The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm).     

Legendrian Subvarieties of Projective Space

I prove that every smooth Legendrian variety generated by quadrics is a homogeneous variety and, further, I give a list of all such Legendrian varieties. Another result is that no complete intersection is a Legendrian variety.Mathematics Subject Classification (2000): Primary: 14M99, secondary: 17B10, 22E46 *The author was partially supported by Marie Curie Programme.     

Structure of a code related to Sp(4, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

We determine the socle and the radical series of the binary code associated with a finite regular generalized quadrangle of even order, considered as a module for the commutator of each of the orthogonal subgroups in the corresponding symplectic group.     

Twisted holomorphic forms on generalized flag varieties

In this paper we prove some vanishing theorems for the twisted Dolbeault cohomology of the complete flag varieties associated to a simple, simply connected algebraic group.     

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.     

Vanishing conditions for the simplicial volume of compact complex varieties

Gromov has defined a notion of simplicial volume: it is a topological invariant for compact manifolds which is closely related to the fundamental group. We investigate here the relevance of this notion in the realm of complex varieties.

     

On compact symplectic manifolds with Lie group symmetries

In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.

     

Actions symplectiques de groupes compacts (Symplectic Actions of Compact Groups)

For any symplectic action of a compact connected group on a compact connected symplectic manifold, we show that the intersection of the Weyl chamber with the image of the moment map is a closed convex polyhedron. This extends Atiyah–Guillemin–Sternberg–Kirwan's convexity theorems to non-Hamiltonian actions. As a consequence, we describe those symplectic actions of a torus which are coisotropic (or multiplicity free), i.e. which have at least one coisotropic orbit: they are the product of an Hamiltonian coisotropic action by an anhamiltonian one. The Hamiltonian coisotropic actions have already been described by Delzant thanks to the convex polyhedron. The anhamiltonian coisotropic actions are actions of a central torus on a symplectic nilmanifold. This text is written as an introduction to the theory of symplectic actions of compact groups since complete proofs of the preliminary classical results are given. An erratum to this article is available at .     

Uncertainty principles for locally compact quantum groups

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.     

Existence and Blow-Up Behavior for Solutions of the Generalized Jang Equation

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up behavior are given, and it is shown that blow-up solutions are not unique.     

The lattice generated by some subvarieties of completely regular semigroups

Using a construction theorem of cryptogroups and congruence methods, we determine an 18-element lattice, generated by {NOBG, ROBG, OBG, NBA, RBA, BA}, of subvarieties of completely regular semigroups. This work was supported by the National Natural Science Foundation of China (Grant No. 10871161) and the Research Foundations for Doctor Programme of Southwest University (Grant Nos. SWNUB2005020, SWNUB2008001)     

Hyperbolic Monopoles and Holomorphic Spheres

We associate to an SU(2) hyperbolic monopole a holomorphicsphere embedded in projective space and use this to uncovervarious features of the monopole.     

四阶整数辛群中扭元的共轭类

本文给出四阶整数辛群中扭元的全部共轭等价类.总共有55个不同的扭元等价类,它们的阶数分别是2, 3, 4, 5, 6, 8, 10, 12.没有其它阶的扭元.     

On Factorization of Harmonic Maps of Surfaces into U(N) and Action of Groups

§ 0 .Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＣｂｅｔｈｅｃｏｍｐｌｅｘｐｌａｎｅ ,ΩａｓｉｍｐｌｅｃｏｎｎｅｃｔｅｄｄｏｍａｉｎｉｎＣ∪ {∞ },ａｎｄＵ(Ｎ)ｔｈｅｒｅａｌｆｏｒｍｏｆｔｈｅｃｏｍｐｌｅｘ ｇｒｏｕｐＧＬ (Ｎ ,Ｃ) .Ｉｆｓ :Ω→Ｕ (Ｎ )ｉｓａｈａｒｍｏｎｉｃｍａｐ ,Ｕｈｌｅｎｂｅｃｋ[1] ｄｅｆｉｎｅｄｔｈｅｅｘｔｅｎｄｅｄｓｏｌｕｔｉｏｎＥλｏｆｓ,λ∈Ｃ \{0 },ｗｈｉｃｈｍａｙｂｅｅｘｐａｎｄｅｄｉｎｔｏａＬａｕｒｅｎｔｓｅｒｉｅｓ ∑+∞α=-∞λαＴα,ｓ=ＱＥ-1,ｗｈｅｒｅＱ∈Ｕ(Ｎ)ｉｓａｃ…     

Cartan Matrix for the Finite Symplectic Group <Emphasis Type="Italic">Sp</Emphasis>(6, 3)

The Cartan matrix for the finite symplectic group S_p(6, 3) over a field of three elements is computed in this paper by using a computer and the MATLAB software.2000 Mathematics Subject Classification: 20C33 20C40, 20G05Supported in part by the NNSFC (10271088) and the Foundation for the University Science and Technology Development of Shanghai.     

传递辛矩阵群收敛于辛Lie群

通过作用量变分原理,给出了Hamilton正则方程离散积分的传递辛矩阵表示,利用Hamilton正则方程给出了其对应的Lie代数．说明了当时间区段长度趋近于0时,离散系统积分的传递辛矩阵群收敛于连续时间Hamilton系统微分方程分析积分得到的辛Lie群．     

Linear Operators Preserving Symplectic Group over a Field Consisting of at Least Four Elements

Suppose F is a field consisting of at least four elements. Let Mn(F) and SP2n(F) be the linear space of all n×n matrices and the group of all 2n×2n symplectic matrices over F, respectively. A linear operator L:M2n(F)→M2n(F) is said to preserve the symplectic group if L(SP2n(F))=SP2n(F). It is shown that L is an invertible preserver of the symplectic group if and only if L takes the form (i) L(X)=QPXP-1 for any X∈M2n(F) or (ii) L(X)=QPXTP-1 for any X∈M2n(F), where Q∈SP2n(F) and P is a generalized symplectic matrix. This generalizes the result derived by Pierce in Canad J. Math., 3(1975), 715-724.     

Invariant pseudo-Kähler metrics on generalized flag manifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider signatures of invariant pseudo-Kähler metrics on generalized flag manifolds from the viewpoint of T-root systems.