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.
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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.
Imprint | Springer-Verlag |
Country of origin | Germany |
Series | Lecture Notes in Mathematics, 1941 |
Release date | May 2008 |
Availability | Expected to ship within 10 - 15 working days |
First published | May 2008 |
Authors | Shai M. J. Haran |
Dimensions | 235 x 155 x 12mm (L x W x T) |
Format | Paperback |
Pages | 222 |
Edition | 2008 ed. |
ISBN-13 | 978-3-540-78378-7 |
Barcode | 9783540783787 |
Categories | |
LSN | 3-540-78378-4 |