On Some Families of Smooth Affine Spherical Varieties of Full Rank

Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify:(a) all such varieties for G = SL(2) × C~×and(b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.     

Involutions of \operatorname{{SL}} (nk), (n>2)

A first characterization of the isomorphism classes of -involutions for any reductive algebraic group defined over a perfect field was given in [7] using three invariants. In this paper we give a simple characterization of the isomorphism classes of involutions of with any field of characteristic not equal to . We classify the isomorphism classes of involutions for algebraically closed, the real numbers, the -adic numbers and finite fields. We also determine in which cases the corresponding fixed point group is -anisotropic. In those cases the corresponding symmetric -variety consists of semisimple elements.Aloysius G. Helminck was partially supported by N.S.F. Grant DMS-9977392.     

Spherical Tropical Geometry: a Survey of Recent Developments

This is a survey of some recent results on spherical tropical geometry.     

A Note on Normality of Cones Over Symmetric Varieties

Let G be a semisimple and simply connected algebraic group, and let H ⁰ be the subgroup of points fixed by an involution of G. Let V be an irreducible representation of G with a nonzero vector v fixed by H ⁰. In this article, we prove a property of the normalization of the coordinate ring of the closure of G·[v] in ?(V).     

On the Duals of Segre Varieties

The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M.M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?     

Restrictions of Symmetric Invariants of a Complex Semisimple Lie Algebra and Some Applications

Let 𝔤 be a finite-dimensional complex semisimple Lie algebra and σ an arbitrary semisimple automorphism of 𝔤. Let 𝔱 be a Cartan subalgebra of 𝔨 = 𝔤^σ and 𝔥 =Z _𝔤(𝔱) be the centralizer of 𝔱 in 𝔤. Then 𝔥 is a σ-invariant Cartan subalgebra of 𝔤 and 𝔱 = 𝔥^σ. Let W(𝔤, 𝔥) be the Weyl group. One knows that Δ(𝔤, 𝔱), the set of roots of 𝔤 in 𝔱, is also a root system. It is proved that the corresponding Weyl group W(𝔤, 𝔱) is isomorphic to W(𝔤, 𝔥)^σ, which is the subgroup of W(𝔤, 𝔥) consisting of those elements commuting with σ. It is also shown that the image of the restriction map S(𝔥*) ^{W(𝔤, 𝔥)} → S(𝔱*) ^{W(𝔨, 𝔱)}, where S(𝔥*) and S(𝔱*) are the polynomial algebras on 𝔥 and 𝔱, respectively, is exactly S(𝔱*) ^{W(𝔤, 𝔱)}. Based on the above result, we also get a complete classification of the pairs (𝔤, σ) such that 𝔤^σ is noncohomologous to zero in 𝔤.     

Landau-Lifshitz方程派生的球面锥对称族及其演化

直至2008年,尚未见到讨论同时具有外磁场和各向异性场的多维Landau-Lifshitz方程的显式动态解的文献.当缺少外加磁场或各向异性场时,许多作者曾经尝试过Hirota的著名方法,Lie代数结构与Backlund变换,双线性变换等,但精确解仍然没有构造出来.本文首先给出了一种求解具有外磁场和各向异性场的Landa...     

Orbits and Invariants of Third-Order Cubic Matrices with Symmetric Fibers

In the note, for an action which is the tensor product of the tautological action of the unimodular group of three-dimensional space and the second symmetric power of the tautological action of the unimodular group of three-dimensional space, the orbits are classified and the generators of the algebra of invariants are described.     

Singularities of Toric Varieties Associated with Finite Distributive Lattices

With each finite lattice L we associate a projectively embedded scheme V(L); as Hibi has shown, the lattice D is distributive if and only if V(D) is irreducible, in which case it is a toric variety. We first apply Birkhoff's structure theorem for finite distributive lattices to show that the orbit decomposition of V(D) gives a lattice isomorphic to the lattice of contractions of the bounded poset of join-irreducibles of D. Then we describe the singular locus of V(D) by applying some general theory of toric varieties to the fan dual to the order polytope of P: V(D) is nonsingular along an orbit closure if and only if each fibre of the corresponding contraction is a tree. Finally, we examine the local rings and associated graded rings of orbit closures in V(D). This leads to a second (self-contained) proof that the singular locus is as described, and a similar combinatorial criterion for the normal link of an orbit closure to be irreducible.     

On Some Varieties of Soluble Lie Algebras

In this paper,we study a class of soluble Lie algebras with variety relations that the commutator of m and n is zero.The aim of the paper is to consider the relationship between the Lie algebra L with ...     

On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and V _p is the variety of associative algebras over F defined by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 and by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 (here [x, y] = xy ? yx). As is known, the free algebras of countable rank of the varieties V _p contain non-finitely generated T-spaces. We prove that the varieties V _p are minimal with respect to this property.     

Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples

Energy conservation of numerical integrators is well understood for symplectic one-step methods. This article provides new insight into energy conservation with non-symplectic methods. Sufficient conditions and counter-examples are presented. AMS subject classification (2000) 65L06, 65P10, 37J99.Submitted June 2004. Accepted October 2004. Communicated by Syvert Nørsett.     

相应于一类对称自对偶李代数的顶点代数结构

设gsp是给定的对称自对偶李代数,给出了相应于gsp的一个顶点代数结构.     

Spherical extension property no longer true for domains in algebraic variety with isolated singularity Dedicated to Professor YANG Lo on the occasion of his 70th birthday

Let D be a domain in Cn. It is known that if D is a simply connected bounded domain in C with spherical real analytic boundary D, then every local biholomorphic map at boundary as above extends to a biholomorphic map from D onto the unit ball in Cn. As a consequence, a local biholomorphic map between D1 and D2 where D1 and D2 are simply connected domains in Cn with spherical real analytic boundaries can extend to a global biholomorphic map from D1 onto D2. If the boundary is algebraic, the simply connected condition in the above result can be dropped. In this note, we show that the above phenomenon is no longer true if domains are in algebraic varieties with isolated singularities.     

球壳中球形夹杂对SH波的三维散射与动应力集中

研究了一般情况下球壳中球形夹杂(包括孔洞)引起的SH波三维散射与动应力集中现象.根据球壳与夹杂的几何特点,分别以球壳和夹杂中心建立球坐标,用于描述球壳中的入射波、散射波和夹杂中的驻波势函数,并采用球波函数的加法公式,实现了不同坐标下球波函数的变换,推导出位移、应力分量的解析解.结合球壳的边界条件和夹杂界面的连续条件,求解了不同材料属性夹杂,以及空洞情况下弹性波的散射和动应力集中因子分布情况,并分析了频率以及夹杂中心位置对动应力集中因子的影响.文中的研究为球壳结构的力学性能分析以及无损检测提供了理论支持.     

预给二面角的单形在球面型空间Sn,r的嵌入

本文获得预给二面角的单形嵌入球面型空间Ｓ_n,r的一个充分必要条件，并利用它得到关于球面单形二面角的两类几何不等式。     

一些特殊变换群的联合不变量和联合微分不变量

The theory of moving frames developed by Peter J Olver and M Fels has importaut applications to geometry,classical invariant theory.We will use this theory to classify joint invariants and joint differential invariants of some transformation groups.     

可解群的若干充分条件

A subgroup if of a group G is called semipermutable if it is permutable with every subgroup K of G with (|H|, |K|) - 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. In this paper, some sufficient conditions for a group to be solvable are obtained in terms of s-semipermutability.     

利用球面坐标及柱面坐标计算曲面积分

介绍了利用球面坐标及柱面坐标计算对面积的曲面积分的方法及其应用 .     

The Moduli Space of Real Abelian Varieties with Level Structure

The moduli space of principally polarized Abelian varieties with real structure and with level N = 4m structure (with m1) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over , and to consist of finitely many copies of the quotient of the space GL(n, )/O(N) (of positive definite symmetric matrices) by the principal congruence subgroup of level N in GL(n, ).