Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.     

Super-biderivations of classical simple Lie superalgebras

In this paper, we characterize super-biderivations of classical simple Lie superalgebras over the complex field \(\mathbb {C}\). Furthermore, we prove that all super-biderivations of classical simple Lie superalgebras are inner super-biderivations. As an application, the super-biderivations of a general linear Lie superalgebra are studied. We find that there exist non-inner and non-super-skewsymmetric super-biderivations. Finally, using the results on biderivations we characterize linear super commuting maps on the classical simple Lie superalgebras and general linear Lie superalgebras.     

On structure and TKK algebras for Jordan superalgebras

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions reduce to one of two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions become nonequivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero.     

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.     

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.     

Multipliers,covers, and stem extensions for Lie superalgebras

Suppose that the underlying field is of characteristic different from 2 and 3. We first prove that the so-called stem deformations of a free presentation of a finite-dimensional Lie superalgebra L exhaust all the maximal stem extensions of L; up to equivalence of extensions. Then we prove that multipliers and covers always exist for a Lie superalgebra and they are unique up to superalgebra isomorphisms. Finally, we describe the multipliers, covers, and maximal stem extensions of Heisenberg superalgebras and model filiform Lie superalgebras.     

δ-Jordan李超代数的交换扩张

通过δ-Jordan李超代数T的表示和上同调理论,构造δ-Jordan李超代数T■V.证明了δ-Jordan李超代数的等价交换扩张给出相同的表示.通过δ-Jordan李超代数的表示和其交换扩张得到2-上圈.     

Left supersymmetric Structures on Lie Superalgebras

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Lie Superautomorphisms on Associative Algebras,II

We consider Lie superautomorphisms of prime associative superalgebras. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.     

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.     

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）.     

李超代数上的不变双线性型

本文通过定义李超代数上的形心和零次形心来考察其性质.证明了二次李超代数(G,B)上的不变数积的集合和其形心中的可逆B-超对称元素的集合之间存在一一对应.而对实单李超代数分为两种不同的类型:或者是一个忽略了复结构的复李超代数或者是一个复单李超代数的实形式.     

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．     

Hopf Superquivers

In this paper, a super version of the Hopf quiver theory is developed. The notion of Hopf superquivers is introduced. It is shown that only the path supercoalgebras of Hopf superquivers admit graded Hopf superalgebra structures. A complete classification of such graded Hopf superalgebras is given. A superquiver setting for general pointed Hopf superalgebras is also built up. In particular, a super version of the Gabriel type theorem and the Cartier-Gabriel decomposition theorem is given.     

Generalized P(N)-graded Lie superalgebras

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but suffcient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie super-algebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov's theorem for type P(N). We also obtain a generalization of Kac's coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.     

Whittaker Categories and Whittaker Modules for Lie Superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, the construction of Whittaker modules from Whittaker modules for the even part.     

New simple modular Lie superalgebras as generalized prolongs

Over algebraically closed fields of characteristic p > 2, —prolongations of simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. We discover several new simple Lie superalgebras, serial and exceptional, including super versions of Brown and Melikyan algebras, and thus corroborate the super analog of the Kostrikin-Shafarevich conjecture. Simple Lie superalgebras with 2 × 2 Cartan matrices are classified.     

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.