模代数的形变理论

本文构作了与模代数HA相适应的“形变复形”CH*(H,A),并利用形变复形的上同调群来刻画模代数HA的形变,证明了HA的无穷小形变的等价类是与形变复形的2阶上同调群HH2(H,A)是一一对应的,且当HH2(H,A)=0时,模代数HA为刚性的.     

m-重代数A~((m))上的倾斜模

设A是代数闭域k上的有限维遗传代数,A~((m))和ζ_m(A)分别是A的m-重代数和m-丛范畴.众所周知,代数A~((m))的投射维数不超过m的基本的(basic)倾斜模与m-丛范畴ζ_m(A)的基本的倾斜对象一一对应,这是本文进一步研究m-重代数的倾斜模的原因.本文综述m-重代数A~((m))的偏倾斜模的补、倾斜箭图、倾斜模的自同态代数以及生成子-余生成子的自同态代数的整体维数的值分布.     

C(X) A上的自同构群

本文研究了连续函数代数C(X)与某个C*-代数 A的张量积C(X) A的自同构群.当 A是有单位元且具有平凡中心的C*-代数时,本文完全刻划了C(X) A的自同构群.利用AF-代数的K-理论,本文还刻划了当X是全不连通的紧致Hausdorff空间时,C(X)与紧算子理想的张量积的自同构群.     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.     

归纳极限与积C~*-代数的α-比较性

?对于C~*-代数归纳极限A=(lim→)(A_n,φ_(m,n)(其中A_n?A_(n-1)?A且φ_(n,n-1):A_n→A_(n+1)为嵌入映射),若A_n人为具有α-比较的单的含单位元的稳定有限的C~*-代数,则A具有α-比较性;若A_λ(?_λ∈Λ)具有α-比较性,则积C~*-代数(Πλ∈A)A_λ具有α-比较性,特别地,和C~*-代数(λ∈A)A_λ具有α-比较性.     

3-李代数的广义导子

给出了3-李代数的广义导子、拟导子、拟型心的定义,研究了他们之间的结构关系,并对具有极大对角环面的3-李代数的拟导子和拟型心结构进行了系统的研究.证明了(1)广义导代数GDer(A)可以分解成拟导子代数QDer(A)和拟型心QΓ(A)的直和;(2)3-李代数A的拟导子可以扩张成一个具有较大维数的3-李代数的导子;(3)拟导子代数QDer(A)包含在拟型心的正规化子中,表示为[QDer(A),QΓ(A)]?QΓ(A);(4)如果A包含极大对角环面T,那么QDer(A)和Qr(A)是T的对角模,也就是(T,T)半单地作用在QDer(A)和QΓ(A)上.     

最简线状n-李代数

定义了一类特殊的幂零n-李代数,即最简线状n-李代数,它是最简线状李代数的推广.确定了m维最简线状n-李代数A的导子代数Der(A)和自同构群Aut(A),定义了n-李代数的全形h(A)=Der(A)( ) A,并证明了当A的基域F的特征P为零或P＞m-n时,Der(A)是不可解的完备李代数,而h(A)的一个子代数是可解的完备李代数,当F的特征为零时,Aut(A)是无中心的不可解群.     

态BL-代数上的微分

在态BL-代数(A,σ)上引入(⊙,∨)-微分,研究了态BL-代数(A,σ)上(⊙,∨)-微分的性质。定义并研究了态BL-代数(A,σ)上正则微分和强微分,给出了正则强微分成为保序微分的等价条件。给出态BL-代数上的主微分的概念,讨论了全体主微分的代数结构,并利用Galois联结给出主微分的微分伴随。最后在态BL-代数(A,σ)上引入了不动点集A_(dσ),得到若d为正则保序的强微分时,A_(dσ)为A的格理想。我们的研究思路旨在将BL-代数上的态结构与微分结构紧密融合,以此来研究BL-代数的结构。     

一类Heisenberg n-李代数的自同构群(英文)

本文主要研究Heisenberg n-李代数的结构.给出了一类(3m+1)-维Heisenberg3-李代数及(nm+1)-维Heisenberg n-李代数的自同构群.且给出了自同构的具体表达式.     

泛阶化李超代数

对于给定的负阶化李超代数K-,本文定义了K-型泛阶化李超代数并证明了它的存在性．进而引出阶化Cartan型李超代数,并且证得阶化Cartan型李超代数 W(m,n),K(m,n,ωA),S(m,n)和H(m,n)分别可以用某种泛阶化李超代数来刻画．     

PrO-C*-代数的顺从性和核性

研究了Pro—C^*-代数的顺从性和核性．主要证明了(1)顺从Pro—C^*代数的闭理想是顺从的；(2)核Pro—C^*代数类对归纳极限封闭；(3)交换σ-C^*-代数和核C^*-代数都是核，σ-C^*-代数并且核σ-C^*-代数类对于商运算、张量积运算和可数逆向极限封闭．进一步得到核，σ-C^*-代数的扩张保持核性的条件。     

On tracial approximation

Let C be a class of unital C*-algebras. The class TAC of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in C is studied (Lin considered the case that C consists of finite-dimensional C*-algebras or the tensor products of such with C([0,1])). In particular, the question is considered whether, for any simple separable A∈TAC, there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in C, such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.)     

Unifying derived deformation theories

We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L_∞-algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toën-Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as André-Quillen cohomology.     

Quasivarieties of distributivep-algebras

A counterexample is given to show that not all quasivarieties of distributivep-algebra lie between two consecutive varieties. It is shown that there are no strict relatively congruence distributive quasivarieties of distributivep-algebras. The relative congruence extension property in the class of distributivep-algebras is studied. It is proved that only quasivarieties which are varieties possess this property.Presented by J. Berman.This research is part of the author's dissertation done at Iowa State University under the supervision of Dr. Clifford Bergman.     

The Bunce-Deddens algebras as crossed products by partial automorphisms

We describe both the Bunce-DeddensC ^*-algebras and their Toeplitz versions, as crossed products of commutativeC ^*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy of the set of natural numbers , fitted together in such a way that is an open dense subset. The partial automorphism is induced by a map that acts like the odometer map on the Cantor set while being the translation by one on . From this we deduce, by taking quotients, that the Bunce-DeddensC ^*-algebras are isomorphic to the (classical) crossed product of the algebra of continuous functions on the Cantor set by the odometer map.     

Algebraic analysis of the logic with the quantifier “there exist uncountably many”

In this paper we define a class of polyadic algebras (Q-algebras) as adequate algebraic structures for the logic with the quantifier there exist uncountably many. The main result is a representation theorem for the countableQ-algebras of countable degree.Presented by B. Jónsson.The author wishes to thank the referee for his helpful suggestions.     

Prime spectra of non-commutative generalizations of MV-algebras

Non-commutative generalizations of MV-algebras were introduced by G. Georgescu and A. Iorgulesco as well as by the author; the generalizations are equivalent and are called GMV-algebras. We show that GMV-algebras can be considered as special cases of Grishin algebras. As MV-algebras are algebraic models of the Łukasiewicz logic and Grishin algebras have the analogous role for the classical bilinear logic, GMV-algebras correspond to a non-commutative logic between the above logics. Further, by A. Dvurečenskij, any GMV-algebra is isomorphic to an interval of an l-group, which in general is not commutative. This generalizes D. Mundici's representation of MV-algebras by means of intervals of abelian l-groups. In the paper (using this representation) we describe the properties of prime ideal spectra of GMV-algebras and of their factor algebras and ideals and prove that the spectrum of closed ideals of any GMV-algebra is homeomorphic to that of a completely distributive GMV-algebra. Received January 4, 2001; accepted in final form May 2, 2002.     

实（AF）—代数的理想

给出实（AF）－代数Bratteli图的理想的概念,讨论了这个理想与实（AF）－代数的理想的关系,是对实（AF）－代数结构理论的进一步完善。     

The weak and strong Gaussian probabilistic realization problem

A classification is given of all σ-algebras that make two given σ-algebras conditionally independent in the case that the σ-algebras are generated by finite dimensional Gaussian random variables. In addition a classification is given of all Gaussian measures that have the conditional independence property and such that restricted to a subspace, they coincide with a given measure.     

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

Introducing Nijenhuis forms on L_∞-algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into an L_∞-algebra. Then we show that Nijenhuis forms on L_∞-algebras also give a short and e?cient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.