Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 19 (2024), 331 -- 359

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

THE PARABOLIC GEOMETRY GENERATED BY THE MÖBIUS ACTION OF SL(2;ℝ) THROUGH THE ERLANGEN PROGRAM

Debapriya Biswas and Sneha Gupta

Abstract. Inspired by the Erlangen Program of Felix Klein, we have studied the SL(2;ℝ)-action on complex, dual and double numbers focusing mainly on dual numbers. Using the Iwasawa decomposition, we have classified SL(2;ℝ) into three one-parameter subgroups denoted by A,N and K and studied their orbits. We have explored the parabolic geometry associated with dual numbers and studied its SL(2;ℝ)-invariant and other geometric properties along with compactification of the space of dual numbers. We have also studied the Cayley transform and deduced equations representing the parabolic unit circle and disk. Lastly, we have studied the applications of Cayley transform on the A,  N and K-orbits using its intertwining property.

2020 Mathematics Subject Classification: Primary: 51B10; Secondary: 22E30; 51B25; 51M15; 51N30; 57S20.
Keywords: Matrix Lie group; Möbius action; Iwasawa decomposition; dual numbers; SL(2;ℝ); invariant properties.

Full text

References

  1. Arfken, G. B. and Weber, H. J. and Harris, F. E. Mathematical Methods for Physicists, Academic Press Inc., 2013. MR1423357. Zbl 1239.00005.

  2. Bisom, T. The Works of Omar Khayyam in the History of Mathematics, The Mathematics Enthusiast, 18 (2021). https://doi.org/10.54870/1551-3440.1524

  3. Biswas, D. Projective coordinates and compactification in elliptic, parabolic and hyperbolic 2-D geometry, Journal of Applied Analysis, 18 (2012). MR2929000. Zbl 1276.30059.

  4. Biswas, D. and Dutta, S. Geometric Invariants under the Möbius action of the group SL(2;R), Kragujevac Journal of Mathematics, 45 (2021), pp. 925-941. MR4369917. Zbl 1499.57023.

  5. Castillo, G. F. T. D. Differentiable Manifolds A Theoretical Physics Approach, Springer Nature Switzerland AG, 2020. MR4215618. Zbl 1237.58001.

  6. Dummit, D. S. and Foote, R. M. Abstract Algebra, John Wiley and Sons Inc., 2004. MR2286236. Zbl 1037.00003.

  7. Fitzpatrick, M. M. Saccheri, Forerunner of Non-Euclidean Geometry The Mathematics Teacher, 57(5) (1964), pp. 323–332. https://doi.org/10.5951/MT.57.5.0323.

  8. Hall, B. C. Lie Groups, Lie Algebras and Representations, An Elementary Introduction, Springer-Verlag, 2003. MR1997306. Zbl 1316.22001.

  9. Kisil, A. V. Isometric Action of SL(2;R) on Homogeneous Spaces, Advances in Applied Clifford Algebras, 20 (2010), pp. 299-312. MR2645350. Zbl 1204.53042.

  10. Kisil, V. V. and Biswas, D. Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains, Trans. Inst. Math. of the NAS of Ukraine, 1 (2004), pp. 100--118. Zbl 1199.30078.

  11. Kisil, V. V. Erlangen Program at large-0: Starting with the group SL(2,R), Notices of the AMS, 54 (2007), pp. 2-9. MR2361159. Zbl 1137.22006.

  12. Kisil, V. V. Erlangen program at large-1: Geometry of invariants, SIGMA Symmetry Integrability Geom. Methods Appl., 6 (2010) 45 pp. MR2766965. Zbl 1218.30136.

  13. Kisil, V. V. Geometry of Möbius transformations: Elliptic, Parabolic and Hyperbolic Actions of SL(2;ℝ), Imperial College Press, 2012. MR2977041. Zbl 1254.30001.

  14. Kisil, V. V. Induced Representations and Hypercomplex Numbers, Advances in Applied Clifford Algebras, 23 (2013), pp. 417-440. MR3068127. Zbl 1269.30052.

  15. Klein, F. Elementary Mathematics from an Advanced Standpoint Geometry, Translated from the 3rd German edition by E.R. Hendrick and C.A. Noble, Dover Publication Inc., 2004. Zbl 1099.51001.

  16. Lang, S. SL(2,ℝ) Graduate text in mathematics, Springer New York, 1985. Zbl 0583.22001.

  17. Selin, H. Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer Netherlands, 2008. Zbl 1333.01007.

  18. Yaglom, I. M. A simple non-Euclidean geometry and its physical basis, Springer-Verlag New York, 1979. MR0520230. Zbl 0393.51013.

  19. Yaglom, I. M. Felix Klein and Sophius Lie. Evolution of the idea of symmetry in 19th century, Birkhäuser Boston Inc. Boston, MA, 1988. MR0919605. Zbl 0627.01010.



Debapriya Biswas
Department of Mathematics, Indian Institute of Technology Kharagpur,
Kharagpur, West Bengal-721302, India.
e-mail: priya@maths.iitkgp.ac.in



Sneha Gupta - corresponding author
Department of Mathematics, Indian Institute of Technology Kharagpur,
Kharagpur, West Bengal-721302, India.
e-mail: snehag863@gmail.com

http://www.utgjiu.ro/math/sma