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.     

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.     

李超三系的分解唯一性

该文首先研究了李超三系的中心、导子超代数和内导子超代数的分解问题, 给出了具有平凡中心的李超三系的分解唯一性定理, 同时讨论了李超三系的自同构的扩张.     

$T^∗$-Extension of Lie Supertriple Systems

In this article, we study the Lie supertriple system (LSTS) $T$ over a field$\mathbb{K}$ admitting a nondegenerate invariant supersymmetric bilinear form (call such a $T$ metrisable). We give the definition of $T^∗_ω$-extension of an LSTS $T$, prove a necessary and sufficient condition for a metrised LSTS ($T$, $ϕ$) to be isometric to a $T^∗$-extension of some LSTS, and determine when two $T^∗$-extensions of an LSTS are "same", i.e., they are equivalent or isometrically equivalent.     

具有非退化Killing型的广义李超代数

设F是特征数为零的域.本文证明了F上的有限维广义李超代数的Killing型是相容的、不变的与上对称的.进而证明了具有非退化Killing型的有限维广义李超代数的 若干性质，最后得出这种广义李超代数必为有限个典型的广义李超代数的直积.     

关于单李超三系的分类(英文)

本文证明了特征零代数闭域上的单李超三系可分为两部分:一是将单李超代数视为李超三系;二是单李超代数的对合自同构的(-1)-特征子空间.这些推广了Lister和Meyberg给出的关于单李三系分类的相应结果.     

Left supersymmetric Structures on Lie Superalgebras

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李超三系的Frattini-子系

In the present paper,we develop initially the Frattini theory for Lie supertriple systems,obtain some properties of the Frattini subsystem and show that the intersection of all maximal subsystems of a solvable Lie supertriple system is its ideal.Moreover,we give the relationship between φ-free and complemented for Lie supertriple system.     

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.     

Invariant bilinear forms on real irreducible representations of real semisimple lie algebras *

In this note, we give a necessary and sufficient condition for a real irreducible representation of a real simple Lie algebra to admit an invariant bilinear form. We also determine when the invariant bilinear form is symmetric or skew-symmetric.     

Some properties of the family Γ of modular Lie superalgebras

In this paper, we continue to investigate some properties of the family Γ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras Γ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.     

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.     

Some properties of generalized reduced Verma modules over ℤ-graded modular Lie superalgebras

We study some properties of generalized reduced Verma modules over ?-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for ?-graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.     

NON-DEGENERATE INVARIANT BILINEAR FORMS ON NONASSOCIATIVE TRIPLE SYSTEMS

§1. IntroductionLie algebras admitting non-degenerate and invariant bilinear forms (i.e. self-dual Liealgebras or pseudo-metric Lie algebras) has been a hot topic in the study of Lie theory. Themotivation for studying these algebras comes from the fact t…     

三次可解型非退化李代数及其导子李代数的结构

本文给出了非退化可解李代数的两个类型:三次可解型非退化李代数和扩充的 Heisenberg李代数,并确定三次可解型非退化李代数及其导子李代数的结构．     

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。     

Detecting cohomology for Lie superalgebras

In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras. These results are used later to affirmatively answer questions, which were originally posed in Boe et al. (2010) [5] and Bagci et al. (2008) [2], about realizing support varieties for Lie superalgebras via rank varieties constructed for the smaller detecting subalgebras.