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.     

反李三系的不变双线性型

本文考察了反李三系和它的标准嵌入李超代数的Killing型之间的关系,并且证明了反李三系的反对称不变双线性型可以被唯一地扩张到它的标准嵌入李超代数。作为扩张定理的一个应用,得到了二次李和反李三系的唯一分解定理。     

有限维非退化可解李代数的顶点算子代数

构造相应于非退化可解李代数g的顶点算子代数分两步进行,首先构造顶点代数.本文是在已经得到的相应于非退化可解李代数g的顶点代数(Vg(l,0),Y(V,1)上构造顶点算子代数.定义了非退化可解李代数g的Casimir算子Ω,给出了在伴随表示下Ω作用在g上是0及相关性质,并应用Ω定义出Vg(l,0)中元素ω,证明了Vg(l,0)关于ω的顶点算子YV(ω,x)的系数构成一个Virasoro代数-模,还证明了ω满足顶点算子代数定义中Virasoro-向量的所有公理.从而证得(Vg(l,0),Yv,1,ω)是一个顶点算子代数.     

Supersymmetric invariance and universal central extensions of lie supertriple systems

In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland's theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.     

SUPERSYMMETRIC INVARIANCE AND UNIVERSAL CENTRAL EXTENSIONS OF LIE SUPERTRIPLE SYSTEMS

In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland＇s theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.     

2-Local Automorphisms on Basic Classical Lie Superalgebras

Let G be a basic classical Lie superalgebra except A(n, n) and D(2, 1, α) over the complex number field C. Using existence of a non-degenerate invariant bilinear form and root space decomposition, we prove that every 2-local automorphism on G is an automorphism. Furthermore, we give an example of a 2-local automorphism which is not an automorphism on a subalgebra of Lie superalgebra spl(3, 3).     

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.     

Dirac Operators on Quadratic Lie Superalgebras

Assume that r is a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, p is a finite dimensional complex super vector space with a nondegenerate super-symmetric bilinear form, and ν : r → osp(p) is a homomorphism of Lie superalgebras.In this paper, we give a necessary and sufficient condition for r ⊕ p to be a quadratic Lie superalgebra.Then, we define the cubic Dirac operator D(g, r) on g and give a formula of(D(g, r))~2. Finally, we get the Vogan's conjecture for quadratic Lie superalgebras by D(g, r).     

Witt’s theorems for Galois Ring valued quadratic forms

We prove Witt’s cancelation and extension theorems for Galois Ring valued quadratic forms. The proof is based on the properties of the invariant I, previously defined by the authors, that classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular Galois Ring valued quadratic forms. Our results extend the Witt’s theorem for mod four valued quadratic forms. On the other hand, the known relation between the invariant I and the Arf invariant of an ordinary quadratic form (if the associated nonsingular bilinear form is alternating) is extended to the nonalternating case by explaining the invariant I in terms of Clifford algebras.     

二次李三系的分解定理

有限维李三系称为二次的,如果它容许一个非退化的不变对称双线性型.无论是李三系分解为不可分解理想的直和,还是二次李三系分解为不可分解非退化理想的正交直和,证明了这两类分解在同构意义下都是唯一的.     

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H²_Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.     

Lax-integrable Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems and its finite-dimensional reduction of Neumann type

A compatibly bi-Hamiltonian Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems is obtained by using a relation for the Casimir functionals of the central extension of a Lie algebra of superconformal even vector fields of two anticommuting variables. Its matrix Lax representation is determined by using the property of the gradient of the supertrace of the monodromy supermatrix for the corresponding matrix spectral problem. For a supersymmetric Laberge–Mathieu hierarchy, we develop a method for reduction to a nonlocal finite-dimensional invariant subspace of the Neumann type. We prove the existence of a canonical even supersymplectic structure on this subspace and the Lax–Liouville integrability of the reduced commuting vector fields generated by the hierarchy.     

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

Let g be a classical simple Lie superalgebra. To every nilpotent orbit O in g₀ we associate a Clifford algebra over the field of rational functions on O. We find the rank, k(O) of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a U(g)-module with O or an orbital subvariety of O as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant k(O) is in many cases, equal to the odd dimension of the orbit G⋅O, where G is a Lie supergroup with Lie superalgebra g.     

Metric Lie algebras with maximal isotropic centre

A metric Lie algebra is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form. It is called indecomposable if it is not the direct sum of two metric Lie algebras. We are interested in describing the isomorphism classes of indecomposable metric Lie algebras. In the present paper we restrict ourselves to a certain class of solvable metric Lie algebras which includes all indecomposable metric Lie algebras with maximal isotropic centre. We will see that each metric Lie algebra belonging to this class is a twofold extension associated with an orthogonal representation of an abelian Lie algebra. We will describe equivalence classes of such extensions by a certain cohomology set. In particular we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre and the classification of metric Lie algebras of index 2.     

Left supersymmetric Structures on Lie Superalgebras

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辛代数的不变双线性型

考察了辛代数和与它相联系的李三系的双线性型之间的关系,并证明了辛代数的反对称不变双线性型可以唯一扩张到与它相联系的李三系中.作为这种关系的一个应用,得到了二次辛代数是单辛代数的一个充要条件,并证明二次辛代数的唯一分解定理.     

POINTED REPRESENTATIONS OFINFINITE DIMENSIONAL LIE ALGEBRAS

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