Arithmetical Investigations : Representation Theory, Orthogonal Polynomials, and Quantum Interpolations / edited by Shai M. J. Haran.

Format:

Book

Contributor:

Haran, M. J. Shai, editor.

SpringerLink (Online service)

Series:

Lecture Notes in Mathematics,. 0075-8434

Lecture Notes in Mathematics, 0075-8434

Language:

English

Subjects (All):

Number theory.

Number Theory.

Local Subjects:

Number Theory.

Physical Description:

1 online resource (XII, 222 pages 23 illustrations).

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg, 2008.

System Details:

text file PDF

Summary:

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.

Contents:

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.

Other Format:

Printed edition:

ISBN:

9783540783794

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