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.     

具有特殊卡当矩阵的胞腔代数

本文研究了胞腔代数的直接构造问题.利用构造箭图并在其上添加关系的方法,获得了一种不可分解胞腔代数的构造方法.证明了总存在不可分解的胞腔代数A(对λ∈S(n))使得其卡当矩阵具有形如{n,1,…,1}的谱,从而拓广了胞腔代数的构造途径.     

参系数零维分片代数簇的实零点

分片代数簇是一些多元样条函数的公共零点集. 文中表明: 解参系数分片代数簇问题可转化为解有限个包含严格不等式的参系数多项式系统. 利用半代数系统的正则分解和柱形代数分解方法, 提出了计算零维参系数分片代数簇无挠实零点数的上确界, 以及达到上确界时实零点在各个胞腔内的数目分布情形的算法. 该算法同时能产生达到上确界的充要条件, 以及达到上确界时实零点数在各个$n$维胞腔内 取得某种分布的充要条件. 也给出了另一算法, 用于产生零维分片代数簇在$n$维复形中的 各个$n$维胞腔内恰有指定数目的相异无挠实零点的充分必要条件.     

刻画排列的连通分支

应用柱代数分解算法和简化的胞腔相邻算法,得到一个刻画R3中由n个紧半代数集所组成排列连通分支的算法.     

三维半群代数

首先给出代数闭域上三维半群代数的幂等元集和Jacobson根,并且刻画了三维半群代数的同构类.通过计算箭图,研究了三维代数的表示型.进一步,证明一个三维(或者二维)半群代数是胞腔的,当且仅当它是交换的.作为推论,得到一个左零带所对应的半群代数是胞腔的,当且仅当这个左零带是一个半格.     

一些无限维完备李代数

本文讨论了无限维完备李代数的一些性质,由Virasoro代数,Kac-Moody代数构造了几类无限维完备李代数.同时给出了Kac-Moody代数及其广义抛物子代数的导子代数的刻划.证明了完备李代数的Loop扩张仍为完备李代数.     

Sweedler四维代数及其子代数上的带权无穷小双代数和预李代数

带权无穷小双代数是非齐次结合经典杨巴方程的代数抽象,在数学和数学物理领域有诸多重要的应用.给出了Sweedler四维代数及其子代数上的带权无穷小双代数的分类.作为应用,得到了Sweedler四维代数上的预李代数,进而诱导出它们的李代数结构.     

一类非Hopf 代数的双Frobenius 代数

本文构造了一类非Hopf 代数的双Frobenius 代数. 特别地, 在某些特殊的情形下, 这里构造的双Frobenius 代数是整体维数为3 的阶1 生成的Artin-Schelter 正则代数的Yoneda 代数.     

Fuzzy锁代数

众所周知,论域X上的全体Fuzzy子集[0,1]~x,在L. A. Zadeh的∪,∩,C(余运算)意义下,由于排中律不成立,而只能构成“软”代数。可是我们还注意到了Giles. R在1976年提出的算子(?),(?)及C运算所构成的代数系统,确定非分配的“硬”代数结构。本文从这一类算子出发,抽象出一个新的代数系统——Fuzzy锁代数。并研究了它的代数结构。     

关于局部顶点李代数的一些注记

本文研究局部顶点李代数与顶点代数之间的关系.利用由局部顶点李代数构造顶点代数的方法,定义局部顶点李代数之间的同态,证明了同态可以唯一诱导出由局部顶点李代数构造所得到的顶点代数之间的同态.     

Inductions and restrictions on towers of cellularly stratified diagram algebras

Hartmann et al. defined the concept of cellularly stratified algebras that combine the features of both cellular algebras and stratified algebras. Many important diagram algebras in mathematics and physics, such as some Brauer, partition and BMW algebras, are cellularly stratified algebras, and each of these forms a tower of algebras. This article gives the concept of towers of cellularly stratified algebras in an axiomatic manner, and studies it in terms of induction and restriction functors. In particular, for certain towers of cellularly stratified algebras, we provide a criterion for semi-simplicity by using the cohomology groups of cell modules.     

Representation theory of q-rook monoid algebras

We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.Supported by EPSRC grant GR/S18151/01     

Graded extended Lie-type algebras

The class of extended Lie-type algebras contains the ones of associative algebras, Lie algebras, Leibniz algebras, dual Leibniz algebras, pre-Lie algebras, and Lie-type algebras, etc. We focus on the class of extended Lie-type algebras graded by an Abelian group G and study its structure, by stating, under certain conditions, a second Wedderburn-type theorem for this class of algebras.     

Generalizing the notion of Koszul algebra

We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.     

关于格上蕴涵代数及其对偶代数

给出了格蕴涵代数、MV代数、R₀代数等一些格上蕴涵代数之间的关系,并建立了它们的对偶代数.其结果描述了这些代数内部结构的特征,同时也为从语义的角度进一步研究格值逻辑系统提供了一个新的途径.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Monoidal Hom–Hopf Algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras, and Lie algebras.