Shai M.J. Haran

Arithmetical Investigations

Representation Theory, Orthogonal Polynomials, and Quantum Interpolations

• Broschiertes Buch

Produktbeschreibung

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.

Produktdetails

• Lecture Notes in Mathematics 1941
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 12231904, 978-3-540-78378-7
• 2008
• Seitenzahl: 236
• Erscheinungstermin: 2. Mai 2008
• Englisch
• Abmessung: 235mm x 155mm x 13mm
• Gewicht: 380g
• ISBN-13: 9783540783787
• ISBN-10: 3540783784
• Artikelnr.: 23526499

Inhaltsangabe

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.