Arithmetical Investigations
Springer Berlin (Verlag)
978-3-540-78378-7 (ISBN)
In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp,w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.
Introduction: Motivations from Geometry.- Gamma and Beta Measures.- Markov Chains.- Real Beta Chain and q-Interpolation.- Ladder Structure.- q-Interpolation of Local Tate Thesis.- Pure Basis and Semi-Group.- Higher Dimensional Theory.- Real Grassmann Manifold.- p-Adic Grassmann Manifold.- q-Grassmann Manifold.- Quantum Group Uq(su(1, 1)) and the q-Hahn Basis.
| Erscheint lt. Verlag | 2.5.2008 |
|---|---|
| Reihe/Serie | Lecture Notes in Mathematics |
| Zusatzinfo | XII, 222 p. 23 illus. |
| Verlagsort | Berlin |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 340 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Schlagworte | 11-02, 11S80, 11S85 • Approximation • arithmetic • Arithmetic Geometry • Beta • DEX • Finite • Fourier transform • manifold • Markov Chain • markov chains • Probability • probability measure • quantum groups • Representation Theory • special funtions |
| ISBN-10 | 3-540-78378-4 / 3540783784 |
| ISBN-13 | 978-3-540-78378-7 / 9783540783787 |
| Zustand | Neuware |
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