辛三代数的Frattini子代数

本文研究了辛三代数的Frattini子代数和基本辛三代数的问题.利用Frattini子代数和基本辛三代数的性质,得到了辛三代数的非嵌入定理,从而推广了李三系中关于Frattini子系的结果.     

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 .     

Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.     

Simple Examples of Symplectic Four-manifolds with Exotic Properties

We construct examples of simply connected nonalgebraic symplectic four-manifolds with a prescribed number of nonintersecting symplectic curves with positive self-intersections.     

Symplectic convexity for orbifolds

Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map for the lifted action on the universal branch covering orbifold, and if the lifted G-action commutes with that of Г, then the symplectic convexity theorem is still true for this kind of lifted Hamiltonian action.     

基于辛理论的Timoshenko梁波散射分析

基于应用力学对偶理论,综合运用了辛Gramme-Schmidt正交化算法,辛本征解的独立性,辛两端边值问题精细积分法等特色理论,分析Timoshenko（铁木辛柯）梁的能带结构,以及端部散射体对于波的散射     

Homologous non-isotopic symplectic surfaces of higher genus

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.

     

The η-form and a generalized Maslov index

Given a family of pairs of transverse Lagrangian subspaces of a hermitean symplectic vector space we define a family of Dirac operators on the unit interval and consider its η-form . To a family of pairwise transverse Lagrangian subspaces we associate the cocycle which is a closed form. We identify its cohomology class with a generalization to families of the triple Maslov index. Received: 6 March 1997     

A remark on symplectic semifield planes and Z 4-linear codes

Kantor and Williams (Trans Am Soc 356:895–938, 2004) introduced a family of non-desarguesian symplectic semifields of even order and studied a number of structures connected with such semifields; namely, symplectic spreads, orthogonal spreads and Z ₄-linear codes. Also, they provided equivalence results concerning such objects, although under certain field restrictions. In this article we will succeed in removing such hypotheses.     

A note on Poisson derivations

Free Poisson algebras are very closely connected with polynomial algebras, and the Poisson brackets are used to solve many problems in affine algebraic geometry. In this note, we study Poisson derivations on the symplectic Poisson algebra, and give a connection between the Jacobian conjecture with derivations on the symplectic Poisson algebra.     

Non-symplectic smooth circle actions on symplectic manifolds

We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or cyclic isotropy sets. All such actions are not compatible with any symplectic form on the manifold in question. In order to cover the case of non-symplectic fixed point sets, we use two smooth 4-manifolds (one symplectic and one non-symplectic) which become diffeomorphic after taking the products with the 2-sphere. The second type of actions is obtained by constructing smooth circle actions on spheres with non-symplectic cyclic isotropy sets, which (by the equivariant connected sum construction) we carry over from the spheres on products of 2-spheres. Moreover, by using the mapping torus construction, we show that periodic diffeomorphisms (isotopic to symplectomorphisms) of symplectic manifolds can provide examples of smooth fixed point free circle actions on symplectic manifolds with non-symplectic cyclic isotropy sets.     

Problems of Lifts in Symplectic Geometry

Let(M,ω)be a symplectic manifold.In this paper,the authors consider the notions of musical(bemolle and diesis)isomorphisms ω~b:T M→T~*M and ω~?:T~*M→TM between tangent and cotangent bundles.The authors prove that the complete lifts of symplectic vector field to tangent and cotangent bundles is ω~b-related.As consequence of analyze of connections between the complete lift ~cω_(T M )of symplectic 2-form ω to tangent bundle and the natural symplectic 2-form dp on cotangent bundle,the authors proved that dp is a pullback o f~cω_(TM)by ω~?.Also,the authors investigate the complete lift ~cφ_T~*_M )of almost complex structure φ to cotangent bundle and prove that it is a transform by ω~?of complete lift~cφ_(T M )to tangent bundle if the triple(M,ω,φ)is an almost holomorphic A-manifold.The transform of complete lifts of vector-valued 2-form is also studied.     

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold(M,ω) into a smooth symplectic manifold . Then we study how the formality and the Lefschetz property of are compared with that of (M,ω). We also study the formality of the symplectic blow-up of (M,ω) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.     

Embedded Surfaces for Symplectic Circle Actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces.More precisely, it is shown that(1) if(M, ω) admits a Hamiltonian S~1-action, then there exists a two-sphere S in M with positive symplectic area satisfying c1(M, ω), [S] 0,and(2) if the action is non-Hamiltonian, then there exists an S~1-invariant symplectic2-torus T in(M, ω) such that c1(M, ω), [T] = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott,Lupton-Oprea, and Ono: Suppose that(M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ· [ω] for some λ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then(1) if λ 0, then G must be trivial,(2) if λ = 0, then the G-action is non-Hamiltonian, and(3) if λ 0, then the G-action is Hamiltonian.     

The geography of spin symplectic 4-manifolds

In this paper we construct a family of simply connected spin non-complex symplectic 4-manifolds which cover all but finitely many allowed lattice points () lying in the region . Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on for all n large enough. Received: 29 August 2000 / in final form: 15 August 2001 / Published online: 28 February 2002     

Invariant linear connections on homogeneous symplectic varieties

We find all homogeneous symplectic varieties of connected semisimple Lie groups that admit an invariant linear connection.The research was supported by the grant INTAS-OPEN-97-1570 of the INTAS Foundation.     

The symplectic ideal and a double centraliser theorem

We interpret a result of Oehms as a statement about the symplecticideal. We use this result to prove a double centraliser theoremfor the symplectic group acting on , where V is the natural module for the symplectic group.This result was obtained in characteristic zero by Weyl. Furthermore,we use this to extend to arbitrary connected reductive groupsG with simply connected derived group the earlier result ofthe author that the algebra K[G] of infinitesimal invariantsin the algebra of regular functions on G is a unique factorisationdomain.     

A remark on equivariance of a moment mapping

It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension $ \tilde G It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension × M → M. Original Russian Text ? I.V. Mikityuk, A.M. Stepin, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 3, pp. 30–33.     

Factorization and symplectic uniton numbers for harmonic maps into symplectic groups

It is proved that any harmonic map ϕ : Ω →Sp(N) from a simply connected domain Ω ⊆R ²⋃ | ∞ | into the symplectic groupSp(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 minimal symplectic uniton number of ϕ is not larger thanN, and the minimal uniton number of ϕ is not larger than 2N - 1. The latter has been shown in literature in a quite different way.