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V. V. Kisil University of Leeds Leeds, United Kingdom kisilv@maths.leeds.ac.uk

Elliptic, parab olic and hyp erb olic actions of S L(2, R) group
The group S L(2, R) of 2 в 2 matrices with real entries and the unit determinant acts on the real line by means of linear-fractional (MЁ obius) transformations: ab cd :x ax + b . cx + d (1)

It is natural to look for an extension of this action into the plane. The most common solution is to consider plane to be a field of complex numbers and define linear-fractional transformations by the same formula (1). This can be naturally identified as an elliptic action. If we consider the same plane as an algebra of either dual or double numbers [6]*App. C then the same linear-fractional transformations (1) in these non-division rings provide us with corresponding parabolic and hyperbolic actions. This opens a possibility to describe corresponding geometries and analytic function theories in the spirit of Erlangen program. A suitable modifications allows to extend many fundamental notions of the representation theory (e.g. character, induced representation, etc.) from complex numbers to the cases of non-division rings [4], [3], [5]. An extension of transformations (1) to Banach algebras allows us to review the fundamental notions of functional calculi and spectra [1], [2]. References obius Transformations and Monogenic Functional Calculus, Elec[1] Kisil, Vladimir V., MЁ tron. Res. Announc. Amer. Math. Soc., 2, (1996), no. 1, ISSN 1079-6762, 26­33, (electronic) MR 98a:47018. [2] Kisil, Vladimir V., Spectrum as the support of functional calculus, Functional analysis and its applications, North-Holland Math. Stud., 197, 133­141, Elsevier, Amsterdam, 2004, arXiv:math.FA/0208249. [3] Kisil, Vladimir V., Erlangen program at large--0: Starting with S L(2, R) group, 2006, arXiv:math.GM/0607387, Preprint LEEDS­MATH­PURE­2006­11. [4] Kisil, Vladimir V., Erlangen program at large--1: Geometry of invariants, 2005, arXiv:math.CV/0512416, Preprint LEEDS­MATH­PURE­2005­28. [5] Kisil, Vladimir V., Erlangen program at large--2: Inventing a wheel. The parabolic one, 2007, arXiv:0707.4024, Preprint LEEDS­MATH­PURE­2007-07. [6] Yaglom, I. M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag, New York, 1979, ISBN 0-387-90332-1, An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon, MR 80c:51007.

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