Transformation groups on real plane and their differential invariants

Nesterenko, Maryna O.

doi:https://doi.org/10.1155/IJMMS/2006/17410

International Journal of Mathematics and Mathematical Sciences

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 017410 | https://doi.org/10.1155/IJMMS/2006/17410

Transformation groups on real plane and their differential invariants

Maryna O. Nesterenko¹

Received03 Apr 2006

Revised29 Jun 2006

Accepted09 Jul 2006

Published18 Sept 2006

Abstract

Complete sets of bases of differential invariants, operators of invariant differentiation, and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of finite-dimensional Lie algebras on the real plane are revisited.

References

A. Andrada, M. L. Barberis, I. G. Dotti, and G. P. Ovando, “Product structures on four dimensional solvable Lie algebras,” Homology, Homotopy and Applications, vol. 7, no. 1, pp. 9–37, 2005.
View at: Google Scholar | MathSciNet
P. Basarab-Horwath, V. Lahno, and R. Zhdanov, “The structure of Lie algebras and the classification problem for partial differential equations,” Acta Applicandae Mathematicae, vol. 69, no. 1, pp. 43–94, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Bianchi, Lezioni sulla teoria dei gruppi continui finiti di transformazioni, Spoerri, Pisa, 1918.
A. Bourlioux, C. Cyr-Gagnon, and P. Winternitz, “Difference schemes with point symmetries and their numerical tests,” Journal of Physics A: Mathematical and General, vol. 39, no. 22, pp. 6877–6896, 2006.
View at: Publisher Site | Google Scholar
D. Burde and C. Steinhoff, “Classification of orbit closures of 4-dimensional complex Lie algebras,” Journal of Algebra, vol. 214, no. 2, pp. 729–739, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. F. Cariñena, J. Grabowski, and G. Marmo, “Some physical applications of systems of differential equations admitting a superposition rule,” Reports on Mathematical Physics, vol. 48, no. 1-2, pp. 47–58, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
T. Cerquetelli, N. Ciccoli, and M. C. Nucci, “Four dimensional Lie symmetry algebras and fourth order ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 9, no. suppl. 2, pp. 24–35, 2002.
View at: Google Scholar | MathSciNet
W. A. de Graaf, “Classification of solvable Lie algebras,” Experimental Mathematics, vol. 14, no. 1, pp. 15–25, 2005.
View at: Google Scholar | MathSciNet
B. M. Doubrov and B. P. Komrakov, “Contact Lie algebras of vector fields on the plane,” Geometry and Topology, vol. 3, no. 1, pp. 1–20, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Fels and P. J. Olver, “Moving coframes. I. A practical algorithm,” Acta Applicandae Mathematicae, vol. 51, no. 2, pp. 161–213, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Fels and P. J. Olver, “Moving coframes. II. Regularization and theoretical foundations,” Acta Applicandae Mathematicae, vol. 55, no. 2, pp. 127–208, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Fialowski and M. Penkava, “Deformations of four dimensional Lie algebras,” http://arxiv.org/abs/math.RT/0512354.
View at: Google Scholar
A. González-López, N. Kamran, and P. J. Olver, “Lie algebras of differential operators in two complex variables,” American Journal of Mathematics, vol. 114, no. 6, pp. 1163–1185, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. González-López, N. Kamran, and P. J. Olver, “Lie algebras of vector fields in the real plane,” Proceedings of the London Mathematical Society. Third Series, vol. 64, no. 2, pp. 339–368, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
N. Jacobson, Lie Algebras, Interscience, New York, 1955.
A. A. Kirillov, Elements of the Theory of Representations, vol. 220 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 1976.
View at: Zentralblatt MATH | MathSciNet
R. D. Levine, “Lie algebraic approach to molecular structure and dynamics,” in Mathematical Frontiers in Computational Chemical Physics (Minneapolis, MN, 1987), D. G. Truhlar, Ed., vol. 15 of IMA Volumes in Mathematics and Its Applications, pp. 245–261, Springer, New York, 1988.
View at: Google Scholar | MathSciNet
S. Lie, “Classification und Integration von gewöhnlichen Differential-gleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten,” Archiv für Mathematik og Naturvidenskab. Christiana, vol. 8, pp. 371–458, 1883.
View at: Google Scholar
S. Lie, “Über Differentiation,” Mathematische Annalen, vol. 24, no. 4, pp. 537–578, 1884.
View at: Publisher Site | Google Scholar
S. Lie, Theorie der Transformationsgruppen, vol. 1, B.G. Teubner, Leipzig, 1888.
S. Lie, Theorie der Transformationsgruppen, vol. 2, B.G. Teubner, Leipzig, 1890.
S. Lie, Theorie der Transformationsgruppen, vol. 3, B.G. Teubner, Leipzig, 1893.
S. Lie, Gesammelte Abhandlungen, vol. 6, B.G. Teubner, Leipzig, 1927.
M. A. H. MacCallum, “On the classification of the real four-dimensional Lie algebras,” in On Einstein's Path: Essays in Honor of Engelbert Schucking (New York, 1996), A. L. Harvey, Ed., pp. 299–317, Springer, New York, 1999, first version circulated as a preprint since 1979.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. V. Morozov, “Classification of nilpotent Lie algebras of sixth order,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 1958, no. 4 (5), pp. 161–171, 1958 (Russian).
View at: Google Scholar | MathSciNet
G. M. Mubarakzyanov, “Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 1963, no. 4 (35), pp. 104–116, 1963 (Russian).
View at: Google Scholar | MathSciNet
G. M. Mubarakzyanov, “On solvable Lie algebras,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 1963, no. 1 (32), pp. 114–123, 1963 (Russian).
View at: Google Scholar | MathSciNet
G. M. Mubarakzyanov, “The Classification of the real structure of five-dimensional Lie algebras,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika, vol. 1963, no. 3(34), pp. 99–106, 1963 (Russian).
View at: Google Scholar
P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, 1993.
View at: Zentralblatt MATH | MathSciNet
P. J. Olver, “Differential invariants and invariant differential equations,” Lie Groups and Their Applications, vol. 1, no. 1, pp. 177–192, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995.
View at: Zentralblatt MATH | MathSciNet
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
View at: Zentralblatt MATH | MathSciNet
J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, “Invariants of real low dimension Lie algebras,” Journal of Mathematical Physics, vol. 17, no. 6, pp. 986–994, 1976.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Patera and P. Winternitz, “Subalgebras of real three- and four-dimensional Lie algebras,” Journal of Mathematical Physics, vol. 18, no. 7, pp. 1449–1455, 1977.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Patera and H. Zassenhaus, “Solvable Lie algebras of dimension $\leq 4$ over perfect fields,” Linear Algebra and Its Applications, vol. 142, pp. 1–17, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Z. Petrov, New Methods in the General Theory of Relativity, Izdat. “Nauka”, Moscow, 1966.
View at: MathSciNet
R. O. Popovych and V. M. Boyko, “Differential invariants and application to Riccati-type systems,” in Symmetry in Nonlinear Mathematical Physics, Part 1, 2 (Kyiv, 2001), vol. 2 of Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, pp. 184–193, Natsīonal. Akad. Nauk Ukraïni Inst. Mat., Kiev, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. O. Popovych, V. M. Boyko, M. O. Nesterenko, and M. W. Lutfullin, “Realizations of real low-dimensional Lie algebras,” Journal of Physics A: Mathematical and General, vol. 36, no. 26, pp. 7337–7360, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Schmucker and G. Czichowski, “Symmetry algebras and normal forms of third order ordinary differential equations,” Journal of Lie Theory, vol. 8, no. 1, pp. 129–137, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Shnider and P. Winternitz, “Nonlinear equations with superposition principles and the theory of transitive primitive Lie algebras,” Letters in Mathematical Physics, vol. 8, no. 1, pp. 69–78, 1984.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Tresse, “Sur les invariants différentiels des groupes continus de transformations,” Acta Mathematica, vol. 18, no. 1, pp. 1–3, 1894.
View at: Publisher Site | Google Scholar
M. A. Tresse, Determination des invariants ponctuels de l'equation differentielle du second ordre $y^{″} = w (x, y, y^{'})$ , S. Hirzel, Leiptzig, 1896.
P. Turkowski, “Solvable Lie algebras of dimension six,” Journal of Mathematical Physics, vol. 31, no. 6, pp. 1344–1350, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

178

Downloads

952

Citations