Group gradings on the Lie and Jordan superalgebras Q(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras Q(n), n ≥ 2, over an algebraically closed field of characteristic different from 2 (and not dividing n + 1 in the Lie case): Fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism.     

δ-Derivations of simple finite-dimensional Jordan superalgebras

We describe non-trivial δ-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero δ-derivations are shown to be missing for δ ≠ 0, 1/2, 1, and we give a complete account of 1/2-derivations. Supported by RFBR grant No. 05-01-00230 and by RF Ministry of Education and Science grant No. 11617. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 585–605, September–October, 2007.     

Witt型单李超代数

设F是特征p〉2的域,A是F上结合的超交换的代数,D是域为F上A的超交换的导子.设A×D=A[D]为Witt型李超代数.从环论的角度得到了Witt型李超代数为单代数的充分必要条件.     

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.     

Lie superderivations of superalgebras

The aim of this paper is to investigate Lie superderivations of superalgebras, using the theory of functional identities in superalgebras. As a consequence, we obtain a definitive result on Lie superderivations of prime superalgebras.     

完备Lie超代数

本文引入了完备Ｌｉｅ超代数和Ｌｉｅ超代数的全形这两个概念，讨论了完备Ｌｉｅ超代数的一些等价条件和结构定理。所得结果是Ｊａｃｏｂｓｏ［１］和ＮｅｎｇＤａｏｊｉ［２］的推广。     

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）.     

Symmetrizers for Schur superalgebras

For the Schur superalgebra $S=S(m|n,r)$ over a ground field K of characteristic zero, we define the symmetrizer $T^{\lambda }[i:j]$ of the ordered pairs of tableaux $(T_i,T_j)$ of the shape λ. We show that the K-span $A_{\lambda ,K}$ of all symmetrizers $T^{\lambda }[i:j]$ has a basis consisting of $T^{\lambda }[i:j]$ for $T_i$ and $T_j$ semistandard. In particular, $A_{\lambda ,K}\ne 0$ if and only if λ is an $(m|n)$-hook partition. In this case, the S-superbimodule $A_{\lambda ,K}$ is identified as $D_{\lambda }{?}_K{D}_{\lambda }^o$, where $D_{\lambda }$ and $D_{\lambda }^o$ are left and right irreducible S-supermodules of the highest weight λ.We define modified symmetrizers $T^{\lambda }\{i:j\}$ and show that their $\mathbb{Z}$-span forms a $\mathbb{Z}$-form $A_{\lambda ,\mathbb{Z}}$ of $A_{\lambda ,\mathbb{Q}}$. We show that every modified symmetrizer $T^{\lambda }\{i:j\}$ is a $\mathbb{Z}$-linear combination of modified symmetrizers $T^{\lambda }\{i:j\}$ for $T_i,T_j$ semistandard. Using modular reduction to a field K of characteristic $p>2$, we obtain that $A_{\lambda ,K}$ has a basis consisting of modified symmetrizers $T^{\lambda }\{i:j\}$ for $T_i$ and $T_j$ semistandard.     

Shapovalov determinant for loop superalgebras

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 378–397, September, 2008.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

Embedding theorem ofZ-graded Lie superalgebras

The embedding theorem ofZ-graded Lie superalgebras is given and proved. As a subsidiary result it is proved that a transitiveZ-graded restricted lie superalgebm must be isomorphic toW(m,n, 1) if the dimension ofG _i satisfies a certain condition. Project supported by the National Natural Science Foundation of China and the Science of the University Doctoral Program of CNEC.     

Identities for Lie superalgebras with a nilpotent commutator subalgebra

It is proved that the exponential growth rate of identities in any superalgebra with a nilpotent commutator subalgebra over a field of zero characteristic is an integer. Supported by RFBR (project Nos. 07-01-00080 and 06-01-00485) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-1983.2008.1). Translated from Algebra i Logika, Vol. 47, No. 5, pp. 617–645, September–October, 2008.     

Uniqueness of the decomposition of lie superalgebras and quadratic lie superalgebras

A Lie superalgebra endowed with a non-degenerate super-symmetric and invariant bilinear form is called a quadratic Lie superalgebra. ln this paper, we consider the decomposition of a Lie superalgebra and the first main result is that the decomposition of a Lie superalgebra into indecomposable graded ideals is unique up to an isomorphism. Next, we obtain the uniqueness of the decomposition of an arbitrary quadratic Lie superalgebra into irreducible graded ideals up to an isometry.     

Hom-Malcev superalgebras

Hom-Malcev superalgebras can be considered as a deformation of Malcev superalgebras. We give the definition of Hom-Malcev superalgebras. Moreover, we characterize the Hom-Malcev operator and the representation of Hom-Malcev superalgebras. Finally, we study the central extension and the double extension of Hom-Malcev superalgebras.     

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。     

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.     

Special odd Lie superalgebras in prime characteristic

Let g be a finite dimensional special odd Lie superalgebra over an algebraically closed field F of characteristic p > 3.The sufficient and necessary condition is given for g possessing a nondegenerate associative form and in this case the second cohomology group H 2 (g,F) is completely determined.     

Relations and deformations of odd Hamiltonian superalgebras

The following (co)homology groups of the Lie superalgebraH(0,n) are calculated: theH ² (H, H) cohomology with coefficients in the adjoint module and theH ₂ (LH) homology of the nitpotent subalgebra. It is shown that dimH ² (H,H) = 1. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 391–401, March, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-01263 and by the International Science Foundation under grant No. RO 4300.