The Decomposition and uniqueness of n-Lie Superalgebras

In this paper, the definitions and some properties of n-Lie superalgebras are presented. Our main aim is to study the decomposition and uniqueness of finite dimensional n-Lie superalgebras with trivial center. Aecoding to the decomposition of n-Lie superalgebras, we obtain the decomposition of inner derivation superalgebras and derivation superalgebras respectively. Furthermore, we discuss some properties about the centroid of n-Lie superalgebras, so we can see its application in the decomposition of n-Lie superalgebras.     

On the Cohomology of Associative Superalgebras

§ 1.ＢａｓｉｃＣｏｎｃｅｐｔｓ　Ｔｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,Ａ =Ａ0 Ａ1ｉｓａｌｗａｙｓａｓｓｕｍｅｄｔｏｂｅａｎａｓｓｏｃｉａｔｉｖｅｓｕｐｅｒａｌｇｅｂｒａｗｉｔｈｕｎｉｔｙｏｖｅｒａｆｉｌｅｄＦ ,ａｎｄＬ =Ｌ0 Ｌ1ａＬｉｅｓｕｐｅｒａｌｇｅｂｒａｏｖｅｒＦ .Ｉｆ |ｘ|ｏｃｃｕｒｓｉｎｓｏｍｅｅｘｐｒｅｓｓｉｏｎ ,ｔｈｅｎｉｔｉｓａｓｓｕｍｅｄｔｈａｔｘｉｓａｈｏｍｏｇｅｎｅｏｕｓｅｌｅｍｅｎｔａｂｏｕｔＺ2 ｇｒａｄｅｄａｎｄ |ｘ|ｄｅｎｏｔｅｓｔｈｅＺ2 ｇｒａｄｅｄｄｅｇｒｅｅ…     

关于完满的Lie超代数

In this paper, some properties of perfect Lie superalgebras are investigated. We prove that the derivation superalgebra of a centerless perfect Lie superalgebra of arbitrary dimension over a field of arbitrary characteristic is complete and we obtain a necessary and sufficient condition for the holomorph of a centerless perfect Lie superalgebra to be complete. Finally, some properties of perfect restricted Lie superalgebras are given.     

N-Crossed Extensions of Crossed Modules

Abstract

We obtain cohomological δ-functors and a five term exact sequence associated to an extension of crossed modules which yields Eilenberg-MacLane cohomology theory and the classic five term exact sequence associated to an extension of groups, when we consider the particular situation of groups.     

Isoclinism between pairs of n-Lie algebras and their properties

In this paper, we investigate the notion of isoclinism on a pair of n-Lie algebras, which forms an equivalence relation. In addition, we prove that each equivalence class contains a stem pair of n-Lie algebras, which has minimal dimension amongst the finite dimensional pairs of n-Lie algebras. Finally, some more results are obtained when two isoclinic pairs of n-Lie algebras are given.     

Some Properties On Isoclinism in n-Lie Algebras

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.     

Superalgebras and operads. I

We construct a theory of multioperator superalgebras and superalgebras over operad.     

强半单的n-李代数的表示

本文主要研究了强半单的n-李代数的表示,证明了强半单的n-李代数的表示(V,ρ)可转化为一个约化李代数Lρ(V)的表示,并证明了不变线性形等其它相关性质．     

李超代数的泛包络代数同构问题

李超代数的一个性质P叫做关于泛包络代数的不变量,如果对于任意李超代数L,H,只要L具有性质P,并且泛包络代数U(L)和U(H)作为结合超代数是同构的,那么H亦具有性质P.通过讨论李超代数关于泛包络代数的不变量证明了:如果L的幂零长度不超过2,那么L和H是同构的.     

n-Lie代数的Frattini子代数及非嵌入定理

In this paper,we prove the nonimbedding theorem in nilpotent n-Liealgebras which is an analogue to the nonimbedding theorem of Burnsids in groupsof prime power order.We also study the properties of Frattini subalgebras of n-Liealgebras over the field with characteristic zero,and prove that the Frattini subalgebraof any k-solvable(k≥2)n-Lie algebra is zero.     

On Restricted Lie Superalgebras with Semisimple Elements

In the present paper, we give some sufficient conditions for the commu tativity of restricted Lie superalgebras and characterize some properties of restricted Lie superalgebras with semisimple elements.     

Borel Subalgebras of Schur Superalgebras

It is proved that any Schur superalgebra is representable as a product of two Borel subalgebras of that superalgebra, which are symmetric w.r.t. its natural anti-isomorphism (Bruhat-Tits decomposition). This readily implies that any simple module is uniquely defined by its highest weight, and all other weights are strictly less than is the highest under the dominant ordering. It is stated that the fundamental theorem of Kempf, which is valid for all classical Schur algebras, might be true for superalgebras only if they are semisimple. Nevertheless, a weaker theorem of Grothendieck holds true for superalgebras since Borel subalgebras are quasihereditary. Also we formulate an analog of the Donkin-Mathieu theorem for Schur superalgebras, and show that it is valid in the elementary non-classical case, that is, for the algebras S(1|1, r).__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 305–334, May–June, 2005.     

关于交换环上带正则基的Hopf-Galois扩张的同构类

本文考虑交换环上带正则基的Ｈｏｐｆ－Ｇａｌｏｉｓ扩张的刻划及其同构类集合的结 构．主要结论是：当Ｂ为一交换环、Ｈ为余交换的有限Ｈｏｐｆ时,上述同构类集合 构成群并与 Ｌ~＊＝（ＢＨ）~＊的 ２次上同调群 Ｈ~２（Ｌ~＊, Ｕ）同构．     

Unipotent Algebraic Affine Supergroups and Nilpotent Lie Superalgebras

We characterize hereditary (as coalgebras) Hopf algebras by the property of ‘equivariant smoothness’, and apply the result to generalize to the super-context, the category equivalence, due to Hochschild, between the unipotent algebraic affine groups and the finite-dimensional nilpotent Lie algebras, in characteristic zero. The global dimension of commutative Hopf algebras, regarded as coalgebras, is also discussed. Presented by S. Montgomery Mathematics Subject Classification (2000) 16W30.     

极大三角代数的交换子和上同调

设M为闭极大三角代数S的σ-弱闭双边模且满足MS,证明了模M交换子C(A,M)= M．进而,如果MAlgLat S,得到H~n(S,AlgLat S)=H~n(S,M)(n≥1)；若dimH〧≤1,则H~n(S,M)=0．     

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.     

四维线状李超代数具有权1的Rota-Baxter算子

根据低维线状李超代数的分类定理,通过计算刻画了复数域C上的四维线状李超代数具有权1的Rota-Baxter算子.     

The Decomposition and Uniqueness of $n$-Lie Superalgebras

In this paper, the definitions and some properties of $n$-Lie superalgebras are presented. Our main aim is to study the decomposition and uniqueness of finite dimensional $n$-Lie superalgebras with trivial center. According to the decomposition of $n$-Lie superalgebras, we obtain the decomposition of inner derivation superalgebras and derivation superalgebras respectively. Furthermore, we discuss some properties about the centroid of $n$-Lie superalgebras, so we can see its application in the decomposition of $n$-Lie superalgebras.