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Author: BOGDAN. SAHI ION (SIDDHARTHA.)
Publisher:
ISBN: 9781470463410
Category :
Languages : en
Pages :
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Author: BOGDAN. SAHI ION (SIDDHARTHA.)
Publisher:
ISBN: 9781470463410
Category :
Languages : en
Pages :
Get Book
Book Description
Author: Bogdan Ion
Publisher: American Mathematical Soc.
ISBN: 1470443260
Category : Education
Languages : en
Pages : 90
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Book Description
The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by auto-morphisms of a finite index subgroup of the Artin group of type A2, which descends to a faithful outer action of a congruence subgroup of SL(2, Z)or PSL(2, Z). This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. It turns out that the structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, even by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products W Q∨ of the Weyl group W with the coroot lattice Q∨. They were defined topologically by van der Lek, and independently, algebraically, by Cherednik. We now describe our results for the adjoint case in greater detail. We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type A2,B2,G2 act by automorphisms on the adjoint DAAG/DAHA associated to affine Lie algebras of twist number r =1, 2, 3, respec-tively. This extends a fundamental result of Cherednik for r =1. We show further that the above rank two Artin group action descends to an outer action of the congruence subgroup Γ1(r). In particular, Γ1(r) acts naturally on the set of isomorphism classes of representations of an adjoint DAAG/DAHA of twist number r, giving rise to a projective representation of Γ1(r)on the spaceof aΓ1(r)-stable representation. We also provide a classification of the involutions of Kazhdan-Lusztig type that appear in the context of these actions.
Author: Ivan Cherednik
Publisher: Cambridge University Press
ISBN: 9781139441254
Category : Mathematics
Languages : en
Pages : 452
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Book Description
This is an essentially self-contained monograph in an intriguing field of fundamental importance for Representation Theory, Harmonic Analysis, Mathematical Physics, and Combinatorics. It is a major source of general information about the double affine Hecke algebra, also called Cherednik's algebra, and its impressive applications. Chapter 1 is devoted to the Knizhnik-Zamolodchikov equations attached to root systems and their relations to affine Hecke algebras, Kac-Moody algebras, and Fourier analysis. Chapter 2 contains a systematic exposition of the representation theory of the one-dimensional DAHA. It is the simplest case but far from trivial with deep connections in the theory of special functions. Chapter 3 is about DAHA in full generality, including applications to Macdonald polynomials, Fourier transforms, Gauss-Selberg integrals, Verlinde algebras, and Gaussian sums. This book is designed for mathematicians and physicists, experts and students, for those who want to master the double Hecke algebra technique. Visit http://arxiv.org/math.QA/0404307 to read Chapter 0 and selected topics from other chapters.
Author: Tom H. Koornwinder
Publisher: Cambridge University Press
ISBN: 1108916554
Category : Mathematics
Languages : en
Pages : 442
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This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.
Author: Philipp Fleig
Publisher: Cambridge Studies in Advanced
ISBN: 1107189926
Category : Mathematics
Languages : en
Pages : 587
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Detailed exposition of automorphic representations and their relation to string theory, for mathematicians and theoretical physicists.
Author: Mima Stanojkovski
Publisher: American Mathematical Society
ISBN: 1470450038
Category : Mathematics
Languages : en
Pages : 117
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Author: András Juhász
Publisher: American Mathematical Society
ISBN: 1470449722
Category : Mathematics
Languages : en
Pages : 174
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Author: Yann Palu
Publisher: American Mathematical Society
ISBN: 1470450046
Category : Mathematics
Languages : en
Pages : 95
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Author: Stuart Margolis
Publisher: American Mathematical Society
ISBN: 1470450429
Category : Mathematics
Languages : en
Pages : 135
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Author: Athanassios S. Fokas
Publisher: American Mathematical Society
ISBN: 1470450984
Category : Mathematics
Languages : en
Pages : 114
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