R. Kolchin, Galois theory of differential fields,, Amer. J. Math., 75 (1953), 753.
doi: 10.2307/2372550. |
[23] |
E. R. Kolchin, "Differential Algebra and Algebraic Groups,'', Pure and Applied Mathematics, 54 (1973).
|
[24] |
A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures,, J. Dynam. Control Systems, 1 (1995), 91.
doi: 10.1007/BF02254657. |
[25] |
S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten,, Math. Ann. Bd., 25 (1885), 71.
doi: 10.1007/BF01446421. |
[26] |
S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'', Lepziger Berichte, (1893). Google Scholar |
[27] |
S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'', (German) Bearbeitet und herausgegeben von Georg Scheffers, (1893).
|
[28] |
S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales,, Compt. Rend. Acad. Sci. Paris, (1893), 1233. Google Scholar |
[29] |
S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'', Reprinted by Chelsea books, (1967). Google Scholar |
[30] |
J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités,, J. Math. Pures Appl., 4 (1839), 423. Google Scholar |
[31] |
B. Malgrange, Le groupoïde de Galois d'un feuilletage,, (French) [The Galois groupoid of a foliation], 38 (2001), 465.
|
[32] |
B. Malgrange, On nonlinear differential Galois theory,, Chinese Ann. Math. Ser. B, 23 (2002), 219.
doi: 10.1142/S0252959902000213. |
[33] |
B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'', (French) [Lie Pseudogroups and Differential Galois Theory], (2010). Google Scholar |
[34] |
P. Malliavin, "Géométrie Différentielle Intrinsèque,'', (French) [Intrinsic Differential Geometry], (1972).
|
[35] |
M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases,, Phys. Rev. A (3), 52 (1995), 936.
doi: 10.1103/PhysRevA.52.936. |
[36] |
J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II,, Methods Appl. Anal., 8 (2001), 33.
|
[37] |
J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113.
|
[38] |
J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'', Progress in Mathematics, 179 (1999).
|
[39] |
J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations,, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845.
|
[40] |
K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants,, Nagoya Math. J., 113 (1989), 173.
|
[41] |
K. Nishioka, General solutions depending algebraically on arbitrary constants,, Nagoya Math. J., 113 (1989), 1.
|
[42] |
K. Nishioka, Lie extensions,, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82.
doi: 10.3792/pjaa.73.82. |
[43] |
K. Nomizu, "Lie Groups and Differential Geometry,'', The Mathematical Society of Japan, (1956).
|
[44] |
J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation,, (French) [Differential Galois theory and resummation], (1990), 117.
|
[45] |
M. Rosenlicht, A remark on quotient spaces,, An. Acad. Brasil. Ci., 35 (1963), 487.
|
[46] |
C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory],, Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16 (2001).
|
[47] |
J.-P. Serre, Géométrie algébrique et géométrie analytique, (French), 6 (): 1955.
|
[48] |
J.-P. Serre, Espaces fibrés algebriques (French),, Séminaire Claude Chevalley, 3 (1958), 1. Google Scholar |
[49] |
Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'', Translated from the Japanese by the author, 82 (1990).
|
[50] |
S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles,, J. Math. Phys., 25 (1984), 3155.
doi: 10.1063/1.526085. |
[51] |
M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory,, IEEE Trans. Automat. Control, 30 (1985), 266.
doi: 10.1109/TAC.1985.1103934. |
[52] |
H. Umemura, On the irreducibility of the first differential equation of Painlevé,, in, (1988), 771.
|
[53] |
H. Umemura, Galois theory of algebraic and differential equations,, Nagoya Math. J., 144 (1996), 1.
|
[54] |
H. Umemura, Differential Galois theory of infinite dimension,, Nagoya Math. J., 144 (1996), 59.
|
[55] |
H. Umemura, Sur l'équivalence des théories de Galois différentielles générales,, (French) [On the equivalence of general differential Galois theories], 346 (2008), 1155.
|
[56] |
M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328 (2003).
|
[57] |
E. Vessiot, Sur l'intégration des equations différentielles linéaires, (French), 9 (1892), 197.
|
[58] |
E. Vessiot, Sur une classe d'équations différentielles,, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53.
|
[59] |
E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires,, Compt. Rend. Acad. Sci. Paris, (1893), 1112. Google Scholar |
[60] |
E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales,, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894).
|
[61] |
E. Vessiot, Sur la théorie de Galois et ses diverses généralisations, (French), 21 (1904), 9.
|
[62] |
E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif, (French), 57 (1940), 1.
|
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