Kadison-Singer代数研究综述

回顾了建立KS-代数的研究背景,系统介绍了KS-代数的定义和性质以及超有限KS-代数、非超有限KS-代数、KS-格的构造和强KS-代数的研究结果,同时分析了KS-代数和经典的不变子空间、Kadison可迁代数、von Neumann代数生成元等问题之间的联系;讨论了非自伴代数的运算,给出了两种不同构造非自伴代数的运算法则;在此基础上,提出了未来学科发展有待研究的16个问题.     

FI代数同构于一族全序FI代数的直积的子代数的条件

解决了王国俊教授提出的关于FI代数的嵌入问题,并进一步研究了FI代数与一族全序FI代数的直积的子代数同构的条件.主要结果是:交换FI代数可同构嵌入一族全序交换FI代数的直积;分配FI代数可同构嵌入一族全序分配FI代数的直积.     

可换BCH-代数

引入了可换BCH-代数的概念,给出了可换BCH-代数的两个充要条件.对偏序可换BCH-代数进行了讨论,给出了偏序BCH-代数是可换的两个充要条件.证明了偏序可换BCH-代数的每个分支是一个下半格,局部有界偏序可换BCH-代数的每个分支是一个格.     

LPNG代数的泛中心扩张(英文）

利用交换结合代数,Lie代数和Novikov代数可以定义LPNG代数.LPNG代数具有三个代数结构,满足四个相容条件,并且可以分别构成Novikov-Poisson代数和Gel'fandDorfman代数.同时,通过对LPNG代数的中心扩张和泛中心扩张的研究,得出LPNG代数有泛中心扩张的充分必要条件是LPNG代数是完备的.最后,得到了自同态和导子提升条件.     

分配的Fuzzy蕴涵代数

给出了分配的Fuzzy蕴涵代数的定义并探讨了其有关性质，接着本文证明了分配的Fuzzy蕴涵代数与Boole代数、正则的HFI代数是相互等价的，从而得到Boole代数的两个等价形式，并且证明了分配的Fuzzy蕴涵代数是BL代数，最后得到了FI代数成为Boole代数的几个充要条件。     

乘积型模糊B-代数

引入乘积型模糊B-代数的概念,提供它们的几个例子,研究它们的一些性质.讨论模糊B-代数与乘积型模B-代数的关系,研究乘积型模糊B-代数的同态象与同态原象的性质,给出B-代数上乘积型模糊B-代数与B-代数的积代数上乘积型模糊B-代数的关系.     

微分算子Lie代数上的Leibniz 2-上循环

中心扩张问题在Leibniz代数的研究中起着非常重要的作用,因此有许多文章研究各种各样Leibniz代数的中心扩张问题.在这篇文章里,我们确定了微分算子Lie代数上的所有一维Leibniz中心扩张.     

双扭HOPF代数的对偶

本文研究了一个双扭Hopf代数的分次对偶空间以及两个双扭Hopf代数的分次对偶关系.利用代数和余代数分次对偶空间的性质,得出一个局部有限的双扭(χ1,χ2)-Hopf代数的分次对偶空间是一个双扭(χ1T,χ2)-Hopf代数,并判定两个双扭Hopf代数的分次对偶可以简化为判定它们作为双扭双代数是分次对偶的.     

BL-代数上的⊙-导子

本文引入了BL-代数的⊙-导子并研究了BL-代数上⊙-导子的相关问题.利用导子的保序性,不动点集和BL-代数的格理想,讨论了BL-代数上的保序∧-导子和保序⊙-导子的关系,并给出了Gdel代数和线性Gdel代数的刻画.这些结果丰富了逻辑代数上的导子理论.     

MV-代数的f-微分

利用MV-代数的自同态在MV-代数上引入并研究了广义微分—f-微分.随之给出MV-代数上保序的f-微分及f-微分的不动点集F_d(M)等概念,得到保序的f-微分的几个等价条件以及每一个f-微分的不动点集F_d(M)和集d~(-1)(0)为M的格理想等重要结论.最后,利用保序f-微分及其不动点集F_d(M)给出了MV-代数成为Boolean-代数的一个等价刻画.     

Nijenhuis algebras, NS algebras, and N-dendriform algebras

In this paper, we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras and related algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.     

A note on the left-symmetric algebraic structures of the Witt algebra

In Tang and Bai [Math Nachr. 2012;285:922–935] classified a class of non-graded left-symmetric algebraic structures on the Witt algebra under a certain rational condition. In this note, we show that this rational condition is not necessary. This leads to a more elegant classification of the left-symmetric algebraic structures and Novikov algebraic structures on the Witt algebra.     

On the exponent of finite-dimensional non-cosemisimple Hopf algebras

In 1999, Kashina introduced the exponent of a Hopf algebra. In this article, we prove that the exponent of a finite-dimensional non-cosemisimple Hopf algebra with Chevalley property in characteristic 0 is infinite, and the exponent of a finite-dimensional non-cosemisimple pointed Hopf algebra in positive characteristic is finite.     

Presenting generalized Schur algebras in types B, C, D

We give explicit presentations by generators and relations of certain generalized Schur algebras (associated with tensor powers of the natural representation) in types B, C, D. This extends previous results in type A obtained by two of the authors. The presentation is compatible with the Serre presentation of the corresponding universal enveloping algebra. In types C, D this gives a presentation of the corresponding classical Schur algebra (the image of the representation on a tensor power) since the classical Schur algebra coincides with the generalized Schur algebra in those types. This coincidence between the generalized and classical Schur algebra fails in type B, in general.     

Flat Nonunimodular Lorentzian Lie Algebras

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (?, +,…, +) such that left multiplications are skew-symmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4.     

Heisenberg Lie Color Algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(?) and the holomorph L of finite dimensional Heisenberg Lie color algebra ? graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(?) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.     

Quotients of maximal class of thin lie algebras. the odd characteristic case

Among thin graded Lie algebras, which are particular instances of Lie algebras of finite width, there are many interesting objects, such as the graded Lie algebra associated to the Nottingham group. Among the factors of a thin algebra with respect to the terms of the lower central series, there is a greatest factor which is of maximal class. In thin Lie algebras associated to groups, this factor is metabelian.

In this paper we show that the same holds in general, provided the characteristic of the underlying field is odd. In another paper by the second author it is shown that this is not the case for characteristic two.     

Hom-Lie quadratic and Pinczon Algebras

Presenting the structure equation of a hom-Lie algebra 𝔤, as the vanishing of the self commutator of a coderivation of some associative comultiplication, we define up to homotopy hom-Lie algebras, which yields the general hom-Lie algebra cohomology with value in a module. If the hom-Lie algebra is quadratic, using the Pinczon bracket on skew symmetric multilinear forms on 𝔤, we express this theory in the space of forms. If the hom-Lie algebra is symmetric, it is possible to associate to each module a quadratic hom-Lie algebra and describe the cohomology with value in the module.     

广义仿射李代数上的非交换Poisson代数结构

非交换的Poisson代数同时具有(未必交换的)结合代数和李代数两种代数结构,且结合代数和李代数之间满足所谓的Leibniz法则.本文确定了一般广义仿射李代数上所有的Poisson代数结构.