SOME RESULTS OF MODULAR LIE SUPERALGEBRAS

In the present article, the authors give some properties on subinvariant subalgebras of modular Lie superalgebras and obtain the derivation tower theorem of modular Lie superalgebras, which is analogous to the automorphism tower theorem of finite groups. Moreover, they announce and prove some results of modular complete Lie superalgebras.

On Perfect Lie Superalgebras

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.

DERIVATION ALGEBRAS OF THE MODULAR LIE SUPERALGEBRAS W AND S OF CARTAN-TYPE

Let F be a field and char F = p > 3. In this paper the derivation algebras of Lie superalgebras W and S of Cartan-type over F are determined by the calculating method.

Induced Modules of Restricted Lie Superalgebras

In this paper we first prove the PBW theorem for reduced universal enveloping algebras of restricted Lie superalgebras. Then the notion of an induced module is introduced and the dimension formula of induced modules is established. Finally, using the results above, we obtain a property of induced modules pertaining to automorphisms of Lie superalgebras and isomorphisms of modules.

A new eight-dimensional Lie superalgebra and two corresponding hierarchies of evolution equations

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.

EMBEDDING THEOREM OF FILTERED LIE SUPERALGEBRAS

The homomorphic realization of Lie superalgebras and the embedding theorem of filtered Lie superalgebras are given and proved.

Infinite-Dimensional Lie Superalgebras SHO' over a Field of Prime Characteristic

The natural filtrations of the infinite-dimensional modular Lie superalgebra SHO＇ are proved to be invariant under automorphisms of SHO＇. The proof involves the investigation of the ad-nilpotent elements of the even part, and the determination of the subalgebras generated by certain ad-nilpotent elements. A property of automorphisms of these Lie superalgebras can be established, and an intrinsic characterization of SHO＇ can be obtained.

COHERENT STATE REPRESENTATION OF THE LIE SUPERALGEBRA OSP(1,2)

We construct coherent states associated with the Lie superalgebra OSP(1. 2) with the aid of two SU(2) coherent states and discuss their properties in detail. We also evaluate the matrix elements of the OSP(1, 2) generators in the coherent-state representation and give a realization for the OSP(1, 2) generators in the coherent state space.

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.

Ideals of Classical Linear Lie Superalgebra P(n) over Commutative Rings

Let R be a commutative ring with identity 1. The relations between the ideals of Lie superalgebra P（n） and the ideals of R are discussed by studying the basis, center and order ideal of P（n）. All ideals of P（n） are proved to be minimal and standard.

On Restricted Lie Superalgebras with Semisimple Elements

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

BOSON-FERMION ALGEBRA AND BOSON-FERMION REPRESENTATION OF LIE SUPERALGEBRA OSP (1, 2)

We describe the establishing process for the Boson-Fermion algebra of OSP(1, 2) starting from the fundamental Wigner operators of OSP(1, 2), and study its structure feature and representation theory. We also give the Boson-Fermion representation of OSP (1, 2), which is an infinite dimensional and grade star representation

A Matrix Representation of Outer Derivations from gl0|2 to the Generalized Witt Lie Superalgebra

Let gl0|2 be a subalgebra of the general linear Lie superalgebra.In this paper,outer derivations from gl0|2 to the generalized Witt Lie superalgebra are completely determined by matrices.

The Derivation Algebra of the Cartan-type Lie Superalgebra H

In this paper, the derivation algebra of Lie superalgebra H of Cartan-type over F are determined by the calculating method in the situations of CharF = p ≥ 3 or m ≥ 2 or n ≥ 1. The main result is following: DerFH = adH(H + Fh) ⊕ {(adDi)pt | i = 1,2,··· ,m, t = 1,2,··· ,ti -1} .

Transitive filtration of Lie superalgebras

Let L be a finite dimensional simple Lie superalgebra, and a sufficient condition is obtained for the length of transitive filtration of L to be greater than zero.

Superbiderivations of Simple Modular Lie Superalgebras of Witt Type and Special Type

Let L be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic p>2.In this paper,all the super-biderivations of L are studied.By means of weight space decompositions with respect to a suitable torus and the standard Z-grading structures of L,we show that the super-symmetric superbiderivations of L are zero.Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case,we find that all the superbiderivations of L are inner.As applications,the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on L are described.

Super-biderivations and Post-Lie Superalgebras on Some Lie Superalgebras

In this paper,all symmetric super-biderivations of some Lie superalgebras are determined.As an application,commutative post-Lie super algebra structures on these Lie super algebras are also obtained.

Left-Symmetric Superalgebra Structures on an Infinite-Dimensional Lie Superalgebra

In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures.

Completable nilpotent Lie superalgebras

We discuss a class of filiform Lie superalgebras Ln,m. From these Lie superalgebras, all the other filiform Lie superalgebras can be obtained by deformations. We have decompositions of Der0（Ln,m） and Dery（Ln,m）. By computing a maximal torus on each Ln,m, we show that Ln,m are completable nilpotent Lie superalgebras. We also view Ln,m as Lie algebras, prove that Ln,m are of maximal rank, and show that Ln,m are completable nilpotent Lie algebras. As an application of the results, we show a Heisenberg superalgebra is a completable nilpotent Lie superalgebra.

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.