03420nam 22005175 450 99646653580331620200706192224.03-540-78379-210.1007/978-3-540-78379-4(CKB)1000000000437232(SSID)ssj0000316330(PQKBManifestationID)11225812(PQKBTitleCode)TC0000316330(PQKBWorkID)10263930(PQKB)11498750(DE-He213)978-3-540-78379-4(MiAaPQ)EBC3068723(PPN)125218338(EXLCZ)99100000000043723220100301d2008 u| 0engurnn|008mamaatxtccrArithmetical Investigations[electronic resource] Representation Theory, Orthogonal Polynomials, and Quantum Interpolations /by Shai M. J. Haran1st ed. 2008.Berlin, Heidelberg :Springer Berlin Heidelberg :Imprint: Springer,2008.xii, 217 p. illLecture Notes in Mathematics,0075-8434Bibliographic Level Mode of Issuance: Monograph3-540-78378-4 Includes bibliographical references (p. [209]-213) and index.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.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.Lecture Notes in Mathematics,0075-8434Number theoryNumber Theoryhttps://scigraph.springernature.com/ontologies/product-market-codes/M25001Number theory.Number Theory.511.42Haran Shai M. Jauthttp://id.loc.gov/vocabulary/relators/aut724875BOOK996466535803316Arithmetical investigations1416595UNISA