胞腔代数和仿射胞腔代数简介

表示论中一个最基本的问题是确定不可约表示的参数集,这个问题至今没有完全解决.对于Graham和Lehrer引入的有限维胞腔代数,这个问题得到了完满解答,并被成功地应用于数学和物理中出现的许多代数.近来,人们引入仿射胞腔代数,将Graham和Lehrer有限维胞腔代数的表示理论框架推广到一类无限维代数上.仿射胞腔代数不仅包括有限维胞腔代数,也包括无限维的仿射Temperley-Lieb代数和Lusztig的A-型仿射Hecke代数.本文将对胞腔代数的发展历史和主要研究成果做一些综述,同时,对新引入的仿射胞腔代数及其最新成果做一点简介.     

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups. Received: July 25, 2001     

Representations of affine Hecke algebras and based rings of affine Weyl groups

In this paper we show that the Deligne-Langlands-Lusztig classification of simple representations of an affine Hecke algebra remains valid if the parameter is not a root of the corresponding Poincaré polynomial. This verifies a conjecture of Lusztig proposed in 1989.

     

A1型扩张仿射Lie代数的分次自同构群

文献[1]从Euclid空间R^v（v≥1）的一个半格S出发,定义了一个Jordan代数J（S）：然后通过Tits—Kantor-Koecher方法由J（S）构造出Lie代数G（J（S））．最后利用G（J（S））得到A1型扩张仿射Lie代数L（J（S））．本文给出v=2,S为格时。A1型扩张仿射Lie代数L（J（S））的Z^2一分次自同构群．     

D_4型广义扭仿射李代数及其模

本文是[3]的继续,将讨论D4型的广义扭仿射李代数及其表示理论;证明作用在其不可约模上的一类算子的局部幂零性.     

The modular branching rule for affine Hecke algebras of type A

For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani (2002) [22].     

The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of cell modules for the extended affine Hecke algebra of type A which are parametrised by SL_n()-conjugacy classes of pairs (s, N), where s SL_n() is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q ² N. When q ²–1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q ²=–1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the standard modules earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]The second author thanks the Australian Research Council and the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.     

Applications of the Frobenius Formulas for the Characters of the Symmetric Group and the Hecke Algebras of Type A

We give a simple combinatorial proof of Ram's rule for computing the characters of the Hecke Algebra. We also establish a relationship between the characters of the Hecke algebra and the Kronecker product of two irreducible representations of the Symmetric Group which allows us to give new combinatorial interpretations to the Kronecker product of two Schur functions evaluated at a Schur function of hook shape or a two row shape. We also give a formula for the regular representation of the Hecke algebra.     

A categorical approach to Weyl modules

Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.     

Hochschild cohomology and graded Hecke algebras

We develop and collect techniques for determining Hochschild cohomology of skew group algebras and apply our results to graded Hecke algebras. We discuss the explicit computation of certain types of invariants under centralizer subgroups, focusing on the infinite family of complex reflection groups to illustrate our ideas. Resulting formulas for Hochschild two-cocycles give information about deformations of and, in particular, about graded Hecke algebras. We expand the definition of a graded Hecke algebra to allow a nonfaithful action of on , and we show that there exist nontrivial graded Hecke algebras for , in contrast to the case of the natural reflection representation. We prove that one of these graded Hecke algebras is equivalent to an algebra that has appeared before in a different form.

     

Weyl型代数的环论性质

设A是代数闭域k上有单位元1的交换结合代数，D是A的交换κ-导子组成的非零k-向量空间，苏育才与赵开明引进Weyl型代数A[D]并且证明了结合代数A[D]是单代数当且仅当A是D-单的且k1[D]在A上的作用为忠实的,通过证明A[D]与smash product A#U(D)同构，我们给出了这一结果的一个纯环论的证明，同时给出了A[D]的一个Ore扩张实现。     

A3型仿射Weyl群最低双边胞腔上的Kazhdan-Lusztig系数

对于Kazhdan-Lusztig多项式Py,w(q),μ(y,w)为它的首项系数(简称KL系数).首项系数在李代数及其表示理论中起着重要的作用.在文章中,W为A3型仿射Weyl群,通过它对应的Hecke代数的性质及其KL基{Cw}的乘积计算,以及不可约模在张量积中的重数公式,给出了A3型仿射Weyl群最低双边胞腔上的KL系数.     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

Functional equations for transfer-matrix operators in open Hecke chain models

We consider integrable open chain models formulated in terms of the generators of affine Hecke algebras. We use the fusion procedure to construct the hierarchy of commutative elements, which are analogues of the commutative transfer matrices. These elements satisfy a set of functional relations generalizing functional relations for a family of transfer matrices in solvable spin chain models of the U_q(gl(n|m)) type. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 219–236, February, 2007.     

An inductive approach to representations of complex reflection groups G(m, 1, n)

We propose an inductive approach to the representation theory of the chain of complex reflection groups G(m, 1, n). We obtain the Jucys-Murphy elements of G(m, 1, n) from the Jucys-Murphy elements of the cyclotomic Hecke algebra and study their common spectrum using representations of a degenerate cyclotomic affine Hecke algebra. We construct representations of G(m, 1, n) using a new associative algebra whose underlying vector space is the tensor product of the group ring ?G(m, 1, n) with a free associative algebra generated by the standard m-tableaux.     

A 类量子群和 Hecke 代数的 Schur-Weyl 对偶

设Vm是量子群Uq（SLm)的标准表示,通过Hecke代数的作用,作者将Vm的张量积V^nm分解成了Uq(SLm)的不可约表示的直和,从而给出了Uq(SLm)与Hecke代数的H（q,n)Schur-Weyl对偶的完整证明,进一步得到,当q是实数时,在Hilbert空间H^n上,Uq(SL∞）和H（q,n)之间存在着Schur-Weyl对偶。