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）.     

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.     

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.     

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。     

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.     

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.     

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.     

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.     

The abelian subalgebras of maximal dimensions for general linear Lie superalgebras

Over an algebraically closed field of characteristic zero, all the abelian subalgebras of maximal dimensions for the general linear Lie superalgebras are classified in the sense of conjugation. As a consequence, the minimal faithful representations are determined for the so-called nice abelian Lie superalgebras.     

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.     

Restricted Kac modules for special contact Lie superalgebras of odd type

We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3: We give a suffcient and necessary condition in terms of typical or atypical weights for restricted Kac modules to be simple. In the process, we also determine the socle for each restricted Kac module and the length for each simple restricted module.     

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.     

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.     

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,     

Crystal graphs for Lie superalgebras and Cauchy decomposition

We discuss Cauchy type decompositions of crystal graphs for general linear Lie superalgebras. More precisely, we consider bicrystal graph structures on various sets of matrices of non-negative integers, and obtain their decompositions with explicit combinatorial isomorphisms.This research was supported by 2005 research fund of University of Seoul.     

The Kac Jordan superalgebra: automorphisms and maximal subalgebras

In this note the group of automorphisms of the Kac Jordan superalgebra is described and used to classify the maximal subalgebras.

     

Strongly typical representations of the basic classical Lie superalgebras

We describe the category of representations with a strongly typical central character of a basic classical Lie superalgebra in terms of representations of its even part.