约化Poisson作用

给出了Poisson Lie商群作用于Poisson商流形成为Poisson作用的充要条件 ,其中 ,商群和商流形的Poisson结构都由Dirac结构诱导 .建立了Poisson齐性空间的左不变Dirac结构与左不变张量两种刻画的等价关系 .     

利用Dirac理论进行Poisson流形及Jacobi流形的约化(英文)

通过将可约的Dirac以及Jacobi-Dirac结构分别分为两种类型,给出对应于Poisson流形和Jacobi流形的约化定理.这些约化定理的证明只需要进行一些直接的计算,而不需要借助于矩映射或者相容函数等复杂概念的引入.另外,给出了一些相应的例子和应用.     

Bretherton方程的对称约化和精确解

运用李群对称方法解决Bretherton方程问题,得到方程的对称约化和群不变解,比如幂级数解,最后得出该问题的守恒率.     

ON THE SYMPLECTIC REDUCTIONS

A symplectic reduction method for symplectic G-spaces is given in this paper without usingthe existence of momentum mappings.By a method similar to the above one,the arthorsgive a symplectic reduction method for the Poisson action of Poisson Lie groups on symplecticmanifolds,also without using the existence of momentum mappings.The symplectic reductionmethod for momentum mappings is thus a special case of the above results.     

具有殆仿切触结构流形的新例子

本文证明了(2n+1)维连通实Lie群具有殆仿切触结构,并给出它们是可积的充要条件,从而给出这类流形一批新的例子。它们与过去已知的这类流形大都是满足某些条件的仿复流形上的主圆丛不同。     

约化枚举及约化方程的Hamilton结构

本文研究了[1]中提出的谱问题:Ψ_χ=UΨ(其中,U=-iλσ_3 P(χ,t) iλ~(-1)Q(x,t))的约化枚举问题,并得到了几族新的约化方程;应用BPT方法研究了约化方程的Hamilton结构.     

约化环的一些刻画

主要研究了约化环的一些性质,给出了约化环的一些刻画,进一步研究了约化环和弱约化环之间的联系.     

Reduction of Volume—preserving Flows on an n—dimensional Manifold

A geometric reduction procedure for volume-preserving flows with a volume-preserving symme-try on an n-dimensional manifold is obtained.Instead of the coordinate-dependent theory and the concrete coordinate transformation,we xhow that a volume-preserving flow with a one-parameter volume-preserving symmetry on an n-dimensional manifold can be reduced to a volume-preserving flow on the corresponding (n-1)-dimensional quotient space.More generally,if it admits an r-parameter volume-preserving commutable symmetry,then the reduced folw preserves the corresponding (n-r)-dimensional volume form.     

PBIB设计的约化

设Ｄ是一个有ｍ个结合类的ＰＢＩＢ设计，且它的参数λ１，λ２，．．．，λｍ中至少有两个是相等，第二节中，给出了ＰＢＩＢ设计Ｄ可以用遂次合并结合类来约化的充分必要条件，第三节给出了ＰＢＩＢ设计Ｄ可以用同时合并结合来约化的充分必要条件，最后，在第四节里我们研究了上述两种方法之间的关系。     

群胚上的Dirac结构及泊松约化

本文详细讨论了李双代数胚中的Dirac结构、群胚上的Dirac结构。利用Dirac结构的特征对的概念,给出了作用不变Dirac结构,拉回Dirac结构等概念的新的刻画。最后利用Dirac结构的有关性质,讨论了泊松齐性空间和泊松群胚作用的约化。     

Poisson Morphisms and Reduced Affine Poisson Group Actions

We establish the concept of a quotient affine Poisson group, and study the reduced Poisson action of the quotient of an affine Poisson group G on the quotient of an affine Poisson-G-variety V. The Poisson morphisms (including equivariant cases) between Poisson affine varieties are also discussed. Received April 5, 1999, Accepted March 5, 2001     

关于群胚作用和李双代数胚态射

在这篇文章中，我们讨论了李双代数胚之间的态射，得到了一些李双代数胚之间态射的性质．研究了泊松群胚在泊松流形上的泊松作用，以及这个泊松作用与被作用流形的切李双代数胚到作用泊松群胚的切李双代数胚之间的态射的关系，得到了一些有用的结论。     

ON THE SYMPLECTIC REDUCTIONS

A symplectic reduction method for symplectic G-spaces is given in this paper without using the existence of momentum mappings. By a method similar to the above one, the arthors give a symplectic reduction method for the Poisson action of Poisson Lie groups on symplectic manifolds, also without using the existence of momentum mappings. The symplectic reduction method for momentum mappings is thus a special case of the above results.     

Poisson deleting derivations algorithm and Poisson spectrum

Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.[Crossref], [Web of Science ®] , [Google Scholar]] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.[Crossref], [Web of Science ®] , [Google Scholar]]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.[Crossref], [Web of Science ®] , [Google Scholar]]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2 Bell, J., Launois, S., Sanchez, O. L., Moosa, R. Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. (to appear). [Google Scholar]]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.     

Poisson Prime Ideals of Poisson Polynomial Rings

Let A be a finitely generated Poisson algebra over a field of characteristic zero. Here we prove that every Poisson prime ideal of A is prime and give a method to find all Poisson prime ideals in an arbitrary Poisson polynomial ring A[x; α, δ].     

Affine Poisson structures

We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all equivalence classes of central extensions of the Lie algebra by ℝ. We characterize all the affine Poisson structures defined on the duals of some lower dimensional Lie algebras. It is shown that under a certain condition every Poisson structure locally looks like an affine Poisson structure. As an application, we show the role played by affine Poisson structures in mechanics. Finally, we prove some involution theorems.     

约束的NLS系统的刚体型Poisson结构

A constrained system associated with a 3×3 matrix spectral problem of the nonlinear Schrodinger(NLS) hierarchy is proposed. It is shown that the constrained system is a Hamiltonian system with the rigid body type Poisson structure on the Poisson manifold R3N. Further, the reduction of the constrained system extended to the common level set of the complex cones is proved to be the constrained AKNS system on C2N.