Automatic determination of optimal systems of Lie subalgebras: the package SymboLie. (English) Zbl 07738954
Krasil’shchik, I. S. (ed.) et al., The diverse world of PDEs. Algebraic and cohomological aspects. Alexandre Vinogradov memorial conference. Diffieties, cohomological physics, and other animals, Independent University of Moscow and Moscow State University, Moscow, Russia, December 13–17, 2021. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 789, 1-17 (2023).
Summary: Lie groups of point symmetries of partial differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of group-invariant solutions. An important goal is a classification in order to have an optimal system of inequivalent group-invariant solutions from which all other solutions can be derived by action of the group itself. In turn, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite-dimensional Lie algebra relies on the use of inner automorphisms. We present a novel effective algorithm that can automatically determine optimal systems of Lie subalgebras of a generic finite-dimensional Lie algebra; here, we limit the analysis to one-dimensional Lie subalgebras, though the same approach still works well for higher dimensional Lie subalgebras. The algorithm is implemented in the computer algebra system Wolfram Mathematica\texttrademark and illustrated by means of some examples.
For the entire collection see [Zbl 1519.35008].
For the entire collection see [Zbl 1519.35008].
MSC:
| 17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |
| 17-08 | Computational methods for problems pertaining to nonassociative rings and algebras |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
Lie algebras; inner automorphisms; optimal systems; group-invariant solutions; symbolic computationReferences:
| [1] | Humphreys, James E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, xii+169 pp. (1972), Springer-Verlag, New York-Berlin · Zbl 0254.17004 |
| [2] | de Graaf, Willem A., Lie algebras: theory and algorithms, North-Holland Mathematical Library, xii+393 pp. (2000), North-Holland Publishing Co., Amsterdam · Zbl 1122.17300 · doi:10.1016/S0924-6509(00)80040-9 |
| [3] | Erdmann, Karin, Introduction to Lie algebras, Springer Undergraduate Mathematics Series, x+251 pp. (2006), Springer-Verlag London, Ltd., London · Zbl 1139.17001 · doi:10.1007/1-84628-490-2 |
| [4] | Ovsiannikov, L. V., Group analysis of differential equations, xvi+416 pp. (1982), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0485.58002 |
| [5] | Olver, Peter J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, xxvi+497 pp. (1986), Springer-Verlag, New York · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2 |
| [6] | Oliveri, Francesco, Lie symmetries of differential equations: classical results and recent contributions, Symmetry, 658-706 (2010) · Zbl 1284.22014 · doi:10.3390/sym2020658 |
| [7] | Wolfram Research, Inc. Mathematica, Version 13.0.0, Champaign, IL, 2021. |
| [8] | L. Margheriti, Lie subalgebras and invariant solutions of differential equations. PhD Thesis, University of Messina, 2008. |
| [9] | L. Amata, F. Oliveri, Optimal Systems of Lie Subalgebras: Computer Assisted Determination. Preprint, 2022. |
| [10] | Patera, J., Subalgebras of real three- and four-dimensional Lie algebras, J. Mathematical Phys., 1449-1455 (1977) · Zbl 0412.17007 · doi:10.1063/1.523441 |
| [11] | Meleshko, S. V., Group classification of two-dimensional stable viscous gas equations, Internat. J. Non-Linear Mech., 449-456 (1999) · Zbl 1342.76105 · doi:10.1016/S0020-7462(98)00028-6 |
| [12] | Ovsiannikov, L. V., Modern group analysis: advanced analytical and computational methods in mathematical physics. The group analysis algorithms, 277-289 (1992), Kluwer Acad. Publ., Dordrecht · Zbl 0832.35004 |
| [13] | Meleshko, S. V., Methods for constructing exact solutions of partial differential equations, Mathematical and Analytical Techniques with Applications to Engineering, xvi+352 pp. (2005), Springer, New York · Zbl 1081.35001 |
| [14] | Chou, Kai-Seng, A note on optimal systems for the heat equation, J. Math. Anal. Appl., 741-751 (2001) · Zbl 0992.35006 · doi:10.1006/jmaa.2001.7579 |
| [15] | Hu, Xiaorui, A direct algorithm of one-dimensional optimal system for the group invariant solutions, J. Math. Phys., 053504, 17 pp. (2015) · Zbl 1316.35011 · doi:10.1063/1.4921229 |
| [16] | Zhang, Lin, A direct algorithm Maple package of one-dimensional optimal system for group invariant solutions, Commun. Theor. Phys. (Beijing), 14-22 (2018) · Zbl 1387.17002 · doi:10.1088/0253-6102/69/1/14 |
| [17] | L. Amata, F. Oliveri, E. Sgroi, Optimal systems of three and four-dimensional real Lie algebras. Sumitted, 2022. |
| [18] | Ovsyannikov, L. V., The program “Submodels”. Gas dynamics, J. Appl. Math. Mech.. Prikl. Mat. Mekh., 30-55 (1994) · Zbl 0890.76070 · doi:10.1016/0021-8928(94)90137-6 |
| [19] | F. Oliveri, ReLie: a Reduce program for Lie group analysis of differential equations. Symmetry, 13, 1826 (1-39), 2021. |
| [20] | Vinogradov, A. M., Particle-like structure of Lie algebras, J. Math. Phys., 071703, 49 pp. (2017) · Zbl 1422.17008 · doi:10.1063/1.4991657 |
| [21] | Vinogradov, A. M., Particle-like structure of coaxial Lie algebras, J. Math. Phys., 011703, 42 pp. (2018) · Zbl 1422.17009 · doi:10.1063/1.5001787 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
