Modular Hecke Algebras and their Hopf Symmetry (original) (raw)
Related papers
Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry
Moscow Mathematical Journal, 2003
We settle in this paper a question left open in our paper ``Modular Hecke algebras and their Hopf symmetry'', by showing how to extend the Rankin-Cohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation corresponding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin-Cohen brackets give rise to associative deformations.
Let W be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras Hq(W) as (left) coideal subalgebras. Our Hecke-Hopf algebras 1 H(W) have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of Hq(W)-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras; for an arbitrary Coxeter group W the " Demazure " part of H(W) is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.
Hecke operators with respect to the modular group of quaternions
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1990
The classical theory of Hecke operators for (entire) elliptic modular forms is a powerful instrument in obtaining number theoretical applications and especially to reveal multiplicative properties of the Fourier coefficients. M. SUGAWA~ [20], [21] developed an approach of Hecke operators for Siegel modular forms, which was extended by H. MA~SS [15]. A comprehensive representation of this theory containing recent results can for instance be found in the books of E. FREITAG [6], chapter IV, or A.N. ANDRIANOV [2], where the more complicated theory for congruence subgroups is included. H.-W. Lu [14] started the investigation of modular forms over the Hurwitz order (cf.
Affine Lie algebras and Hecke modular forms
Bulletin of the American Mathematical Society, 1980
The character of a highest weight representation of an affine lie algebra can be written as a finite sum of products of classical 0-functions and certain modular functions, called string functions. We find the transformation law for the string functions, which allows us to compute them explicitly in many interesting cases. Finally, we write an explicit formula for the partition function, in the simplest case A[ x \ and compute the string functions directly. After multiplication by the cube of the T?-function, they turn out to be Hecke modular forms! 1. (See [3] or [7] for details.) Let g be a complex finite-dimensional simple lie algebra, § a Cartan subalgebra of g. A the set of roots of § in g. A + a set of positive roots, II = {OL X ,..., a ; } the corresponding set of simple roots, 0 the highest root. Let (,) be an invariant symmetric bilinear form on g normalized by (6,6) = 2. For a€^* with (a, a) * 0 define H a G § by 0(# a) = 2(0, a)/(a, a) for j8 G §*. Let W be the Weyl group of § in g. Denote by M the Z-span of W6 (long roots). Let C[t, t" 1 ] be the algebra of Laurent polynomials over C in an indeterminate t. We regard g' := C[t, t~x] ® c g as an (infinite-dimensional) complex lie algebra. Define the affine Lie algebra g as follows. Let g = 'g © Cc © Cd and define the bracket by (dx \ dx for x, y G 'gf. The algebra g is an important example of a Kac-Moody algebra [5], [10]. Note that Cc is the center of the algebra g. The subalgebra % = i) <B Cc 0 Giis called the Cartan subalgebra of g. For a G §* set g a = {x G g | [ft, x] = a(h)x for ft G §}; then we have the root space decomposition 8 = ®9<r Detine a nondegenerate symmetric bilinear form (,) on § by (ft, ft') is unchanged if ft, ft' G § C 6, (ft, c) = (ft, d) = 0 for ft G fc>, (c, c) = (tf, J) = 0, (c, rf) = 1. We identify § with §* by this form; then §* is identified with a subspace in %* by a(c) = a(<2) = 0 for a G §*. For a G §* set a = aL " so that
Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1
Advances in Mathematics, 2007
We develop intrinsic tools for computing the periodic Hopf cyclic cohomology of Hopf algebras related to transverse symmetry in codimension 1. Besides the Hopf algebra found by Connes and the first author in their work on the local index formula for transversely hypoelliptic operators on foliations, this family includes its 'Schwarzian' quotient, on which the Rankin-Cohen universal deformation formula is based, the extended Connes-Kreimer Hopf algebra related to renormalization of divergences in QFT, as well as a series of cyclic coverings of these Hopf algebras, motivated by the treatment of transverse symmetry for nonorientable foliations.
1987
In the present paper the elementary divisor theory over the Hurwitz order of integral quaternions is applied in order to determine the structure of the Hecke-algebras related to the attached unimodular and modular group of degree n. In the case n = 1 the Hecke-algebras fail to be commutative. If n > 1 the Hecke-algebras prove to be commutative and coincide with the tensor product of their primary components. Each primary component turns out to be a polynomial ring in n resp. n + 1 resp. 2n resp. 2n + 1 algebraically independent elements. In the case of the modular group of degree n, the law of interchange with the Siegel ~b-operator is described. The induced homomorphism of the Hecke-algebras is surjective except for the weights r = 4n -4 and r = 4n -2.
Modular Representations and Branching Rules for Wreath Hecke Algebras
International Mathematics Research Notices, 2010
We introduce a generalization of degenerate affine Hecke algebra, called wreath Hecke algebra, associated to an arbitrary finite group G. The simple modules of the wreath Hecke algebra and of its associated cyclotomic algebras are classified over an algebraically closed field of any characteristic p ≥ 0. The modular branching rules for these algebras are obtained, and when p does not divide the order of G, they are further identified with crystal graphs of integrable modules for quantum affine algebras. The key is to establish an equivalence between a module category of the (cyclotomic) wreath Hecke algebra and its suitable counterpart for the degenerate affine Hecke algebra. Contents 1. Introduction 1 2. Definition and properties of the wreath Hecke algebra 3 3. An equivalence of module categories 8 4. Classification of simple modules and modular branching rules 12 5. Cyclotomic wreath Hecke algebras and crystals 16 References 22 Partially supported by NSF grant DMS-0800280.
Homological representations of the Hecke algebra
Communications in Mathematical Physics, 1990
In this paper a topological construction of representations of the series of Hecke algebras, associated with 2-row Young diagrams will be given. This construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection. The fibres of the vector bundle are homology spaces of configuration spaces of points in C, with a suitable twisted local coefficient system. It is also shown that there is a close correspondence between this construction and the work of Tsuchiya and Kanie, who constructed Hecke algebra representations from the monodromy of n-point functions in a conformal field theory on P 1. This work has significance in relation to the one-variable Jones polynomial, which can be expressed in terms of characters of the Iwahori-Hecke algebras associated with 2-row Young diagrams; it gives rise to a topological description of the Jones polynomial, which will be discussed elsewhere [L2].
A realization of the Hecke algebra on the space of period functions for Γ0 (n)
Journal für die reine und angewandte Mathematik (Crelles Journal), 2000
The standard realization of the Hecke algebra on classical holomorphic cusp forms and the corresponding period polynomials is well known. In this article we consider a nonstandard realization of the Hecke algebra on Maass cusp forms for the Hecke congruence subgroups Γ 0 (n). We show that the vector valued period functions derived recently by Hilgert, Mayer and Movasati as special eigenfunctions of the transfer operator for Γ 0 (n) are indeed related to the Maass cusp forms for these groups. This leads also to a simple interpretation of the "Hecke like" operators of these authors in terms of the aforementioned nonstandard realization of the Hecke algebra on the space of vector valued period functions.
2016
We use Bernstein's presentation of the Iwahori-Matsumoto Hecke algebra to obtain a simple proof of the Satake isomorphism and, in the same stroke, compute the center of the Iwahori-Matsumoto Hecke algebra.