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Author Haran, M. J. Shai.

Title Arithmetical investigations : representation theory, orthogonal polynomials, and quantum interpolations / Shai M.J. Haran
Published Berlin : Springer, [2008]
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Description 1 online resource (xii, 217 pages) : illustrations
Series Lecture notes in mathematics, 0075-8434 ; 1941
Lecture notes in mathematics (Springer-Verlag) ; 1941
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
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
Bibliography Includes bibliographical references and index
Notes English
Print version record
Subject Interpolation.
p-adic numbers.
Representations of quantum groups.
Form Electronic book
LC no. 2008921367
ISBN 3540783792
3540783784 (paperback; alk. paper)
9783540783787 (paperback; alk. paper)