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Induced Representations and Hypercomplex Numbers

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Abstract

In the search for hypercomplex analytic functions on the halfplane, we review the construction of induced representations of the group \({G = {\rm SL}_2(\mathbb{R})}\) . Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of \({{\rm SL}_2(\mathbb{R})}\) , is a linearfractional transformation on hypercomplex numbers. Thus, we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group \({{\rm SL}_2(\mathbb{R})}\) and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of \({{\rm SL}_2(\mathbb{R})}\) on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well.

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References

  1. Damir Z. Arov and Harry Dym, J-contractive matrix valued functions and related topics. Encyclopedia of Mathematics and its Applications, vol. 116. Cambridge University Press, Cambridge, 2008.

  2. Christian Bauer, Alexander Frink, Richard Kreckel and Jens Vollinga, GiNaC is Not a CAS, 2001. URL: http://www.ginac.de/.

  3. John L. Bell, A primer of infinitesimal analysis. Cambridge University Press, Cambridge, Second, 2008.

  4. Dino Boccaletti, Francesco Catoni, Roberto Cannata, Vincenzo Catoni, Enrico Nichelatti and Paolo Zampetti, The mathematics of Minkowski space-time and an introduction to commutative hypercomplex numbers. Birkhäuser Verlag, Basel, 2008.

  5. Peter Butkovič, Max-linear systems: theory and algorithms. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London, 2010.

  6. Francesco Catoni, Roberto Cannata and Enrico Nichelatti, The parabolic analytic functions and the derivative of real functions. Adv. Appl. Clifford Algebras 14 (2) (2004), 185–190.

  7. P. Cerejeiras, U. Kähler and F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains. Math. Methods Appl. Sci. 28 (14) (2005), 1715–1724.

    Google Scholar 

  8. Martin Davis, Applied nonstandard analysis. Wiley-Interscience John Wiley & Sons., New York, 1977.

  9. Gromov N.A., Kuratov V.V.: Noncommutative space-time models. Czechoslovak J. Phys. 55(11), 1421–1426 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  10. N. A. Gromov and V. V. Kuratov, Possible quantum kinematics. J. Math. Phys. 47 (1) (2006), 013502, 9.

    Google Scholar 

  11. Francisco J. Herranz, Ramón Ortega and Mariano Santander, Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry. J. Phys. A 33 (24) (2000), 4525–4551. E-print: arXiv:mathph/ 9910041.

  12. Francisco J. Herranz and Mariano Santander, Conformal compactification of spacetimes. J. Phys. A 35 (31) (2002), 6619–6629. E-print: arXiv:mathph/ 0110019.

  13. Roger Howe and Eng-Chye Tan, Nonabelian harmonic analysis. Applications of SL(2,R). Springer-Verlag, New York, 1992.

  14. Andrei Khrennikov and Gavriel Segre, Hyperbolic quantization. In L. Accardi, W. Freudenberg, and M. Schürman (eds.) Quantum probability and infinite dimensional analysis, pages 282–287, World Scientific Publishing, Hackensack, NJ, 2007.

  15. A. A. Kirillov, Elements of the theory of representations. Springer-Verlag, Berlin, 1976. Translated from the Russian by Edwin Hewitt, Grundlehren der Mathematischen Wissenschaften, Band 220.

  16. Anastasia V. Kisil, Isometric action of \({{SL}_2(\mathbb{R})}\) on homogeneous spaces. Adv. App. Clifford Algebras 20 (2) (2010), 299–312. E-print: arXiv:0810.0368.

  17. Vladimir V. Kisil, Möbius transformations and monogenic functional calculus. Electron. Res. Announc. Amer. Math. Soc. 2 (1) (1996), 26–33. On-line.

  18. Vladimir V. Kisil, How many essentially different function theories exist? Clifford algebras and their application in mathematical physics (Aachen, 1996), pages 175–184, KluwerAcademic Publishers, Dordrecht, 1998.

  19. Vladimir V. Kisil, Analysis in R 1,1 or the principal function theory. Complex Variables Theory Appl. 40 (2) (1999), 93–118. E-print: arXiv:funct-an/9712003.

  20. Vladimir V. Kisil, Two approaches to non-commutative geometry. Complex methods for partial differential equations (Ankara, 1998), pages 215–244, Kluwer Acad. Publ., Dordrecht, 1999. E-print: arXiv:funct-an/9703001.

  21. Vladimir V. Kisil, Meeting Descartes and Klein somewhere in a noncommutative space. In A. Fokas, J. Halliwell, T. Kibble, and B. Zegarlinski (eds.) Highlights of mathematical physics (London, 2000), pages 165–189, Amer. Math. Soc., Providence, RI, 2002. E-print: arXiv:math-ph/0112059.

  22. Vladimir V. Kisil, Spectrum as the support of functional calculus. Functional analysis and its applications, pages 133–141, Elsevier, Amsterdam, 2004. Eprint: arXiv:math.FA/0208249.

  23. Vladimir V. Kisil, Erlangen program at large-0: Starting with the group \({{SL}_2(\mathbb{R})}\) . Notices Amer. Math. Soc. 54 (11) (2007), 1458–1465. E-print: arXiv:math/0607387, On-line.

  24. Vladimir V. Kisil, Fillmore-Springer-Cnops construction implemented in GiNaC. Adv. Appl. Clifford Algebr. 17 (1) (2007), 59–70. Updated full text and source files: E-print: arXiv:cs.MS/0512073, On-line.

  25. Vladimir V. Kisil, Two-dimensional conformal models of space-time and their compactification. J. Math. Phys. 48 (7) (2007), 073506, 8. E-print: arXiv:mathph/ 0611053.

  26. Vladimir V. Kisil, Erlangen program at large-2: Inventing a wheel. The parabolic one. Trans. Inst. Math. of the NAS of Ukraine, pages 89–98, 2010. E-print: arXiv:0707.4024.

  27. Vladimir V. Kisil, Erlangen program at large-1: Geometry of invariants. SIGMA, Symmetry Integrability Geom. Methods Appl. 6 (076) (2010), 45. E-print: arXiv:math.CV/0512416. MR2011i:30044.

  28. Vladimir V. Kisil, Covariant transform. Journal of Physics: Conference Series 284 (1) (2011), 012038. E-print: arXiv:1011.3947.

  29. Vladimir V. Kisil, Erlangen program at large-2 1/2: Induced representations and hypercomplex numbers. Izvestiya Komi nauchnogo centra UrO RAN 5 (1) (2011), 4–10. E-print: arXiv:0909.4464.

  30. Vladimir V. Kisil, Erlangen Programme at Large 3.2: Ladder operators in hypercomplex mechanics. Acta Polytechnica 51 (4) (2011), 44–53. E-print: arXiv:1103.1120.

  31. Vladimir V. Kisil, Classical/quantum=commutative/noncommutative? 2012. Eprint: arXiv:1204.1858.

  32. Vladimir V. Kisil, Erlangen programme at large: an Overview. In S.V. Rogosin and A.A. Koroleva (eds.) Advances in applied analysis, pages 1–78, Birkhäuser Verlag, Basel, 2012. E-print: arXiv:1106.1686.

  33. Vladimir V. Kisil, Geometry of Möbius transformations: Elliptic, parabolic and hyperbolic actions of SL 2(R). Imperial College Press, London, 2012. Includes a live DVD.

  34. Vladimir V. Kisil, Hypercomplex representations of the Heisenberg group and mechanics. Internat. J. Theoret. Phys. 51 (3) (2012), 964–984. E-print: arXiv:1005.5057.

  35. János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge, 2008. With the collaboration of C. H. Clemens and A. Corti. Paperback reprint of the hardback edition 1998. Zbl1143.14014.

  36. Nadiia Konovenko, Projective structures and algebras of their differential invariants. Acta Applicandae Mathematicae 109 (1) (2010), 87–99.

  37. Vladislav V. Kravchenko, Applied pseudoanalytic function theory. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2009. With a foreword by Wolfgang Sproessig.

  38. Serge Lang, SL 2(R). Graduate Texts in Mathematics, vol. 105. Springer- Verlag, New York, 1985. Reprint of the 1975 edition.

  39. M. A. Lavrent’ev and B. V. Shabat, Problems of hydrodynamics and their mathematical models. Izdat. “Nauka”, Moscow, Second, 1977.

  40. G. L. Litvinov, The Maslov dequantization, and idempotent and tropical mathematics: a brief introduction. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 326 (Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13) (2005), 145–182, 282. E-print: arXiv:math/0507014.

  41. Volodymyr Mazorchuk, Lectures on sl2-modules. World Scientific, 2009.

  42. A. E. Motter and M. A. F. Rosa, Hyperbolic calculus. Adv. Appl. Clifford Algebras 8 (1) (1998), 109–128.

  43. Valery N. Pilipchuk, Nonlinear dynamics. Between linear and impact limits. Lecture Notes in Applied and Computational Mechanics, vol. 52. Springer, Berlin, 2010.

  44. Valery N. Pilipchuk, Non-smooth spatio-temporal coordinates in nonlinear dynamics. January 2011. E-print: arXiv:1101.4597.

  45. R. I. Pimenov, Unified axiomatics of spaces with maximal movement group. Litov. Mat. Sb. 5 (1965), 457– 486. Zbl0139.37806.

    Google Scholar 

  46. L. S. Pontryagin, Generalisations of numbers. Library “Kvant”, vol. 54. “Nauka”, Moscow, 1986.

  47. Norman Ramsey, Noweb - a simple, extensible tool for literate programming. URL: http://www.eecs.harvard.edu/nr/noweb/.

  48. Garret Sobczyk, The hyperbolic number plane. College Math Journal 26 (4) (1995), 268–280.

  49. Michael E. Taylor, Noncommutative harmonic analysis. Mathematical Surveys and Monographs, vol. 22. American Mathematical Society, Providence, RI, 1986.

  50. S. Ulrych, Relativistic quantum physics with hyperbolic numbers. Phys. Lett. B 625 (3–4) (2005), 313–323.

  51. S. Ulrych, Considerations on the hyperbolic complex Klein-Gordon equation. J. Math. Phys. 51 (6) (2010), 063510, 8.

    Google Scholar 

  52. V. A. Uspenskiĭ, What is non-standard analysis? “’Nauka”, Moscow, 1987. With an appendix by V. G. Kanoveĭ.

  53. J. C. Vignaux and A. Durañona y Vedia, Sobre la teoría de las funciones de una variable compleja hiperbólica On the theory of functions of a complex hyperbolic variable.. Univ. Nac. La Plata. Publ. Fac. Ci. fis. mat. 104 (1935), 139–183. Zbl62.1122.03. 2

  54. I. M. Yaglom, A simple non-Euclidean geometry and its physical basis. Heidelberg Science Library. Springer-Verlag, New York, 1979. Translated from the Russian by Abe Shenitzer, with the editorial assistance of Basil Gordon.

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  1. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Vladimir V. Kisil

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Kisil, V.V. Induced Representations and Hypercomplex Numbers. Adv. Appl. Clifford Algebras 23, 417–440 (2013). https://doi.org/10.1007/s00006-012-0373-1

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