In this article, the investigation into the Lie symmetry algebra of the geodesic equations of the canonical connection on a Lie group was continued. The key ideas of Lie group, Lie algebra, linear connection, and symmetry were quickly reviewed. The focus was on those Lie groups whose Lie algebra was six-dimensional solvable and indecomposable and for which the nilradical was abelian and had a one-dimensional center. Based on the list of Lie algebras compiled by Turkowski, there were eight algebras to consider that were denoted by $ A_{6, 20} $–$ A_{6, 27} $. For each Lie algebra, a comprehensive symmetry analysis of the system of geodesic equations was carried out. For each symmetry Lie algebra, the nilradical and a complement to the nilradical inside the radical, as well as a semi-simple factor, were identified.
Citation: Nouf Almutiben, Ryad Ghanam, G. Thompson, Edward L. Boone. Symmetry analysis of the canonical connection on Lie groups: six-dimensional case with abelian nilradical and one-dimensional center[J]. AIMS Mathematics, 2024, 9(6): 14504-14524. doi: 10.3934/math.2024705
In this article, the investigation into the Lie symmetry algebra of the geodesic equations of the canonical connection on a Lie group was continued. The key ideas of Lie group, Lie algebra, linear connection, and symmetry were quickly reviewed. The focus was on those Lie groups whose Lie algebra was six-dimensional solvable and indecomposable and for which the nilradical was abelian and had a one-dimensional center. Based on the list of Lie algebras compiled by Turkowski, there were eight algebras to consider that were denoted by $ A_{6, 20} $–$ A_{6, 27} $. For each Lie algebra, a comprehensive symmetry analysis of the system of geodesic equations was carried out. For each symmetry Lie algebra, the nilradical and a complement to the nilradical inside the radical, as well as a semi-simple factor, were identified.
| [1] | E. Cartan, J. A. Schouten, On the geometry of the group-manifold of simple and semi-simple groups, Proc. Akad. Wetensch., 29 (1926), 803–815. |
| [2] | R. Ghanam, G. Thompson, Symmetry algebras for the canonical Lie group geodesic equations in dimension three, Math. Aeterna, 8 (2018), 37–47. |
| [3] |
R. Ghanam, G. Thompson, Lie symmetries of the canonical geodesic equations for six-dimensional nilpotent Lie groups, Cogent Math. Stat., 7 (2020), 1781505. http://doi.org/10.1080/25742558.2020.1781505 doi: 10.1080/25742558.2020.1781505
|
| [4] | H. Almusawa, R. Ghanam, G. Thompson, Symmetries of the canonical geodesic equations of five-dimensional nilpotent Lie algebras, J. Generalized Lie Theory Appl., 13 (294), 1–5. http://doi.org/10.4172/1736-4337.1000294 |
| [5] |
J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Math. Phys., 17 (1976), 986–994. https://doi.org/10.1063/1.522992 doi: 10.1063/1.522992
|
| [6] |
N. Almutiben, E. L. Boone, R. Ghanam, G. Thompson, Classification of the symmetry Lie algebras for six-dimensional co-dimension two abelian nilradical Lie algebras, AIMS Math., 9 (2024), 1969–1996. https://doi.org/10.3934/math.2024098 doi: 10.3934/math.2024098
|
| [7] |
P. Turkowski, Solvable Lie algebras of dimension six, J. Math. Phys., 31 (1990), 1344–1350. https://doi.org/10.1063/1.528721 doi: 10.1063/1.528721
|
| [8] | M. Spivak, A comprehensive introduction to differential geometry, Amer. Math. Mon., 1999. https://doi.org/10.2307/2319112 |
| [9] | V. S. Varadarajan, Lie algrbras, Lie groups and their representations, Springer-Verlag, 2013. https://doi.org/10.1007/978-1-4612-1126-6 |
| [10] | F. W. Warner, Foundations of differentiable manifolds and Lie groups, Springer-Verlag, 1983. https://doi.org/10.1007/978-1-4757-1799-0 |
| [11] | S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, 2001. https://doi.org/10.1090/gsm/034 |
| [12] | J. E. Humphreys, Introduction to Lie algebras and their representations, Springer-Verlag, 1997. https://doi.org/10.1007/978-1-4612-6398-2 |
| [13] | D. J. Arrigo, Symmetry analysis of differential equations, John Wiley & Sons, Inc., 2015. |
Nouf Almutiben, Ryad Ghanam, G. Thompson, Edward L. Boone. Symmetry analysis of the canonical connection on Lie groups: six-dimensional case with abelian nilradical and one-dimensional center[J]. AIMS Mathematics, 2024, 9(6): 14504-14524. doi: 10.3934/math.2024705
DownLoad: