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Double Affine Hecke Algebras
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Book Synopsis Double Affine Hecke Algebras by : Ivan Cherednik
Download or read book Double Affine Hecke Algebras written by Ivan Cherednik and published by Cambridge University Press. This book was released on 2005-03-21 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an essentially self-contained monograph centered on the new double Hecke algebra technique.
Book Synopsis Double Affine Hecke Algebras by : Ivan Cherednik
Download or read book Double Affine Hecke Algebras written by Ivan Cherednik and published by Cambridge University Press. This book was released on 2005-03-24 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Double Affine Hecke Algebras and Congruence Groups by : Bogdan Ion
Download or read book Double Affine Hecke Algebras and Congruence Groups written by Bogdan Ion and published by American Mathematical Soc.. This book was released on 2021-06-18 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Affine Hecke Algebras and Orthogonal Polynomials by : I. G. Macdonald
Download or read book Affine Hecke Algebras and Orthogonal Polynomials written by I. G. Macdonald and published by Cambridge University Press. This book was released on 2003-03-20 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: First account of a theory, created by Macdonald, of a class of orthogonal polynomial, which is related to mathematical physics.
Book Synopsis Double Affine Hecke Algebras and Noncommutative Geometry by : Alexei Oblomkov
Download or read book Double Affine Hecke Algebras and Noncommutative Geometry written by Alexei Oblomkov and published by . This book was released on 2005 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t, tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl, t2, t3, t4; q) of algebras.
Book Synopsis Double Affine Hecke Algebras and Their Representations by : Vincent Gajda
Download or read book Double Affine Hecke Algebras and Their Representations written by Vincent Gajda and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Representations of Affine Hecke Algebras by : Nanhua Xi
Download or read book Representations of Affine Hecke Algebras written by Nanhua Xi and published by Springer. This book was released on 2006-11-15 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt: Kazhdan and Lusztig classified the simple modules of an affine Hecke algebra Hq (q E C*) provided that q is not a root of 1 (Invent. Math. 1987). Ginzburg had some very interesting work on affine Hecke algebras. Combining these results simple Hq-modules can be classified provided that the order of q is not too small. These Lecture Notes of N. Xi show that the classification of simple Hq-modules is essentially different from general cases when q is a root of 1 of certain orders. In addition the based rings of affine Weyl groups are shown to be of interest in understanding irreducible representations of affine Hecke algebras. Basic knowledge of abstract algebra is enough to read one third of the book. Some knowledge of K-theory, algebraic group, and Kazhdan-Lusztig cell of Cexeter group is useful for the rest
Book Synopsis Double Affine Hecke Algebras and Macdonald's Conjectures by : Ivan Cherednik
Download or read book Double Affine Hecke Algebras and Macdonald's Conjectures written by Ivan Cherednik and published by . This book was released on 1993 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Double Affine Hecke Algebras and Macdonald's Operators by : Ivan Cherednik
Download or read book Double Affine Hecke Algebras and Macdonald's Operators written by Ivan Cherednik and published by . This book was released on 1992 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Induced Representations of Double Affine Hecke Algebras and Applications by : Ivan Cherednik
Download or read book Induced Representations of Double Affine Hecke Algebras and Applications written by Ivan Cherednik and published by . This book was released on 1993 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Double Affine Hecke Algebra of General Parameters by : Wille Liu
Download or read book Double Affine Hecke Algebra of General Parameters written by Wille Liu and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present thesis work focuses on the study of the category O of degenerate double affine Hecke algebras (dDAHA) with the point of view of Springer theory and perverse sheaves. In the first two chapiters we study algebraically the dDAHAs and their generalisations, quiver double Hecke algebras (QDHA). We in introduce the Knizhnik--Zamolodchikov (KZ) functor for the QDHA and prove that it satisfies the double centraliser property in chapter 2. Chapters 3 and 4 are devoted to the study of perverse sheaves on a Lie algebra equipped with a cyclic grading and the Springer theory for the dDAHAs with certain families of parameters. In chapter 5, we explain how the KZ functor can be realised in terms of perverse sheaves and we show how finer structures on the category O can be deduced from the sheaf-theoretic analysis on cyclically graded Lie algebras.
Book Synopsis Intertwining Operators of Double Affine Hecke Algebras by : Ivan Cherednik
Download or read book Intertwining Operators of Double Affine Hecke Algebras written by Ivan Cherednik and published by . This book was released on 1996 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Lie Groups, Geometry, and Representation Theory by : Victor G. Kac
Download or read book Lie Groups, Geometry, and Representation Theory written by Victor G. Kac and published by Springer. This book was released on 2018-12-12 with total page 540 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, dedicated to the memory of the great American mathematician Bertram Kostant (May 24, 1928 – February 2, 2017), is a collection of 19 invited papers by leading mathematicians working in Lie theory, representation theory, algebra, geometry, and mathematical physics. Kostant’s fundamental work in all of these areas has provided deep new insights and connections, and has created new fields of research. This volume features the only published articles of important recent results of the contributors with full details of their proofs. Key topics include: Poisson structures and potentials (A. Alekseev, A. Berenstein, B. Hoffman) Vertex algebras (T. Arakawa, K. Kawasetsu) Modular irreducible representations of semisimple Lie algebras (R. Bezrukavnikov, I. Losev) Asymptotic Hecke algebras (A. Braverman, D. Kazhdan) Tensor categories and quantum groups (A. Davydov, P. Etingof, D. Nikshych) Nil-Hecke algebras and Whittaker D-modules (V. Ginzburg) Toeplitz operators (V. Guillemin, A. Uribe, Z. Wang) Kashiwara crystals (A. Joseph) Characters of highest weight modules (V. Kac, M. Wakimoto) Alcove polytopes (T. Lam, A. Postnikov) Representation theory of quantized Gieseker varieties (I. Losev) Generalized Bruhat cells and integrable systems (J.-H. Liu, Y. Mi) Almost characters (G. Lusztig) Verlinde formulas (E. Meinrenken) Dirac operator and equivariant index (P.-É. Paradan, M. Vergne) Modality of representations and geometry of θ-groups (V. L. Popov) Distributions on homogeneous spaces (N. Ressayre) Reduction of orthogonal representations (J.-P. Serre)
Book Synopsis Iwahori-Hecke Algebras and Their Representation Theory by : Ivan Cherednik
Download or read book Iwahori-Hecke Algebras and Their Representation Theory written by Ivan Cherednik and published by Springer Science & Business Media. This book was released on 2002-12-19 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori-Hecke algebras and their representation theory, given during the CIME summer school which took place in 1999 in Martina Franca, Italy.
Book Synopsis Calogero-Moser Systems and Representation Theory by : Pavel I. Etingof
Download or read book Calogero-Moser Systems and Representation Theory written by Pavel I. Etingof and published by European Mathematical Society. This book was released on 2007 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc. The goal of the present lecture notes is to give an introduction to the theory of Calogero-Moser systems, highlighting their interplay with these fields. Since these lectures are designed for non-experts, the author gives short introductions to each of the subjects involved and provides a number of exercises.
Book Synopsis Affine Hecke Algebras and Quantum Symmetric Pairs by : Zhaobing Fan
Download or read book Affine Hecke Algebras and Quantum Symmetric Pairs written by Zhaobing Fan and published by American Mathematical Society. This book was released on 2023-01-18 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Book Synopsis Degenerate Double Affine Hecke Algebra and Conformal Field Theory by : Tomoyuki Arakawa
Download or read book Degenerate Double Affine Hecke Algebra and Conformal Field Theory written by Tomoyuki Arakawa and published by . This book was released on 1998 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We introduce a class of induced representations of the degenerate double affine Hecke algebra H of gl[subscript N](C) and analyze their structure mainly by means of intertwiners of H. We also construct them from ŝl[subscript m](C)-modules using Knizhnik-Zamolodchikov connections in the conformal field theory. This construction provides natural quotients of the induced modules, which correspond to the integrable ŝl[subscript m](C)-modules. Some conjectural formulas are presented for the symmetric part of them."
