一个代数不等式与一类几何不等式的指数推广

本文介绍一个代数不等式,应用它直接将一类常见的几何不等式进行指数推广．定理若ａ,ｂ,ｃ∈Ｒ＋,ｎ∈Ｎ且ｎ≥２,则ａｎ＋ｂｎ＋ｃｎ３≥（ａ＋ｂ＋ｃ３）ｎ（＊）当且仅当ａ＝ｂ＝ｃ时等号成立．证当ｎ＝２时,∵ａ２＋ｂ２＋ｃ２３－（ａ＋ｂ＋ｃ３）２＝（ａ－ｂ...     

余代数和Morita-Takeuchi关系

本文证明了若余代数Ｃ和Ｄ是Ｍｏｒｉｔａ Ｔａｋｅｕｃｈｉ等价的，则Ｃ的子余代数格和Ｄ的子余代数格同构．设ＭΓ，ＮΓ是拟有限右Γ 余模，则（ｈ－Γ（Ｍ，Ｍ），ｈ－Γ（Ｎ，Ｎ）；ｈ－Γ（Ｎ，Ｍ），ｈ－Γ（Ｍ，Ｎ）；ｆ，ｇ）有Ｍｏｒｉｔａ Ｔａｋｅｕｃｈｉ关系．并给出了此Ｍ Ｔ关系是Ｍ Ｔ等价条件的及由已知的Ｍ Ｔ关系构造新的Ｍ Ｔ关系的方法．     

二维向量丛的最大子线丛

本文中我们利用 Ａ．Ｂｅｒｔｒａｍ和 Ｂ． Ｆｅｉｂｅｒｇ证明的在 ｇ＝５的当 Ｓ（Ｅ）＜２时的一般代数曲线上二维特殊稳定向量丛的存在定理作为反例，说明进一步的Ｍａｒｕｙａｍａ猜想和Ａｒｒｏｎｄｏ－Ｓｏｌｓ猜想在ｇ＝５的一般代数曲线上均不能成立．     

正规实型黎曼对称空间上Riesz平均的几乎处处收敛性

设Ｘ＝Ｇ０／Ｋ０是非紧致黎曼对称空间，Ｇ０的李代数ｇ０是其复化李代数ｇ的正规实型．若记ｎ０＝ｄｉｍＸ，则我们在该文中证明：对于ｆ∈Ｌｐ（Ｘ），１≤Ｐ≤２，当复数ｚ满足Ｒｅｚ＞δ（ｎ０，Ｐ）＝（ｎ０－１）（１／Ｐ—１／２）时，ｆ关于拉普拉斯算子本征函数展开的ｚ阶Ｒｉｅｓｚ平均几乎处处收敛于ｆ．这一结论与经典的欧氏空间，以及复的和一秩的非紧致黎曼对称空间中完全相同．     

奇异M─矩阵的图相容弱正则分裂

设Ａ是奇异Ｍ－矩阵，Ａ＝Ｍ－Ｎ是Ａ的图相容弱正则分裂．本文研究迭代矩阵Ｍ－１Ｎ的谱性质，得到与迭代矩阵的指数有关的一个定理：ｉｎｄ０（Ａ）＝ｉｎｄ１（Ｍ－１Ｎ）．它推广了Ｈ．Ｓｃｈｎｅｉｄｅｒ和作者的结果．     

无限维李代数L(α,β)的导子李代数

本文讨论了无限维李代数Ｌ（α，β）的导子李代数的结构．分三种情况：（１）当α，β在Ｑ上线性无关时，ＤｅｒＬ（α，β）＝ＣＤｆ０ＣＤｇ０ａｄＬ（α，β），其中Ｄｆ０，Ｄｇ０是由ｆ０，ｇ０决定的导子，ｆ０，ｇ０是定义在Ｚ×Ｚ上的线性函数；（２）当α，β在Ｑ上线性相关且不同时为０时，ＤｅｒＬ（α，β）ｄｅｒＬ（α′，０）（α′≠０），ｄｅｒＬ（α，０）＝ＣＤ－α０ＣＤ－αｇ０ＣＤｆ０ａｄＬ（α，０），（α≠０），其中Ｄ－α０是某一个固定的导子，Ｄ－αｇ０，Ｄｆ０是由ｇ０，ｆ０决定的导子；（３）当α＝β＝０时，ＤｅｒＬ（０，０）＝ＣＤｆ０ＣＤｇ０ａｄＬ（０，０）．     

奇异M—矩阵的图相容弱正则分裂

设Ａ是奇异Ｍ－矩阵，Ａ＝Ｍ－Ｎ是Ａ的图相容弱正则分裂。本文研究迭代矩阵Ｍ＾－１Ｎ的谱性质，得到与迭代矩阵的指数有关的一个定理：ｉｎｄ０（Ａ）＝ｉｎｄ１（Ｍ＾－１Ｎ）．它推广了Ｈ．Ｓｃｈｎｅｉｄｅｒ和作者的结果。     

一个代数不等式及其应用

一个代数不等式及其应用王福楠（江苏省昆山震川高级中学２１５３００）本文将利用算术———几何平均值不等式，导出一个形式简洁、内涵深刻的代数不等式．定理设ｍ，ｎ∈Ｎ，ｍｉ＝１ａｉ１，ｍｉ＝１ｂｉ１，ａｉ０，ｂｉ＞０（ｉ＝１，２，…，ｍ，ｍ２）...     

新题征展(3)

题组新编１．（１）设Ｍ＝｛ｘ｜ｆ（ｘ）＝０｝、Ｎ＝｛ｘ｜ｇ（ｘ）＝０｝,则｛ｘ｜ｆ（ｘ）·ｇ（ｘ）＝０｝为（　　）;（Ａ）Ｍ　（Ｂ）Ｎ　（Ｃ）Ｍ∪Ｎ　（Ｄ）以上都不对（２）设ｆ（ｘ）＝ｘ－１ｘ＋３,ｇ（ｘ）＝ｘ＋３ｘ－１,则集合｛ｘ｜ｆ（ｘ）·ｇ（ｘ）＝０｝＝　　;（３）设函数ｆ（ｘ）、ｇ（ｘ）的定义域依次是Ｆ、Ｇ,且Ｍ＝｛ｘ｜ｆ（ｘ）＝０｝、Ｎ＝｛ｘ｜ｇ（ｘ）＝０｝,则｛ｘ｜ｆ（ｘ）·ｇ（ｘ）＝０｝＝　　．２．（１）设ｍ、ｋ∈Ｎ,则Ｃｎｎ＋Ｃｎｎ＋１＋Ｃｎｎ＋２＋…＋Ｃｎｎ＋ｋ＝　　;（２）求…     

F_3上代数曲线的有理点个数

如果用Ｎｑ（ｇ）表示Ｆｑ上亏格为ｇ的所有光滑不可约代数曲线的Ｆｑ－有理点个数最大值，则确定Ｎｑ（ｇ）的值是一个困难问题．Ｓｅｒｒｅ确定了当ｑ＝２，１≤ｇ≤１０及ｑ＝３，１≤ｇ≤３时的全部Ｎｑ（ｇ），本文确定了ｑ＝３，４≤ｇ≤６的Ｎｑ（ｇ）值．     

so(5,k)主不可分解模的Loewy序列

设k是特征为素数的代数闭域,李代数g=so(5,k).当p-特征函数χ为次正则幂零且具有标准Levi型时,得到g的主不可分解模的Lowey序列.     

A note on strongly Lie nilpotency

In this note the authors studies strongly Lie nilpotent rings and proves that if a ringR is strongly Lie nilpotent thenR ⁽²⁾, the ideal generated by all commutators, is nilpotent.     

Moduli of Einstein and non-Einstein nilradicals

A nilpotent Lie algebra is called an Einstein nilradical if the corresponding Lie group admits a left-invariant Ricci soliton metric. While these metrics are of independent interest, their existence is intimately related to the existence of Einstein metrics on solvable Lie groups. In this note we are concerned with the following question: How are the Einstein and non-Einstein nilradicals distributed among nilpotent Lie algebras? A full answer to this question is not known and we restrict to the class of 2-step nilpotent Lie groups. Within this class, it is known that a generic group admits a Ricci soliton metric. Using techniques from Geometric Invariant Theory, we study the set of non-generic algebras to learn more about the distribution of non-Einstein nilradicals. Many new (continuous) families of non-isomorphic, non-Einstein nilradicals are constructed. Moreover, the dimension of these families can be arbitrarily large (depending on the dimension of the underlying Lie group). To show such large classes of Lie groups are pairwise non-isomorphic, a new technique is developed to distinguish between Lie algebras.     

Systeme de poids sur une algebre de Lie nilpotente

Let g be anilpotent Lie algebra (of finite dimensionn over an algebraically closed field of characteristic zero) and let Der(g) be the algebra of derivations of g. Thesystem of weights of g is defined as being that of the standard representation of a maximal torus in Der(g) (see l.l). For a fixed integern, it is well-known that there are in general uncountably many isomorphism classes of nilpotent Lie algebra of dimensionn; but we show that there arefinitely many systems of weights, and each of them is explicitely constructed. The class of those Lie algebras having a given (arbitrary) system of weights is also studied.The first chapter is a setting for the study of nilpotent Lie algebras, used to prove some general theorems. In the second chapter, attention is restricted to a class of nilpotent Lie algebras for which our setting is particularly well adapted.

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Ce papier est extrait de mon travail de thèse [5] effectué sous la direction du Professeur Jean de Siebenthal que je remercie vivement.     

The classification of two step nilpotent complex Lie algebras of dimension 8

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.     

Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

For any complex 6-dimensional nilpotent Lie algebra \mathfrakg,\mathfrak{g}, we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain \mathfrakg\mathfrak{g} by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras.     

Completable nilpotent Lie superalgebras

We discuss a class of filiform Lie superalgebras L^n,m. From these Lie superalgebras, all the other filiform Lie superalgebras can be obtained by deformations. We have decompositions of Der0ˉ(L^n,m) and Der1 (L^n,m). By computing a maximal torus on each L^n,m, we show that L^n,m are completable nilpotent Lie superalgebras. We also view L^n,m as Lie algebras, prove that L^n,m are of maximal rank, and show that L^n,m are completable nilpotent Lie algebras. As an application of the results, we show a Heisenberg superalgebra is a completable nilpotent Lie superalgebra.     

Filters compatible with isomorphism testing

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring can be trivial or arbitrarily large. We determine properties of filters such that every isomorphism between groups is induced by an isomorphism between graded Lie rings.     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

Gelfand pairs attached to representations of compact Lie groups

For each compact Lie algebra g and each real representationV of g we construct a two-step nilpotent Lie groupN(g, V), endowed with a natural left-invariant riemannian metric. The main goal of this paper is to show that this construction produces many new Gelfand pairs associated with nilpotent Lie groups. Indeed, we will give a full classification of the manifoldsN(g, V) which are commutative spaces, using a characterization in terms of multiplicity-free actions.Supported by a fellowship from CONICET and research grants from CONICOR and SeCyT UNC (Argentina).