单的n+1维n-李代数的表示(英)

本文证明了不可约的$L(A)$-模是$A$-模的充要条件,给出了单的$n+1$-维$n$-李代数的有限维不可约表示的分类.     

非交换群上一类代数的不可约表示

在一类无限维非交换Hopf代数上,借助其Hopf理想,构造出商Hopf代数,讨论了此商代数上的有限维不可约模,得出此非平凡不可约模的维数一定是2.     

素特征的“李定理”

设L为代数闭域F上有限维李代数,著名的李定理说:若char F=0,则L为可解当且仅当L的任一有限维不可约模为1维的.在这里特征为0及模为有限维两个条件都是本质的.(1)若charF=P>0,则L为交换当且仅当L的任一(有限维)不可约模为1维的;(2)若char F=0,则L为交换当且仅当L的任一(有限维或无限维)不可约模为1维的; (3)若char F=P>7,L为李代数(限制李代数),则L为可解当且仅当L的任一不可约模(限制模)的维数为p的幂.     

素特征的"李定理"

设L为代数闭域F上有限维李代数,著名的李定理说若char F=0,则L为可解当且仅当L的任一有限维不可约模为1维的.在这里特征为0及模为有限维两个条件都是本质的.(1)若charF=p＞o,则L为交换当且仅当L的任一(有限维)不可约模为1维的;(2)若char F=0,则L为交换当且仅当L的任一(有限维或无限维)不可约模为1维的;(3)若charF=p＞7,L为李代数(限制李代数),则L为可解当且仅当L的任一不可约模(限制模)的维数为p的幂.     

Special代数S(3,1)的不可约模

该文通过确定生成既约包络代数的极小左理想的极大权向量来确定不可约模,给出了特征p=2上的Special代数S(3,1)的特征标x的高度≤0的不可约模同构类的代表元以及它们的维数.     

一类超W-代数权空间维数有限的不可约模

本文研究了一类超W-代数上某一权空间维数有限的不可约权模,证明了该权模必是Harish-Chandra模.     

Weyl代数的表示

设Ｋ是一个域．证明了，若ｃｈＫ＝０，那么ｎ－ｔｈＷｅｙｌ代数Ａ_ｎ（ｋ）没有有限维表示．还给出了Ａ_ｎ（ｋ）的不可约Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模的分类．当Ｋ是一个特征非零的代数闭域时，给出了有限维不可约Ａ_ｎ（Ｋ）－模的分类．     

曲线上微分算子环及其上的模

我们首先讨论了解析不可约曲线Ｘ／ｋ上微分算子环的右模Ｄ（（ｘ），Ｉ）的性质及应用，然后讨论曲线上微分算子环的有限维向量空间模和ｈｏｌｏｎｏｍｉｃ模．     

超Schr?dinger代数■(1/1)的上同调

本文具体计算了系数在超Schr?dinger代数■(1/1)的平凡模和有限维不可约模中的第一阶上同调群与第二阶上同调群,并给出了系数在通用包络代数U(■(1/1))中■(1/1)的第一阶与第二阶上同调群的维数是无限维的.     

超Schrödinger代数J(1/1)的上同调

本文具体计算了系数在超Schrödinger代数J（1/1）的平凡模和有限维不可约模中的第一阶上同调群与第二阶上同调群,并给出了系数在通用包络代数U（J（1/1））中J（1/1）的第一阶与第二阶上同调群的维数是无限维的.     

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.     

Nodal degenerations of plane curves and galois covers

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois covers.      

sl_2(R)上Harish-Chandra模及其应用

将ｓｌ２（Ｒ）上不可约Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模及ｓｌ２（Ｒ）上不可分解的Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ模进行了完全分类，得到了与ｓｌ２（Ｃ）上模分类的不同形式．作为应用，又构造了实Ｖｉｒａｓｏｒｏ代数的一类新的不可约表示．     

Symmetry of eigenvalues of Sylvester matrices and tensors

In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous PerronFrobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more natural spectral symmetry perspective. A k-th order tensor has symmetric spectrum if the set of eigenvalues is symmetric under a group action with the group being a subgroup of the multiplicative group of k-th roots of unity. A sufficient condition, in terms of linear equations over the quotient ring, for a tensor possessing symmetric spectrum is given, which becomes also necessary when the tensor is nonnegative, symmetric and weakly irreducible, or an irreducible nonnegative matrix. Moreover, it is shown that for a weakly irreducible nonnegative tensor, the spectral symmetries are the same when either counting or ignoring multiplicities of the eigenvalues. In particular, the spectral symmetry(index of imprimitivity) of an irreducible nonnegative Sylvester matrix is completely resolved via characterizations with the indices of its positive entries. It is shown that the spectrum of an irreducible nonnegative Sylvester matrix can only be 1-symmetric or 2-symmetric, and the exact situations are fully described. With this at hand, the spectral symmetry of a nonnegative two-dimensional symmetric tensor with arbitrary order is also completely characterized.     

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

既约随机矩阵的一些性质

本文讨论了既约广义随机矩阵特征值的性质,得到了双随机矩阵的益为既约矩阵的充要条件,以及P类矩阵的一些性质．     

Irreducible Representations of Wreath Products of Association Schemes

The wreath product of finite association schemes is a natural generalization of the notion of the wreath product of finite permutation groups. We determine all irreducible representations (the Jacobson radical) of a wreath product of two finite association schemes over an algebraically closed field in terms of the irreducible representations (Jacobson radicals) of the two factors involved.     

The geometry of homogeneous submanifolds of hyperbolic space

We prove, in a purely geometric way, that there are no connected irreducible proper subgroups of SO(N,1). Moreover, a weakly irreducible subgroup of SO(N,1) must either act transitively irreducible subgroup of SO(N,1) must either act transitively on the hyperbolic space or on a horosphere. This has obvious consequences for Lorentzian holonomy and in particular explains clasification results of Marcel Berger's list (e.g. the fact that an irreducible Lorentzian locally symmetric space has constant curvatures). We also prove that a minimal homogeneous submanifold of hyperbolic space must be totally-geodesic. Received August 10, 1999; in final form November 23, 1999 / Published online March 12, 2001