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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 560178, 16 pages

http://dx.doi.org/10.1155/2013/560178

Research Article

## Invariant Operators of Five-Dimensional Nonconjugate Subalgebras of the Lie Algebra of the Poincaré Group P(1,4)

^{1}Institute of Mathematics, Pedagogical University, 2 Podchorążych Street, 30-084 Cracow, Poland^{2}Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3b Naukova Street, Lviv 79601, Ukraine

Received 1 July 2013; Revised 17 September 2013; Accepted 19 September 2013

Academic Editor: Emrullah Yaşar

Copyright © 2013 Vasyl Fedorchuk and Volodymyr Fedorchuk. 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.

#### Linked References

- H. B. G. Casimir, “Ueber die Konstruktion einer zu den irreduzibelen Darstellungen halbeinfacher kontinuierlicher Gruppen gehorigen Differentialgleichung,”
*Proceedings of the Royal Academy Amsterdam*, vol. 34, pp. 844–846, 1931. - E. G. Beltrametti and A. Blasi, “On the number of Casimir operators associated with any Lie group,”
*Physics Letters B*, vol. 20, pp. 62–64, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Pauri and G. M. Prosperi, “On the construction of the invariant operators for any finite-parameter lie group,”
*Nuovo Cimento A*, vol. 43, no. 2, pp. 533–537, 1966. View at Publisher · View at Google Scholar · View at Scopus - W. I. Fushchich and I. Yu. Krivsky,
*On Wave Equations in the Minkowski Five-Space*, Preprint No. ITF-68-72, Institute for Theoretical Physics, Ukrainian Academy of Sciences, Kiev, Ukraine, 1968 (Russian). - V. I. Fushchich and I. Yu. Krivsky, “On representations of the inhomogeneous de Sitter group and equations in five-dimensional Minkowski space,”
*Nuclear Physics B*, vol. 14, pp. 573–585, 1969. - L. Abellanas and L. Martínez Alonso, “A general setting for Casimir invariants,”
*Journal of Mathematical Physics*, vol. 16, pp. 1580–1584, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, “Subgroups of the Poincaré group and their invariants,”
*Journal of Mathematical Physics*, vol. 17, no. 6, pp. 977–985, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Zassenhaus, “On the invariants of a Lie group. I: computers in nonassociative rings and algebras,” in
*Special Session of the 82nd Annual Meeting of the American Mathematical Society (San Ontario, 1976)*, R. E. Beck and B. Kolman, Eds., pp. 139–155, Academic Press, New York, NY, USA, 1977. - M. Perroud, “The fundamental invariants of inhomogeneous classical groups,”
*Journal of Mathematical Physics*, vol. 24, no. 6, pp. 1381–1391, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lemke, Y. Neeman, and J. Pecina-Cruz, “Wigner analysis and Casimir operators of SA(4, R),”
*Journal of Mathematical Physics*, vol. 33, no. 8, pp. 2656–2659, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - “The Casimir effect 50 years later,” in
*Proceedings of the Fourth Workshop on Quantum Field Theory under the Influence of External Conditions (Leipzig, September 14–18, 1998)*, M . Bordag, Ed., World Scientific, River Edge, NJ, USA, 1999. - S. Tremblay and P. Winternitz, “Invariants of the nilpotent and solvable triangular Lie algebras,”
*Journal of Physics A*, vol. 34, no. 42, pp. 9085–9099, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. C. Ndogmo, “Invariants of a semi-direct sum of Lie algebras,”
*Journal of Physics A*, vol. 37, no. 21, pp. 5635–5647, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. J. Echarte, J. Núñez, and F. Ramírez, “Relations among invariants of complex filiform Lie algebras,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 365–376, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. G. Kalnins, Z. Thomova, and P. Winternitz, “Subgroup type coordinates and the separation of variables in Hamilton-Jacobi and Schrödinger equations,”
*Journal of Nonlinear Mathematical Physics*, vol. 12, no. 2, pp. 178–208, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. N. Pecina-Cruz, “On the Casimir of the group $ISL(n,R)$ and its algebraic decomposition,”
*Journal of Mathematical Physics*, vol. 46, no. 6, Article ID 063503, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Šnobl and P. Winternitz, “A class of solvable Lie algebras and their Casimir invariants,”
*Journal of Physics A*, vol. 38, no. 12, pp. 2687–2700, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Ancochea, R. Campoamor-Stursberg, and L. Garcia Vergnolle, “Solvable Lie algebras with naturally graded nilradicals and their invariants,”
*Journal of Physics A*, vol. 39, no. 6, pp. 1339–1355, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Boyko, J. Patera, and R. Popovych, “Computation of invariants of Lie algebras by means of moving frames,”
*Journal of Physics A*, vol. 39, no. 20, pp. 5749–5762, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Boyko, J. Patera, and R. Popovych, “Invariants of Lie algebras with fixed structure of nilradicals,”
*Journal of Physics A*, vol. 40, no. 1, pp. 113–130, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Boyko, J. Patera, and R. Popovych, “Invariants of triangular Lie algebras,”
*Journal of Physics A*, vol. 40, no. 27, pp. 7557–7572, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Boyko, J. Patera, and R. Popovych, “Invariants of triangular Lie algebras with one nil-independent diagonal element,”
*Journal of Physics A*, vol. 40, no. 32, pp. 9783–9792, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Boyko, J. Patera, and R. Popovych, “Invariants of solvable Lie algebras with triangular nilradicals and diagonal nilindependent elements,”
*Linear Algebra and Its Applications*, vol. 428, no. 4, pp. 834–854, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Boyko, J. Patera, and R. Popovych, “Invariants of Lie algebras via moving frames,” in
*Group Analysis of Differential Equations and Integrable Systems*, pp. 36–44, [s.n.], [s.l.], 2009. View at Zentralblatt MATH · View at MathSciNet - R. Campoamor-Stursberg,
*Structural Data and Invariants of Nine Dimensional Real Lie Algebras with Nontrivial Levi Decomposition*, Nova Science, New York, NY, USA, 2009. View at MathSciNet - R. Campoamor-Stursberg and S. G. Low, “Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators,”
*Journal of Physics A*, vol. 42, no. 6, Article ID 065205, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Fuščič, “Representations of the total inhomogeneous de Sitter group, and equations in the five-dimensional approach. I,”
*Teoreticheskaya i Matematicheskaya Fizika*, vol. 4, no. 3, pp. 360–382, 1970 (Russian). - V. G. Kadyševskiĭ, “A new approach to the theory of electromagnetic interactions,”
*Fizika Elementarnykh Chastits i Atomnogo Yadra*, vol. 11, no. 1, pp. 5–39, 1980 (Russian). - V. I. Fushchich and A. G. Nikitin,
*Symmetry of Equations of Quantum Mechanics*, Nauka, Moscow, Russia, 1990 (Russian). - W. I. Fushchych and I. Yu. Krivsky, “On a possible approach to the variable-mass problem,”
*Nuclear Physics B*, vol. 7, no. 1, pp. 79–82, 1968. - V. M. Fedorčuk, “Continuous subgroups of the inhomogeneous de Sitter group P(1, 4),”
*Preprint No. 78.18*, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine, 1978 (Russian). - V. M. Fedorčuk, “Split subalgebras of the Lie algebra of the generalized Poincaré group P(1,4),”
*Ukrainian Mathematical Journal*, vol. 31, no. 6, pp. 554–558, 1979. View at MathSciNet - V. M. Fedorchuk and W. I. Fushchich, “On subgroups of the generalized Poincaré group,” in
*Proceedings of the International Seminar on Group Theoretical Methods in Physics (Zvenigorod, 1979)*, vol. 1, pp. 61–66, Nauka, Moscow, Russia, 1980 (Russian). - V. M. Fedorchuk, “Nonsplit subalgebras of the Lie algebra of the generalized Poincaré group P(1,4),”
*Ukrainian Mathematical Journal*, vol. 33, no. 5, pp. 535–538, 1981. View at MathSciNet - W. I. Fushchich, A. F. Barannik, L. F. Barannik, and V. M. Fedorchuk, “Continuous subgroups of the Poincaré group P(1,4),”
*Journal of Physics A*, vol. 18, no. 15, pp. 2893–2899, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Fushchich and A. G. N
*ǐ*k*ǐ*t*ǐ*n, “Reduction of the representations of the generalized Poincaré algebra by the Galilei algebra,”*Journal of Physics A*, vol. 13, no. 7, pp. 2319–2330, 1980. - V. M. Fedorchuk, “Invariant operators of splittable subgroups of the generalized Poincaré group P(1, 4),” in
*Symmetry and Solutions of Equations of Mathematical Physics*, pp. 90–92, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine, 1989 (Russian). - V. M. Fedorchuk, “Operators invariant under nonsplittable subgroups of the generalized Poincaré group P(1, 4),” in
*Algebra-Theoretic Analysis of Equations of Mathematical Physics*, pp. 98–100, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine, 1990 (Russian). - M. Léveillé, “Casimir invariants for the eight-dimensional subgroups of the Poincaré group P(1, 4),”
*Journal of Mathematical Physics*, vol. 25, no. 11, pp. 3331–3333, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. M. Fedorchuk and V.
*Ī*. Fedorchuk, “On invariant operators of low-dimension nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1, 4),”*Matematicheskie Metody i Fiziko-Mekhanicheskie Polya*, vol. 50, no. 1, pp. 16–23, 2007 (Ukrainian). - V. M. Fedorchuk and V.
*Ī*. Fedorchuk, “Invariant operators for four-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1, 4),”*Matematicheskie Metody i Fiziko-Mekhanicheskie Polya*, vol. 53, no. 4, pp. 17–27, 2010 (Ukrainian), English translation in*Journal of Mathematical Sciences*, vol. 181, no. 3, pp. 305–319, 2012. - G. M. Mubarakzjanov, “On solvable Lie algebras,”
*Izvestija Vysših Učebnyh Zavedeniĭ Matematika*, vol. 32, no. 1, pp. 114–123, 1963 (Russian). - G. M. Mubarakzjanov, “Classification of real structures of Lie algebras of fifth order,”
*Izvestija Vysših Učebnyh Zavedeniĭ Matematika*, vol. 34, no. 3, pp. 99–106, 1963 (Russian). - L. V. Ovsiannikov,
*Group Analysis of Differential Equations*, Academic Press, New York, NY, USA, 1982. View at MathSciNet - J. Patera, P. Winternitz, and H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group,”
*Journal of Mathematical Physics*, vol. 16, no. 8, pp. 1597–1614, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. I. Fushchich, L. F. Barannik, and A. F. Barannik,
*Subgroup Analysis of Galilei and Poincaré Groups and the Reduction of Nonlinear Equations*, Naukova Dumka, Kiev, Ukraine, 1991 (Russian).