IX_r(a)的有限型IX_r~°(a)的未定Weyl群

本文首先给出Kac-Moody代数IXr(a)的有限型I(?)r(a)的未定Weyl群的定义,然后对a≥5证明了不定型李代数,IXr(a)的Weyl群W同构于有限型I(?)r(a)的未定Weyl群.     

D_l和E_6型Weyl群的扭子群的扩群

本文决定了 Dl 和 E6 型 Weyl群扭子群的所有扩群 ,这为确定相应 Chevalley群扭子群的所有扩群奠定了基础 .     

Weyl群的子群结构

对各种类型的不可约根系,讨论了其Ｗｅｙｌ群一类子群的极大性。     

D1和E6型Weyl群的扭子群的扩群

本决定了D1和E6型Weyl群扭子群的所有扩群，这为确定相应Chevalley群扭子群的所有扩群奠定了基础。     

(A)3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?3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?3型仿射Weyl群最低双边胞腔上的...     

_n型仿射Weyl群a值5的A_2×A_(11)×A_(12)型左胞腔

描述了_n型仿射Weyl群a值为5的A_2×A_(11)×A_(12)型左胞腔的个数.并计算出当n=7时,这样的左胞腔个数为20个;当n≥8时,左胞腔个数为1/6(2n~3-21n~2+417n-510)个.     

关于Weyl群的某些研究

记 Φ为欧氏空间 V中某不可约根系 ,具有 Weyl群 W,记 σ为 W中满足条件 w( Φ+ ) =Φ-的唯一元 .本文考虑如何将 σ分解成反射之积 ;σ在 Φ上的作用方式如何 .作为应用确定了 W的中心 ;进一步确定了 V的一类子空间在 W中的固定子群 .     

(B)n型仿射Weyl群a值5的A2×A11×A12型左胞腔

描述了(B)n型仿射Weyl群a值为5的A2×A11×A12型左胞腔的个数.并计算出当n=7时,这样的左胞腔个数为20个;当n≥8时,左胞腔个数为1/6(2n3-21n2+ 417n-510)个.     

关于Weyl群的某些研究

记Φ为低欧氏空间Ｖ中某不可约根系,具有Ｗｅｙｌ群Ｗ,记σ为Ｗ中满足条件ω（Φ＾＋）＝Φ＾－的唯一元。本考虑如何将σ分解成反射之积;σ在Φ上的作用方式如何。作为应用确定了Ｗ的中心;进一步确定了Ｖ的一类子空间在Ｗ中的固定子群。     

Lie Bialgebra Structures on Lie Algebras of Generalized Weyl Type

Lie bialgebra structures on Lie algebras of generalized Weyl type are studied. They are shown to be triangular coboundary.     

On the Automorphism Group of the First Weyl Algebra

Let A ₁: = 𝕜[t, ?] be the first algebra over a field 𝕜 of characteristic zero. Let Aut_𝕜(A ₁) be the automorphism group of the ring A ₁. One can associate to each right ideal I of A ₁ a subgroup of Aut_𝕜(A ₁) called the isomorphism subgroup of I. In this article, we show that each such isomorphism subgroup is equal to its normalizer. For that, we study when the isomorphism subgroup of a right ideal of A ₁ contains a given isomorphism subgroup.     

Some Kazhdan-Lusztig Coefficients of Affine Weyl Group of Type (B)2

Let (W,S) be the affine Weyl group of type (B)2,on which we consider the length function e from W to N and the Bruhat order ≤.For y ＜ w in W,let μ(y,w) be the coefficient of q1/2(e(w)-e(y)-1) in Kazhdan-Lusztig polynomial Py,w ∈ Z[q].We determine some μ(y,w) for y ∈ c0 and w ∈ c2,where c0 is the lowest two-sided cell of (B)2 and c2 is the higher one.Furthermore,we get some consequences using left or right strings and some properties of leading coefficients.     

Rank 2 Nichols Algebras with Finite Arithmetic Root System

The concept of arithmetic root systems is introduced. It is shown that there is a one-to-one correspondence between arithmetic root systems and Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators. This has strong consequences for both objects. As an application all rank 2 Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators are determined. Supported by the European Community under a Marie Curie Intra-European Fellowship.     

Finite Majid Algebras Over the Klein Group

We provide a classification of finite-dimensional connected coradically graded pointed Majid algebras, generated in degrees 0 and 1, over the Klein group on the field of complex numbers.     

Dixmier's Problem 6 for the Weyl Algebra (The Generic Type Problem)

ABSTRACT

In Dixmier (1968 Dixmier , J. ( 1968 ). Sur les algèbres de Weyl . Bull. Soc. Math. France 96 : 209 – 242 . [CSA] [Crossref] , [Google Scholar]), the author posed six problems for the Weyl algebra A ₁ over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975 Joseph , A. ( 1975 ). The Weyl algebra—semisimple and nilpotent elements . Amer. J. Math. 97 ( 3 ): 597 – 615 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and Bavula (2005a Bavula , V. V. ( 2005a ). Dixmier's Problem 5 for the Weyl algebra . J. Algebra 283 ( 2 ): 604 – 621 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\{0}. The same results are true for other popular algebras.     

Weyl代数研究简介

本文简要综述Ｗｅｙｌ代数诞生７０余年来的一系列重要研究成果。