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Factorization and symplectic uniton numbers for harmonic maps into symplectic groups

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Science China Mathematics 2001年第10期44卷 1225-1235页

作者：贺群沈一兵 Department of Mathematics West-Brook Campus Zhejiang University Hangzhou China Department of Applied Mathematics Shanghai Tongji University Shanghai China

It is proved that any harmonic map (?): Ω→Sp(N) from a simply connected domain Ω (?) R2∪ {∞} into the symplectic group Sp(N)(?)U(2N) with finite uniton number can be factorized into a product of a finite number of symplectic unitons. Based on this factorization, it is proved that the mini-mal symplectic uniton number of (?) is not larger than N, and the minimal uniton number of (?) is not larger than 2N - 1. The latter has been shown in literature in a quite different way.

关键词： harmonic map symplectic group factorization uniton number.

Differentiable structure of ω-subsets of symplectic groups

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Science China Mathematics 2001年第5期44卷 545-554页

作者：龙以明朱朝锋 Nankai Institute of Mathematics Nankai University Tianjin China Key Laboratory of Pure Mathematics and Combinatorics Ministry of Education Tianjin China

In this paper, we study the differentiable structure of the ω-subset of Sp(2n), which is formed by all matrices in Sp(2n) possessing ω as an eigenvalue, for ω on the unit circle in the complex plane. Based on this ... 详细信息

关键词： symplectic group ω-subset stratification differentiable structure transversal structure.

Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

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Acta Mathematica Sinica,English Series 2021年第6期37卷 926-940页

作者： Yu Lei WANG He Guo LIU Department of Mathematics He’nan University of TechnologyZhengzhou 450001P.R.China Department of Mathematics Hubei UniversityWuhan 430062P.R.China

Let p be an odd prime,and let k be a nonzero nature *** that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Then G is a central product of an extraspecial pkgroup E andζ***|E|=p(2n+1)k and|ζG|=p(m+1)*** that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤*** AutG’G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G’,let AutG/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the centerζG and let AutG/ζG,ζG/G’G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially onζG/G’.Then(ⅰ)The group extension 1→Aut G’→Aut G→Aut G’→1 is split.(ⅱ)AutG’G/AutG/ζG,ζG G≌G1×G2,where Sp(2n-2,Zpk)■H≤G1≤Sp(2n,Zpk),H is an extraspecial pk-group of order p(2n-1)k and(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)≤G2≤GL(m,Zpk)■Zpk(m).In particular,G1=Sp(2n-2,Zpk)■H if and only if l=k and r=0;G1=Sp(2n,Zpx)if and only if l≤r;G2=(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)if and only if r=k;G2=GL(m,Zpk)■Zpk((m))if and only if r=0.(ⅲ)AutG’G/Aut G/ζG,ζG/G’G≌G1×G3,where G1 is defined in(ⅱ);GL(ml,Zpk)■Zpk(m-1)≤G3≤GL(n,Zpk).In particular,G3=GL(m-1,Zpk)■Zpk(m-1)if and only if r=k;G3=GL(m,Zpk)if and only if r=0.(ⅳ)AntG/ζG,ζG/G’G≌AutG/ζG,ζG/G’G■Zpk(m),If m=0,then AntG/ζG,ζG/G’G=Inn G≌Zpk(2n);If m>0,then AntG/ζG,ζG/G’G≌Zpk(2nm)×Zpk-r(2n),and AutG/ζG,ζG G/Inn G≌Zpk((2n(m-1))×Zpk-r(2n).

关键词： Automorphism p-group cyclic derived subgroup symplectic group

NEW NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON Sp(n)

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Acta Mathematica Scientia 2021年第3期41卷 887-898页

作者：张绍祥陈慧斌邓少强 School of Mathematical Sciences and LPMC Nankai UniversityTianjin 300071China School of Mathematical Sciences Nanjing Normal UniversityNanjing 210046China

In this paper, we consider a class of left invariant Riemannian metrics on Sp(n),which is invariant under the adjoint action of the subgroup Sp(n-3) × Sp(1) × Sp(1) × Sp(1).Based on the related formulae in the literature, we show that the existence of Einstein metrics is equivalent to the existence of solutions of some homogeneous Einstein equations. Then we use a technique of the Gr?bner basis to get a sufficient condition for the existence, and show that this method will lead to new non-naturally reductive metrics.

关键词： Einstein metric non-naturally reductive metric compact Lie group symplectic group

The Automorphism group of a Class of Nilpotent groups with Infinite Cyclic Derived Subgroups

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Acta Mathematica Sinica,English Series 2018年第7期34卷 1151-1158页

作者： He Guo LIU Yu Lei WANG Ji Ping ZHANG Department of Mathematics Hubei University Wuhan 430062 P. R. China Department of Mathematics Henan University of Technology Zhengzhou 450001 P. R. China The School of Mathematical Sciences Peking University Beijing 100871 P. R. China

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 E-group E and a free abelian group A with rank m, where E={{1 kα1 kα2…kαn aα＋1 0 1 0 … 0 αn＋2 0 0 0 … 1 α2n＋1 0 0 0 …0 1}}αi∈Z,i=1,2,…,2n＋1},where k is a positive integer. Let AutG＇G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G＇ of G, and Autc G/ζG,ζGG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then （i） The extension →AutG＇G→AutG→AutG＇→1 is split.（ii）AutG＇G/AutG/ζG,ζGG≈Sp（2n,Z）×（GL（m,Z）×（Z）m）,（iii）Aut GζG,ζGG/InnG≈（Zk）2n＋（Z）2nm.

关键词： Generalized extraspecial Z-group symplectic group automorphism group

The Relation of the Morse Index of Closed Geodesics with the Maslov-type Index of symplectic Paths

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Acta Mathematica Sinica,English Series 2005年第2期21卷 237-248页

作者： ChunGenLIU DepartmentofMathematics NankaiUniversityTianjin300071P.R.China

In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), weconstruct a symplectic path γ(t) starting from identity I and ending at P, such that the Morseindex of the closed geodesic c equals the Maslov-type index of γ. As an application of this result,we study the parity of the Morse index of any closed geodesic.

关键词： riemannian manifold morse index maslov-type index symplectic group

ON GENERALIZED HAMILTONIAN SYSTEMS

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Acta Mathematicae Applicatae Sinica 2001年第4期17卷 475-483页

作者：程代展薛伟民廖立志蔡大用 Institute of System Sciences Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing China Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong China Department of Applied Mathematics Tsinghua University Beifing China

The purpose of this paper is to explore an extension of some fundamental properties of the Hamiltonian systems to a more general case. We first extend symplectic group to a general N- group, GN, and prove that it has certain similar properties. A particular property of GN is that as a Lie group dim (GN)≥1. Certain properties of its Lie-algebra 9N are investigated. The results obtained are used to investigate the structure-preserving systems, which generalize the property of symplectic form preserving of Hamiltonian system to a covariant tensor field preserving of certain dynamic systems. The results provide a theoretical benchmark of applying symplectic algorithm to a considerably larger class of structure-preserving systems.

关键词： Hamiltonian systems Hamiltonian control systems symplectic group symplectic algebra symplectic algorithm

Another Construction of Cartesian Authentication Codes from Geometry of Clasical groups

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Communications in Mathematical Research 1999年第1期 103-114页

作者： Feng Rongquan (School of Mathematical Sciences, Peking University, Beijing, 100871)

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＳ，ＥａｎｄＭｂｅｔｈｒｅｅｎｏｎ－ｅｍｐｔｙｓｅｔｓａｎｄｌｅｔｆ：Ｓ×Ｅ→Ｍｂｅａｍａｐ．Ｔｈｅｆｏｕｒｔｕｐｌｅ（Ｓ，Ｅ，Ｍ；ｆ）ｉｓｃａｌｅｄａｎａｕｔｈｅｎｔｉｃａｔｉｏ... 详细信息

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＳ，ＥａｎｄＭｂｅｔｈｒｅｅｎｏｎ－ｅｍｐｔｙｓｅｔｓａｎｄｌｅｔｆ：Ｓ×Ｅ→Ｍｂｅａｍａｐ．Ｔｈｅｆｏｕｒｔｕｐｌｅ（Ｓ，Ｅ，Ｍ；ｆ）ｉｓｃａｌｅｄａｎａｕｔｈｅｎｔｉｃａｔｉｏｎｃｏｄｅ（ｓｅｅ［１］），ｉｆ（...

关键词： Authentication Code Finite Field symplectic group Unitary group Orthogonal group Pesudo symplectic group

symplectic Runge-Kutta Methods for The Linear Quadratic Regulator Problem

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Symplectic Runge-Kutta Methods for The Linear Quadratic Regu...