The Even Part of Finite-Dimensional Modular Lie Superalgebra Г

Let F be the underlying base field of characteristic p ＞ 3 and denote by Γō the even part of the finite-dimensional Lie superalgebra Γ.We give the generator sets of the Lie algebra Γō.Using certain properties of the canonical tori of Γō,we describe explicitly the derivation algebra of Γō.     

Derivations for even part of finite-dimensional modular Lie superalgebra \tilde \Omega

Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321: 3601?C3619] constructed a new family of finite-dimensional modular Lie superalgebra $\tilde \Omega $ . Let ?? denote the even part of the Lie superalgebra $\tilde \Omega $ .We first give the generator sets of the Lie algebra ??. Then, we reduce the derivation of ?? to a certain form. With the reduced derivation and a torus of ??, we determine the derivation algebra of ??.     

无限维奇Hamilton模李超代数的导子

本文首先确定了无限维奇Hamilton模李超代数的生成元集,然后确定了奇Hamilton模李超代数到广义Witt模李超代数的导子空间,进而确定了无限维奇Hamilton模李超代数的导子代数.     

Malcev superalgebras with trival nucleus

The nucleus of a Malcev superalgebra M measures how far it is from being a Lie superalgebraM being a Lie superalgebra if and only if its nucleus is the whole M. This paper is devoted to study Malcev superalgebras in the opposite direction, that is, with trivial nucleus. The odd part of any finite-dimensional Malcev superalgebra with trivial nucleus is shown to be contained in the solvable radical. For algebraically closed fields, any such superalgebra splits as the sum of its solvable radical and a semisimple Malcev algebra contained in the even part, which is a direct sum of copies of sl(2, F) and the seven-dimensional simple non-Lie Malcev algebra, obtained from the Cayley-Dickson algebra.     

奇Contact李超代数偶部的导子

在特征p>2的情况下,利用奇Contact李超代数偶部的生成元集,通过计算导子在其生成元集上的作用的方法,确定了奇Contact李超代数偶部的-1次数的导子.     

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.     

Simple Jordan superalgebras with associative nil-semisimple even part

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation.     

A Two-Parameter Quantum (2+1)-Superspace and its Deformed Derivation Algebra as Hopf Superalgebra

As is well known, a Hopf algebra setting is an efficient tool to study some geometric structures such as the Maurer-Cartan invariant forms and the corresponding vector fields on a noncommutative space. In this study we introduce a two-parameter quantum (2+1)-superspace with a Hopf superalgebra structure.We also define some derivation operators acting on this quantum superspace, and we show that the algebra of these derivations is a Hopf superalgebra. Furthermore it will be shown how the derivation operators lead to a bicovariant differential calculus on the two- parameter quantum (2+1)-superspace. In conclusion, based on the bicovariant differential calculus, the Maurer-Cartan right invariant differential forms and the corresponding quantum Lie superalgebra are given.     

The Zhang Transformation and U q (osp(1,2l))-Verma Modules Annihilators

R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l+1) and the Lie superalgebra osp(1,2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebras à la Drinfeld and Jimbo and to show how this construction can explain the main theorem of Gorelik and Lanzmann: the annihilator of a Verma module over the Lie superalgebra osp(1,2l) is generated by its intersection with the centralizer of the even part of the enveloping algebra.     

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.     

Universal Enveloping Algebras of Lie Antialgebras

Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically related to a Lie superalgebra and prove that its enveloping algebra is a quotient of the enveloping algebra of the corresponding Lie superalgebra.     

Structure of some Unital Simple Jordan Superalgebras with Associative Even Part

Studying the unital simple Jordan superalgebras with associative even part, we describe the unital simple Jordan superalgebras such that every pair of even elements induces the zero derivation and every pair of two odd elements induces the zero derivation of the even part. We show that such a superalgebra is either a superalgebra of nondegenerate bilinear form over a field or a four-dimensional simple Jordan 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.     

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra ■(m; 1) ■∧(n), where ■(m; 1) is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type Ⅰ.     

Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

     

Laurent多项式代数C[x_1~(±1),x_2~(±1)]上的李三系

设A为交换变元x_1,x_2的罗朗多项式代数,记A的导子代数Der A为M.本文确定了A,M的对合自同构.利用M的对合自同构给出了一类无限维单李三系,并且通过讨论M的自同构与对合自同构的关系,确定这些单李三系的自同构.     

交换环上上三角矩阵的李三导子

Let T(n,R) be the Lie algebra consisting of all n × n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T(n,R)-bimodule.In this paper,we prove that every Lie triple derivation d : T(n,R) → M is the sum of a Jordan derivation and a central Lie triple derivation.     

Omni-Lie superalgebras and Lie 2-superalgebras

We introduce the notion of omni-Lie superalgebras as a super version of an omni-Lie algebra introduced by Weinstein. This algebraic structure gives a nontrivial example of Leibniz superalgebras and Lie 2-superalgebras. We prove that there is a one-to-one correspondence between Dirac structures of the omni-Lie superalgebra and Lie superalgebra structures on a subspace of a super vector space,     

交换环上一个线性李超代数的导子

本文研究了交换环上一个李超代数的导子.利用构造几类特殊的导子,获得了此李超代数的任意导子是几类特殊导子的和.推广了交换环上李代数的导子.     

有限维模李超代数W(m,n,l,(t))的θ-型导子与导子超代数

In this paper F always denotes a field of characteristic P＞2.We construct the finitedimensional modular Lie superalgebra W(m,n,l,(t))over a field F,define θ-type derivation and determine the derivation superalgebra of w(m,n,l,(t)).