A new characterization of semisimple Lie algebras

Using Casimir elements, we characterize the semisimple Lie algebras among the quadratic Lie algebras. This characterization gives, in particular, a generalization of a consequence of Cartan's second criterion.

     

微分算子代数的一种不可约Harish-Chandra模的分类

本文首先讨论了微分算子Ｌｉｅ代数的单性，然后确定出了微分算子Ｌｉｅ代数的权重数都是１的所有不可约Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模。     

系数在基本Harish—Chandra模中的Virasoro代数的上同调群

设ｇ为特征为０的二次闭域上的Ｖｉｒａｓｏｒｏ代数。本文决定了Ｈ＾１（ｇ，Ｍ（４，１）），Ｈ＾２（ｇ，Ｍ（４，２））及Ｈ＾２（ｇ，ＭＬ）的结构，其中Ｍ（４，１），Ｍ（４，２）为两类基本Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模，ＭＬ为无常数项单变量Ｌａｕｒｅｎｔ多项式。     

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? S^ε(V), where S^ε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained.     

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set S of simple roots of SL(n,C) is an equivariant fibration X→XS of the complete flag variety X of Cn. To each such fibration we associate an algebra JS of operators on L²(X), or more generally on L²-sections of vector bundles over X. This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X→XS and X→XT, is a compact operator if S∪T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as ‘essential orthogonality of subrepresentations’.     

On Homomorphism of Valuation Algebras

In this paper, firstly, a necessary condition and a sufficient condition for an isomorphism between two semiring-induced valuation algebras to exist are presented respectively. Then a general valuation homomorphism based on different domains is defined, and the corresponding homomorphism theorem of valuation algebra is proved.     

Homomorphism bounds for oriented planar graphs

If is a class of oriented graphs (directed graphs without opposite arcs), then an oriented graph is a homomorphism bound for if there is a homomorphism from each graph in to H. We find some necessary conditions for a graph to be a homomorphism bound for the class of oriented planar graphs and prove that such a graph must have maximum degree at least 16; thus there exists an oriented planar graph with oriented chromatic number at least 17. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 175–190, 2007     

Contact and Frobenius solvable Lie algebras with abelian nilradical

The aim of this work is to characterize the families of Frobenius (respectively, contact) solvable Lie algebras that satisfies the following condition: 𝔤 = 𝔥?V, where 𝔥?𝔤𝔩(V), |dim V?dim 𝔤|≤1 and NilRad(𝔤) = V, V being a finite dimensional vector space. In particular, it is proved that every complex Frobenius solvable Lie algebra is decomposable, whereas that in the real case there are only two indecomposable Frobenius solvable Lie algebras.     

<Emphasis Type="Italic">K</Emphasis>-finite matrix elements of irreducible Harish-Chandra modules are hypergeometric functions

We show that each K-finite matrix element of an irreducible infinite-dimensional representation of a semisimple Lie group can be obtained from spherical functions by a finite collection of operations. In particular, each matrix element admits a finite expression via the Heckman-Opdam hypergeometric functions.     

Isomorphisms of Some Complex Poisson Brackets

Given a complex manifold M_i, structures of Poisson algebras on (M_i)=C(M_i,C) which are associated with a nondegenerate -closed (2,0)-form _i on M_i are considered. It is shown that every isomorphism of Poisson structures (M₁) (M₂) is generated by a biholomorphic map :M₂ M₁ such that ₂ = *₁     

c-Graded filiform Lie algebras

In this paper we generalize naturally graded filiform Lie algebras as well as filiform Lie algebras admitting a connected gradation of maximal length, by introducing the concept of c-graded complex filiform Lie algebras. We deal with the particular case of 3-graded filiform Lie algebras and we obtain their classification in arbitrary dimension. We finally show a link among derived algebras, graded filiform and rigid solvable Lie algebras.     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Let be a reductive Lie algebra over C. We say that a -module M is a generalized Harish-Chandra module if, for some subalgebra , M is locally -finite and has finite -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when is a Cartan subalgebra. We also review the recent determination of which reductive in subalgebras are essential to a classification. Finally, we present in detail the emerging picture for the case when is a principal 3-dimensional subalgebra.     

关于环同态扩张的注记

本文指出环同态扩张的几个新的充分条件．     

与一类unit型相联系的有限维单李代数的结构

Let n ≥ 4. The complex Lie algebra, which is attached to the unit form q(x1, x2,..., xn)■ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type Dn, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.