构造相应于有限维非退化可解李代数的顶点代数

设g是带有非退化不变对称双线性型的有限维可解李代数，该文首先应用g的仿射李代数g的表示理论，构造出一类水平为l的限制g-模Vg（l，0）．然后应用顶点算子的局部理论在hom（Vg（l，0），Vg（l，0）（（x）））中找到一类顶点代数Lvg（l，0）．建立了LVg（l，0）到Vg（l，0）的映射，最后证明了这类映射是顶点代数同构．     

构造相应于有限维非退化可解李代数的顶点代数

设g是带有非退化不变对称双线性型的有限维可解李代数, 该文首先应用g的仿射李代数{\heiti $\hat{g}$}的表示理论,构造出一类水平为l的限制$\hat{g}$ -模$V_{\hat{g}}(l,0)$.然后应用顶点算子的局部理论在hom$(V_{\hat{g}}(l,0),V_{\hat{g}}(l,0)((x)))$中 找到一类顶点代数$L_{V_{\hat{g}}(l,0)}$.建立了$L_{V_{\hat{g}}(l,0)}$到 $V_{\hat{g}}(l,0)$的映射,最后证明了这类映射是顶点代数同构.     

一般扭算子李代数

由扭算子构成的扭算子李代数在李代数理论中占有重要的位置,首先构造了一般形式的扭顶点算子Z~σ(E_(ij),α,β,z),然后给出了一般扭算子李代数g(G,l)[σ],研究了一般扭顶点算子所具有的性质.     

顶点代数V_(l,0)的极小生成问题

设g是有限维单李代数,是相应于g的无扭仿型Kac-Moody代数的导代数.讨论了相应于的顶点代数V_(l,0)的极小生成问题,证明了V_(l,0)作为顶点代数由a,h两个元素生成,其中a,h∈g.     

带有非退化不变对称双线性型的有限维可解李代数

本文讨论复数域上带有非退化不变对称双线性型的,可裂的有限维可解李代数的性质及结构.给出了不可分解的非退化可解李代数的定义.证明了本文所讨论的李代数可以分解成不可分解的非退化可解理想的正交直和.对于不可分解的非退化可解李代数,给出了它关于极大环面子代数的根空间分解;讨论了根空间的结构及运算关系;证明了它的 Cartan 子代数的交换性,并给出了 Cartan子代数的结构.     

关于局部顶点李代数的一些注记

本文研究局部顶点李代数与顶点代数之间的关系.利用由局部顶点李代数构造顶点代数的方法,定义局部顶点李代数之间的同态,证明了同态可以唯一诱导出由局部顶点李代数构造所得到的顶点代数之间的同态.     

最简线状n-李代数

定义了一类特殊的幂零n-李代数,即最简线状n-李代数,它是最简线状李代数的推广.确定了m维最简线状n-李代数A的导子代数Der(A)和自同构群Aut(A),定义了n-李代数的全形h(A)=Der(A)( ) A,并证明了当A的基域F的特征P为零或P＞m-n时,Der(A)是不可解的完备李代数,而h(A)的一个子代数是可解的完备李代数,当F的特征为零时,Aut(A)是无中心的不可解群.     

项链李代数的性质

研究项链李代数的性质,给出了其中心元的表示形式,证明了项链李代数非半单、非可解,通过构造项链李代数的可解非幂零子代数,证明了当箭图中有长度大于1的循环时,项链李代数非幂零.还给出了没有圈的箭图上项链李代数的分解.     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.     

Tits-Kantor-Koecher李代数的Wakimoto表示

每一个Jordan代数都对应了一个Tits-Kantor-Koecher李代数.在扩张仿射李代数的分类中[1],A1型李代数的分类依赖于欧氏空间上半格给出的Tits-Kantor-Koecher李代数.另外在相似的意义下,二维欧氏空间R2中只有两个半格.设S是R2上的任一半格,Τ(S)为半格S对应的Jordan代数,(g)(Τ(S))为相应的Tits.Kantor-Koecher李代数.利用Wakimoto自由场的方法给出李代数(g)(Τ(S))的一类顶点表示.     

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.     

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.     

On Borcherds type Lie algebras

For each even lattice \({\mathcal L}\), there is a canonical way to construct an infinite-dimensional Lie algebra via lattice vertex operator algebra theory, we call this Lie algebra and its subalgebras the Borcherds type Lie algebras associated to \({\mathcal L}\). In this paper, we apply this construction to even lattices arising from representation theory of finite-dimensional associative algebras. This is motivated by the different realizations of Kac-Moody algebras by Borcherds using lattice vertex operators and by Peng-Xiao using Ringel-Hall algebras respectively. For any finite-dimensional algebra \(A\) of finite global dimension, we associate a Borcherds type Lie algebra \(\mathfrak {BL}(A)\) to \(A\). In contrast to the Ringel-Hall Lie algebra approach, \(\mathfrak {BL}(A)\) only depends on the symmetric Euler form or Tits form but not the full representation theory of \(A\). However, our results show that for certain classes of finite-dimensional algebras whose representation theory is ’controlled’ by the Euler bilinear forms or Tits forms, their Borcherds type Lie algebras do have close relations with the representation theory of these algebras. Beyond the class of hereditary algebras, these algebras include canonical algebras, representation-directed algebras and incidence algebras of finite prinjective types.     

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.     

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.     

半单李代数的一个特征性质

本文给出了中心为零的带非退化对称不变双线性型的有限维李代数的若干性质,并由此给出了半单李代数的一个新刻划。     

Invariant symmetric bilinear forms for reflection groups

In this paper we describe a connection between Vinberg's criterion for the existence of an invariant symmetric bilinear form for a geometric representation of a Coxeter groups and other criteria which are formulated in terms of conjugation invariant sets of reflections generating a given group. Similar methods lead to the result that every non-symmetrizable Kac--Moody Lie algebra contains a non-symmetrizable subalgebra of rank 3. Finally we explain how the results for symmetric bilinear forms can also be obtained for skew-symmetric forms. Received 3 March 2000.     

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.     

Full field algebras, operads and tensor categories

We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted R×R-graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over VL⊗VR, where VL and VR are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over VL⊗VR equipped with a nondegenerate invariant bilinear form. By assuming slightly stronger conditions on VL and VR, we show that a conformal full field algebra over VL⊗VR equipped with a nondegenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of VL⊗VR-modules. The so-called diagonal constructions [Y.-Z. Huang, L. Kong, Full field algebras, arXiv: math.QA/0511328] of conformal full field algebras are given in tensor-categorical language.