Erlangen Program at Large-1: Geometry of Invariants

Vladimir V. Kisil

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of $SL_2(\mathbb R)$ group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore–Springer–Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach.

Détail

Mots-clés : analytic function theory; semisimple groups; elliptic; parabolic; hyperbolic; Clifford algebras; complex numbers; dual numbers; double numbers; split-complex numbers; Möbius transformations.

Bibliographie

[1] Arveson W., An invitation to $C^*$-algebras, Graduate Texts in Mathematics, 39, Springer-Verlag, New York, Heidelberg, 1976 | MR | Zbl

[2] Baird P., Wood J. C., “Harmonic morphisms from Minkowski space and hyperbolic numbers”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 52:100 (2009), 195–209 | MR | Zbl

[3] Barker W., Howe R., Continuous symmetry. From Euclid to Klein, American Mathematical Society, Providence, RI, 2007 | MR | Zbl

[4] Bauer C., Frink A., Kreckel R., Vollinga J., GiNaC is Not a CAS, http://www.ginac.de/

[5] Beardon A. F., The geometry of discrete groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1995 | MR

[6] Beardon A. F., Algebra and geometry, Cambridge University Press, Cambridge, 2005 | MR | Zbl

[7] Bekkara E., Frances C., Zeghib A., “On lightlike geometry: isometric actions, and rigidity aspects”, C. R. Math. Acad. Sci. Paris, 343 (2006), 317–321 | DOI | MR | Zbl

[8] Benz W., Classical geometries in modern contexts. Geometry of real inner product spaces, 2nd ed., Birkhäuser Verlag, Basel, 2007 | MR

[9] Berger M., Geometry, v. II, Springer-Verlag, Berlin, 1987

[10] Boccaletti D., Catoni F., Cannata R., Catoni V., Nichelatti E., Zampetti P., The mathematics of Minkowski space-time and an introduction to commutative hypercomplex numbers, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2008 | MR | Zbl

[11] Catoni F., Cannata R., Nichelatti E., “The parabolic analytic functions and the derivative of real functions”, Adv. Appl. Clifford Algebr., 14 (2004), 185–190 | DOI | MR | Zbl

[12] Catoni F., Cannata R., Catoni V., Zampetti P., “$N$-dimensional geometries generated by hypercomplex numbers”, Adv. Appl. Clifford Algebr., 15 (2005), 1–25 | DOI | MR | Zbl

[13] Cerejeiras P., Kähler U., Sommen F., “Parabolic Dirac operators and the Navier–Stokes equations over time-varying domains”, Math. Methods Appl. Sci., 28 (2005), 1715–1724 | DOI | MR | Zbl

[14] Chern S.-S., “Finsler geometry is just Riemannian geometry without the quadratic restriction”, Notices Amer. Math. Soc., 43 (1996), 959–963 | MR | Zbl

[15] Cnops J., Hurwitz pairs and applications of Möbius transformations, Habilitation Dissertation, University of Gent, 1994

[16] Cnops J., An introduction to Dirac operators on manifolds, Progress in Mathematical Physics, 24, Birkhäuser Boston, Inc., Boston, MA, 2002 | MR | Zbl

[17] Coxeter H. S. M., Greitzer S. L., Geometry revisited, Random House, New York, 1967 | Zbl

[18] Davis M., Applied nonstandard analysis, John Wiley Sons, New York, 1977 | MR

[19] Eelbode D., Sommen F., “Taylor series on the hyperbolic unit ball”, Bull. Belg. Math. Soc. Simon Stevin, 11 (2004), 719–737 | MR | Zbl

[20] Eelbode D., Sommen F., “The fundamental solution of the hyperbolic Dirac operator on $\mathbf R^{1,m}$: a new approach”, Bull. Belg. Math. Soc. Simon Stevin, 12 (2005), 23–37 | MR | Zbl

[21] Fjelstad P., Gal S. G., “Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers”, Adv. Appl. Clifford Algebr., 11 (2001), 81–107 | DOI | MR | Zbl

[22] Garas'ko G. I., Elements of Finsler geometry for physicists, TETRU, Moscow, 2009 available at (in Russian) http://hypercomplex.xpsweb.com/articles/487/ru/pdf/00-gbook.pdf

[23] Gromov N. A., Contractions and analytic extensions of classical groups. Unified approach, Akad. Nauk SSSR Ural. Otdel. Komi Nauchn. Tsentr, Syktyvkar, 1990 (in Russian) | MR | Zbl

[24] Gromov N. A., Kuratov V. V., “Noncommutative space-time models”, Czechoslovak J. Phys., 55 (2005), 1421–1426, arXiv: hep-th/0507009 | DOI | MR

[25] Herranz F. J., Santander M., “Conformal compactification of spacetimes”, J. Phys. A: Math. Gen., 35 (2002), 6619–6629, arXiv: math-ph/0110019 | DOI | MR | Zbl

[26] Herranz F. J., Santander M., “Conformal symmetries of spacetimes”, J. Phys. A: Math. Gen., 35 (2002), 6601–6618, arXiv: math-ph/0110019 | DOI | MR | Zbl

[27] Howe R., Tan E.-C., Non-abelian harmonic analysis. Applications of $SL(2,\mathbf R)$, Springer-Verlag, New York, 1992 | MR

[28] Khrennikov A. Yu., “Hyperbolic quantum mechanics”, Dokl. Akad. Nauk, 402 (2005), 170–172 (in Russian) | MR

[29] Khrennikov A., Segre G., “Hyperbolic quantization”, Quantum Probability and Infinite Dimensional Analysis, QP-PQ: Quantum Probab. White Noise Anal., 20, World Sci. Publ., Hackensack, NJ, 2007, 282–287 | MR | Zbl

[30] Kirillov A. A., Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, 220, Springer-Verlag, Berlin, New York, 1976 | MR | Zbl

[31] Kirillov A. A., A tale of two fractals, see http://www.math.upenn.edu/textasciitilde kirillov/MATH480-F07/tf.pdf

[32] Kisil A. V., “Isometric action of $\mathrm{SL}_2(\mathbf R)$ on homogeneous spaces”, Adv. Appl. Clifford Algebr., 20 (2010), 299–312, arXiv: 0810.0368 | DOI | Zbl

[33] Kisil V. V., “Construction of integral representations for spaces of analytic functions”, Dokl. Akad. Nauk, 350 (1996), 446–448 (Russian) | MR | Zbl

[34] Kisil V. V., “Möbius transformations and monogenic functional calculus”, Electron. Res. Announc. Amer. Math. Soc., 2 (1996), 26–33 | DOI | MR | Zbl

[35] Kisil V. V., “Towards to analysis in $\mathbf R^{pq}$”, Proceedings of Symposium Analytical and Numerical Methods in Quaternionic and Clifford Analysis (Seiffen, Germany, 1996), eds. W. Sprößig and K. Gürlebeck, TU Bergakademie Freiberg, Freiberg, 1996, 95–100

[36] Kisil V. V., “How many essentially different function theories exist?”, Clifford Algebras and Their Application in mathematical physics (Aachen, 1996), Fund. Theories Phys., 94, Kluwer Acad. Publ., Dordrecht, 1998, 175–184 | MR | Zbl

[37] Kisil V. V., “Analysis in $\mathbf R^{1,1}$ or the principal function theory”, Complex Variables Theory Appl., 40 (1999), 93–118, arXiv: funct-an/9712003 | DOI | MR | Zbl

[38] Kisil V. V., “Relative convolutions. I. Properties and applications”, Adv. Math., 147 (1999), 35–73, arXiv: funct-an/9410001 | DOI | MR | Zbl

[39] Kisil V. V., “Two approaches to non-commutative geometry”, Complex Methods for Partial Differential Equations (Ankara, 1998), Int. Soc. Anal. Appl. Comput., 6, Kluwer Acad. Publ., Dordrecht, 1999, 215–244, arXiv: funct-an/9703001 | MR | Zbl

[40] Kisil V. V., “Wavelets in Banach spaces”, Acta Appl. Math., 59 (1999), 79–109, arXiv: math.FA/9807141 | DOI | MR | Zbl

[41] Kisil V. V., Spaces of analytical functions and wavelets – Lecture notes, 2000–2002, arXiv: math.CV/0204018 | Zbl

[42] Kisil V. V., “Meeting Descartes and Klein somewhere in a noncommutative space”, Highlights of Mathematical Physics (London, 2000), Amer. Math. Soc., Providence, RI, 2002, 165–189, arXiv: math-ph/0112059 | MR | Zbl

[43] Kisil V. V., “Spectrum as the support of functional calculus”, Functional Analysis and Its Applications, North-Holland Math. Stud., 197, Elsevier, Amsterdam, 2004, 133–141, arXiv: math.FA/0208249 | MR | Zbl

[44] Kisil V. V., “An example of Clifford algebras calculations with GiNaC”, Adv. Appl. Clifford Algebr., 15 (2005), 239–269, arXiv: cs.MS/0410044 | DOI | MR | Zbl

[45] Kisil V. V., “Starting with the group $\rm{SL}_2(\bf R)$”, Notices Amer. Math. Soc., 54 (2007), 1458–1465, arXiv: math.GM/0607387 | MR | Zbl

[46] Kisil V. V., “Fillmore–Springer–Cnops construction implemented in GiNaC”, Adv. Appl. Clifford Algebr., 17 (2007), 59–70, arXiv: cs.MS/0512073 | DOI | MR | Zbl

[47] Kisil V. V., Erlangen program at large-2 1/2: induced representations and hypercomplex numbers, arXiv: 0909.4464

[48] Kisil V. V., Erlangen program at large-2: inventing a wheel. The parabolic one, Proc. Inst. Math. of the NAS of Ukraine (to appear) , arXiv: 0707.4024

[49] Kisil V. V., Erlangen programme at large 3.1: hypercomplex representations of the Heisenberg group and mechanics, arXiv: 1005.5057

[50] Kisil V. V., Erlangen program at large: outline, arXiv: 1006.2115

[51] Kisil V. V., Biswas D., “Elliptic, parabolic and hyperbolic analytic function theory-0: geometry of domains”, Complex Analysis and Free Boundary Flows, Proc. Inst. Math. of the NAS of Ukraine, 1, no. 3, 2004, 100–118, arXiv: math.CV/0410399 | MR | Zbl

[52] Konovenko N., “Projective structures and algebras of their differential invariants”, Acta Appl. Math., 109 (2010), 87–99 | DOI | MR | Zbl

[53] Konovenko N. G., Lychagin V. V., “Differential invariants of nonstandard projective structures”, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 11 (2008), 10–13 (in Russian) | MR | Zbl

[54] Kurucz A., Wolter F., Zakharyaschev M., “Modal logics for metric spaces: open problems”, We Will Show Them: Essays in Honour of Dov Gabbay, v. 2, eds. S. Artemov, H. Barringer, A. S. d'Avila Garcez, L. C. Lamb and J. Woods, College Publications, 2005, 193–208

[55] Lang S., $\rm{SL}_2(\bf R)$, Graduate Texts in Mathematics, 105, Springer-Verlag, New York, 1985 | MR | Zbl

[56] Lavrent'ev M. A., Shabat B. V., Problems of hydrodynamics and their mathematical models, 2nd ed., Nauka, Moscow, 1977 (in Russian) | MR

[57] McRae A. S., “Clifford algebras and possible kinematics”, SIGMA, 2007, 079, 29 pp., arXiv: 0707.2869 | DOI | MR | Zbl

[58] Mirman R., “Quantum field theory, conformal group theory, conformal field theory”, Mathematical and Conceptual Foundations, Physical and Geometrical Applications, Nova Science Publishers, Inc., Huntington, NY, 2001 | MR

[59] Motter A. E., Rosa M. A. F., “Hyperbolic calculus”, Adv. Appl. Clifford Algebr., 8 (1998), 109–128 | DOI | MR | Zbl

[60] Olver P. J., Classical invariant theory, London Mathematical Society Student Texts, 44, Cambridge University Press, Cambridge, 1999 | MR

[61] Pavlov D. G., “Symmetries and geometric invariants”, Hypercomplex Numbers Geom. Phys., 3:2 (2006), 21–32 (in Russian)

[62] Pimenov R. I., “Unified axiomatics of spaces with maximal movement group”, Litovsk. Mat. Sb., 5 (1965), 457–486 (in Russian) | MR | Zbl

[63] Porteous I. R., Clifford algebras and the classical groups, Cambridge Studies in Advanced Mathematics, 50, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[64] Ramsey N., Noweb – a simple, extensible tool for literate programming, available at http://www.eecs.harvard.edu/~nr/noweb/

[65] Rozenfel'd B. A., Zamakhovskiĭ M. P., Geometry of Lie groups. Symmetric, parabolic, and periodic spaces, Moskovskiĭ Tsentr Nepreryvnogo Matematicheskogo Obrazovaniya, Moscow, 2003 (in Russian) | MR

[66] Segal I. E., Mathematical cosmology and extragalactic astronomy, Pure and Applied Mathematics, 68, Academic Press, New York, 1976 | MR

[67] Sharpe R. W., Differential geometry. Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997 | MR | Zbl

[68] Taylor M. E., Noncommutative harmonic analysis, Mathematical Surveys and Monographs, 22, American Mathematical Society, Providence, RI, 1986 | MR | Zbl

[69] Uspenskiĭ V. A., What is non-standard analysis?, Nauka, Moscow, 1987 (in Russian) | MR

[70] Vignaux J. C., Durañona y Vedia A., “Sobre la teoría de las funciones de una variable compleja hiperbólica”, Univ. nac. La Plata. Publ. Fac. Ci. fis. mat., 104 (1935), 139–183 (in Spanish) | Zbl

[71] Wildberger N. J., Divine proportions. Rational trigonometry to universal geometry, Wild Egg, Kingsford, 2005 | MR | Zbl

[72] Yaglom I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag, New York, 1979 | MR | Zbl

[73] Zejliger D. N., Complex lined geometry. Surfaces and congruency, GTTI, Leningrad, 1934 (in Russian)