Super-biderivations of Lie superalgebras

In this paper, we attempt to investigate the super-biderivations of Lie superalgebras. Furthermore, we prove that all super-biderivations on the centerless super-Virasoro algebras are inner super-biderivations. Finally, we study the linear super commuting maps on the centerless super-Virasoro algebras.     

Structures of Not-finitely Graded Lie Algebras Related to Generalized Virasoro Algebras

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.     

Structures of Not-finitely Graded Lie Algebras Related to Generalized Heisenberg–Virasoro Algebras

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg–Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.     

非有限阶化无中心Virasoro超代数的表示

介绍非有限阶化无中心Virasoro超代数的概念,并通过具体计算给出了非有限阶化无中心Virasoro超代数的一类中间序列模.     

一类广义Heisenberg-Virasoro代数的导子

主要研究了一类非有限分次的广义Heisenberg-Virasoro代数HV,并给出了HV导子的具体结构.为了计算HV的导子Der HV,可以把它分解成内导子adHV和阶为零次的导子(Der HV)_0之和.     

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.     

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.     

Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type

In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including small quantum groups and multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of bosonizations of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite root system.     

Left-symmetric superalgebra structures on the N = 2 superconformal algebras

The super-Virasoro algebras, also known as the superconformal algebras, are nontrivial graded extensions of the Virasoro algebra to Lie superalgebra version. In this paper, we classify the compatible left-symmetric superalgebra structures on the N = 2 Ramond and Neveu–Schwarz superconformal algebras under certain conditions, which generalizes the corresponding results for the Witt, Virasoro and super Virasoro algebras.     

LIE SUPER-BIALGEBRA STRUCTURES ON GENERALIZED SUPER-VIRASORO ALGEBRAS

In this article,Lie super-bialgebra structures on generalized super-Virasoro algebras L are considered.It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras if and only if H1(L,LL)=0.     

Modules of the Intermediate Series over Not-finitely Graded Lie Algebras W(Γ)

Let W(Γ)be a class of not-finitely graded Lie algebras related to gener-alized Virasoro algebras with basis{Lα,i,C | α ∈ Γ,i ∈ Z+},which satisfies relations[Lα,i,Lβ,j]=(β-α)Lα+β,i+j+(j-i)Lα+β,i+j+1+δα+β,0δi+j,0α3-α/12C and[C,Lα,i]=0.In this paper,W(Γ)-modules of the intermediate series satisfying some conditions are con-structed and classified.We also obtain modules of the intermediate series over the related Lie superalgebra.     

李超代数与Frobenius代数

利用限制李超代数的新定义,给出了李超代数的p-包络的一些相关结果,并将李代数中表示理论的一些结果推广到李超代数上,进而研究了限制李超代数与Frobenius代数的关系．     

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.     

ON THE PRIMARY DECOMPOSITION THEOREM OF MODULAR LIE SUPERALGEBRAS

This gives some identities of associative Lie superalgebras and some properties of modular Lie superalgebras. Furthermore, the primary decomposition theorem of modular Lie superalgebras is shown. It is well known that the primary decomposition theorem of modular Lie algebras has played an important role in the classification of the finite-dimensional simple modular Lie algebras (see [5, 6]). Analogously, the primary decomposition theorem of modular Lie superalgebras may play an important role in the open classification of the finite dimensional simple modular Lie superalgebras.     

The existence of superinvolutions

Superinvolutions on graded associative algebras constitute a source of Lie and Jordan superalgebras. Graded versions of the classical Albert and Albert–Riehm Theorems on the existence of superinvolutions are proven. Surprisingly, the existence of superinvolutions of the first kind is a rare phenomenon, as nontrivial central division superalgebras are never endowed with this kind of superinvolutions.     

Vertex Algebras, Lie Algebras, and Superstrings

We construct Lie algebras from vertex superalgebras and study their structure. They are sometimes generalized Kac–Moody algebras. In some special cases we can calculate the multiplicities of the roots.     

Kostant homology formulas for oscillator modules of Lie superalgebras

We provide a systematic approach to obtain formulas for characters and Kostant u-homology groups of the oscillator modules of the finite-dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite-dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned u-homology computation are worked out.     

Superderivations for Modular Graded Lie Superalgebras of Cartan-type

Superderivations for the eight families of finite or infinite dimensional graded Lie superalgebras of Cartan-type over a field of characteristic p?>?3 are completely determined by a uniform approach: The infinite dimensional case is reduced to the finite dimensional case and the latter is further reduced to the restrictedness case, which proves to be far more manageable. In particular, the structures and dimension formulas are clearly described for the outer superderivation algebras of those Lie superalgebras. Certain known results are also covered.     

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.     

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．