The abelian subalgebras of maximal dimensions for general linear Lie superalgebras

Over an algebraically closed field of characteristic zero, all the abelian subalgebras of maximal dimensions for the general linear Lie superalgebras are classified in the sense of conjugation. As a consequence, the minimal faithful representations are determined for the so-called nice abelian Lie superalgebras.     

Minimal prime ideals in enveloping algebras of Lie superalgebras

Let be a finite dimensional Lie superalgebra over a field of characteristic zero. Let be the enveloping algebra of . We show that when , then is not semiprime, but it has a unique minimal prime ideal; it follows then that when is classically simple, has a unique minimal prime ideal. We further show that when is a finite dimensional nilpotent Lie superalgebra, then has a unique minimal prime ideal.

     

A Note on Restricted Representations of the Witt Superalgebras

Let F be an algebraically closed field of prime characteristic p 〉 3, and W（n） the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W（n） are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W（n） is proved to have one block.     

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．     

Quadratic Lie Superalgebras Generalized by Balinsky–Novikov Superalgebras

Balinsky–Novikov superalgebras were introduced by Balinsky for constructing super-Virasoro type Lie superalgebras. In this paper, we give sufficient and necessary conditions for a Lie superalgebra generalized by a Balinsky–Novikov superalgebra with dimension 2|2 to be a quadratic Lie superalgebra.     

Irreducible Wakimoto-like Modules for the Lie Superalgebra <Emphasis Type=\"Italic\">D</Emphasis>(2, 1;α)

By using the idea of Wakimoto's free field, we construct a class of representations for the Lie superalgebra D(2, 1; α) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter λ. Then we obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter λ satisfies (λ + m)(λ-(1+α/α)≠m)=0 for any m ∈ Z₊.     

ON SOLVABLE COMPLETE LIE SUPERALGEBRAS

The authors discuss the properties of solvable complete Lie superalgebra, proving that solvable Lie superalgebras of maximal rank are complete.     

A class of generalized odd Hamiltonian Lie superalgebras

We study class of finite-dimensional Cantan-type Lie superalgebras HO（m） over a field of prime characteristic, which can be regarded as extensions of odd Hamiltonian superalgebra HO. And we also determine the derivation superalgebras of Lie superalgebras HO（m）.     

局限李超代数的局限表示

本文对如下结果给出一个构造性、初等的、简洁的证明:有限维局限李超代数的有限维局限表示均完全可约当且仅当它的奇部为0而偶部是一个环面.这使得李代数中相应结果成为特例.     

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.     

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.     

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.     

On Blocks in Restricted Representations of Lie Superalgebras of Cartan Type

Let g be a restricted Lie superalgebra of Cartan type W (n), S(n) or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac's definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero.     

线状李超代数Ln,m上的Yang-Baxter 方程

在特征零的代数闭域上,首先做出Ln,m 的一个空间的直和分解,从而将Ln,m 上的Yang-Baxter 方程的解分为若干情形。然后分别在每种情形下对Yang-Baxter 方程进行求解,进而得到了Ln,m 上的所有的Yang-Baxter方程的解的矩阵形式。     

一类广义W型李超代数

构造了一类有限维广义Cartan型模李超代数W,并证明了它是李超代数W(n)的一个扩张,进而决定了它的导子超代数.     

On the Cohomology of Associative Superalgebras

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

一类Weyi型单李超代数

本文研究了单李超代数的构造理论.借助于张量积方法,定义了一类Weyl型结合超代数和一类Weyl型李超代数,并且证明了这类Weyl型结合超代数和Weyl型李超代数是单的充分必要条件.