4p阶二面体群连通3度Cayley有向图的正规性

设G是一个有限群,S是G的不包含单位元1的非空子集,定义群G关于S的Cayley(有向)图X:=Cay(G,S)如下:V(X)=G,E(X)={(g,sg)|g∈G,s∈S}.Cayley(有向)图X:=Cay(G,S)称为正规的,如果G的右正则表示R(G)在X的自同构群Aut(X)中是正规的.设G是4p阶二面体群(p为素数).考察了Cay(G,S)连通3度的正规性,并给出了这些图的全自同构群.     

二面体群的Grothendieck环结构

二面体群的表示范畴为对称半单monoidal范畴,因而其Grothendieck环为有限多个元素生成的交换环.本文确定了该Grothendieck环的极小生成元,并且进一步证明了该Grothendieck环与某一多项式环的商环同构.     

The Partition Algebra as a Centralizer Algebra of the Alternating Group

ABSTRACT

In this article, we focus on the result of V.F.R. Jones which says that the partition algebra is the algebra of all transformations commuting with the action of the symmetric group on tensor products of its permutation representation. In particular, we restrict the action of the symmetric group to the action of the alternating group. In this context, we compute a basis for the centralizer algebra and show when the centralizer is isomorphic to the partition algebra.     

Virasoro—相似代数的导子代数

本文给出了Ｖｒａｓｏｒｏ－相似 代数的导子代数及导子代的自同构群。     

Hermitian Adjacency Spectrum of Cayley Digraphs over Dihedral Group

We first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X(D 2n,S)on dihedral group D2n with|S|=3.Then we show that all Cayley digraphs X(D2P,S)with|S|=3 and p odd prime are Cay-DS,namely,for any Cayley digraph X(D2P,T),X(D2P,T)and X(D2P,S)having the same H-spectrum implies that they are isomorphic.     

The Group of Outer Automorphisms and the Picard Group of an Algebra

The primary goal of this work is to develop a strategy to compute Pic_K(A) and Out_K(A) where A is a finite-dimensional algebra over a field K. The basic idea is to put the normal subgroup Inn^*(A) of the inner automorphisms of A induced by elements of 1 + J(A), as a common denominator in the fraction Aut_K(A) / Inn(A). The new numerator Aut_K(A)/Inn^*(A) and the new denominator Inn(A)/Inn^*(A) are much easier to deal with than the original Aut_K(A) and Inn(A). The main ingredient to study Out(A) is now the appearance of an Abelian group Ch(, K), the group of acyclic characters of the quiver of A, that can be completely calculated. We show how to apply these results to compute the Picard group of a split finite-dimensional algebra in several cases.     

On the Restricted Lie Algebra Structure of the Witt Lie Algebra in Finite Characteristic

The article contains an explicit formula for the restricted Lie algebra structure in the Witt Lie algebra over a field of finite characteristic. Some combinatorial lemmas can be of independent interest.     

A Note on the Connectedness of the Invertible Group of a Nest Algebra

The connectedness of the invertibles question for arbitrary nest has been reduced to the case of the lower triangular operators with respect to a fixed orthonormal basis en for n 1. For each f ∈ H∞, let Tf be the Toeplitz operator. In this paper we prove that Tf can be connected to the identity through a path in the invertible group of the lower triangular operators if f satisfies certain conditions.     

First Cohomology Group from the Virasoro-Like Algebra to its Larsson Functor Module

Let 𝒲 be the Virasoro-like algebra, and d ₁, d ₂ be the degree derivations of 𝒲. Set ? = W⊕C d ₁⊕C d ₂. Let F ^α(V) be the ?-module defined by Larsson's functor applied to a finite dimensional sl ₂-module V. In this article, the derivations from ? to ?-modules F ^α(V) and the first cohomology group H ¹(?, F ^α(V)) are given.     

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。     

Projective Resolutions and Yoneda Algebras for Algebras of Dihedral Type

This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson. This work was partially supported by the grant 06-01-00200 of the Russian Foundation for Basic Research.     

Automorphisms Group of an Invariant Subalgebra of a Reductive Lie Algebra

We prove that any automorphism of an invariant subalgebra of a reductive Lie algebra over a field of zero characteristic is a standard automorphism. Mathematics Subject Classifications (2000) 17B20, 17B40.     

On Relative Difference Sets in Dihedral Groups

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon–Hadamard difference sets. In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.     

几类非Abel群之增广商群的结构

记ZG为有限群G的整群环,△n(G)为增广理想△(G)的n次幂,Qn(G)=△(G)/△n 1(G)为G的增广商群.本文考虑了二面体群D2tk(k 奇)和m次对称群Sm,证明了Qn(D2tk)为秩不超过2t 1的基本2-群以及Qn(Sm)≌Z2.     

On the Generalized Enveloping Algebra of a Color Lie Algebra

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle , such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L. Supported by the EC project Liegrits MCRTN 505078.     

二面体群上Hopf路余代数的结构分类

从Hopf quiver出发,借助于右kZ_u(c)-模的直积范畴■ Mkz_(u(C))与kG-Hopf双模范畴kG/kG M kG/kG之间的同构,当G是二面体群D_3时,给出了Hopf路余代数kQ~c的同构分类及其子Hopf代数kG[kQ_1]结构.     

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.     

A Group Algebra for Inductive Limit Groups. Continuity Problems of the Canonical Commutation Relations

Given an inductive limit group where each is locally compact, and a continuous two-cocycle , we construct a C^*-algebra group algebra is imbedded in its multiplier algebra , and the representations of are identified with the strong operator continuous of G. If any of these representations are faithful, the above imbedding is faithful. When G is locally compact, is precisely , the twisted group algebra of G, and for these reasons we regard in the general case as a twisted group algebra for G. Applying this construction to the CCR-algebra over an infinite dimensional symplectic space (S,\,B),we realise the regular representations as the representation space of the C^*-algebra , and show that pointwise continuous symplectic group actions on (S,\, B) produce pointwise continuous actions on , though not on the CCR-algebra. We also develop the theory to accommodate and classify 'partially regular' representations, i.e. representations which are strong operator continuous on some subgroup H of G (of suitable type) but not necessarily on G, given that such representations occur in constrained quantum systems.     

The Peak Algebra of the Symmetric Group

The peak set of a permutation is the set {i : (i – 1) < (i) > (i + 1)}. The group algebra of the symmetric group S _n admits a subalgebra in which elements are sums of permutations with a common descent set. In this paper we show the existence of a subalgebra of this descent algebra in which elements are sums of permutations sharing a common peak set. To prove the existence of this peak algebra we use the theory of enriched (P, )-partitions and the algebra of quasisymmetric peak functions studied by Stembridge (Trans. Amer. Math. Soc. 349 (1997) 763–788).