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Erlangen Program at Large-1: Geometry of Invariants
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2010, том 6
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Erlangen Program at Large-1: Geometry of Invariants
Інший ID:
2010 Mathematics Subject Classification: 30G35; 22E46; 30F45; 32F45
DOI:10.3842/SIGMA.2010.076
DOI:10.3842/SIGMA.2010.076
Посилання:
Erlangen Program at Large-1: Geometry of Invariants / V.V. Kisil // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 73 назв. — англ.
Підтримка:
This paper has some overlaps with the paper [51] written in collaboration with D. Biswas.
However the present paper essentially revises many concepts (e.g. lengths, orthogonality, the parabolic Cayley transform) introduced in [51], thus it was important to make it an independent reading to avoid confusion with some earlier (and na¨ıve!) guesses made in [51].
The author is grateful to Professors S. Plaksa, S. Blyumin and N. Gromov for useful discussions and comments. Drs. I.R. Porteous, D.L. Selinger and J. Selig carefully read the previous paper [51] and made numerous comments and remarks helping to improve this paper as well. I am also grateful to D. Biswas for many comments on this paper.
The extensive graphics in this paper were produced with the help of the GiNaC [4, 44] computer algebra system. Since this tool is of separate interest we explain its usage by examples from this article in the separate paper [46]. The noweb [64] wrapper for C++ source code is included in the arXiv.org files of the papers [46].
Дата:
2010
Переглядів:
471
Завантажень:
246
Анотація:
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 SL₂(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.
