{"product_id":"9783540783787","title":"Arithmetical Investigations : Representation Theory, Orthogonal Polynomials, and Quantum Interpolations (Lecture Notes in Mathematics) \u003cVol. 1941\u003e","description":"\u003cp\u003eIn 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 Z\u003csub\u003ep\u003c\/sub\u003e which are the inverse limit of the finite rings Z\/p\u003csup\u003en\u003c\/sup\u003e. This gives rise to a tree, and probability measures w on Z\u003csub\u003ep\u003c\/sub\u003e correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L\u003csub\u003e2\u003c\/sub\u003e(Z\u003csub\u003ep\u003c\/sub\u003e,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 L\u003csub\u003e2\u003c\/sub\u003e([-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 GL\u003csub\u003en\u003c\/sub\u003e(q)that interpolates between the p-adic group GL\u003csub\u003en\u003c\/sub\u003e(Z\u003csub\u003ep\u003c\/sub\u003e), and between its real (and complex) analogue -the orthogonal O\u003csub\u003en\u003c\/sub\u003e (and unitary U\u003csub\u003en\u003c\/sub\u003e )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.\u003c\/p\u003e 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.","brand":"Springer","offers":[{"title":"Default Title","offer_id":49168380297443,"sku":"00000_00000_00000_00000","price":49.95,"currency_code":"USD","in_stock":true}],"url":"https:\/\/usa.kinokuniya.com\/zh\/products\/9783540783787","provider":"Books Kinokuniya USA","version":"1.0","type":"link"}