On the connection between harmonic maps and the self-dual Yang—Mills and the sine-Gordon equations

This is an algebraic paper which demonstrates some algebraic connections between the Yang—Mills and the sine-Gordon equation.     

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.     

非交换Lipschitz-φ算子代数

本文引入由紧距离空间(K,d)到给定Banach代数A中的Lipschitz-φ算子构成的非交换Banach代数L~φ(K,A)与l~φ(K,A),证明了它们都是由K到A的全体连续算子构成的非交换Banach代数C(K,A)的子代数,并且关于范数||f||φ=L_φ(f)+||f||∞是Banach代数,研究了不同 Lipschitz尺度函数φ对应的大(小)Lipschitz代数之间的关系。特别当φ(t)=t~α时,引入了极限代数lim_(α→0+)l~α(K,A),lim_(α→+∞)l~α(K,A),lim_(α→0+)L~α(K,A)与lim_(α→+∞)L~α(K,A)以及距离空间的Lipschitz连通性,得到了lim_(α→+∞)l~α(K,A)=A的充要条件,也给出了lim_(α→0+)L~α(K,A)=C(K,A)的条件。     

Lax方程与耦合Poisson流形的动力r-矩阵

对于一个与Poisson流形耦合的动力r-矩阵,我们在相应的Lie双代数胚上构造出一类Lax方程和一族守恒量,希望利用该方法进一步研究可积Hamilton系统.     

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

Lax可积系统的有限变换对称群及严格解

本文用一种简单直接的方法研究了非线性物理系统Davey-Stewartson方程及（２＋１）维的Camassa-Holm方程，并给出了它们的有限变换对称群及对称。     

On the local and asymptotic behavior of Brownian motion on simply connected nilpotent Lie groups

The aim of this paper is to study the local and asymptotic behavior of Brownian motion on simply connected nilpotent Lie groups. We carry over a qualitative version of the Erdös-Rényi law of large numbers for Brownian motion to simply connected step 2-nilpotent Lie groups. The method applied gives rise to a proof for qualitative results concerning the modulus of continuity of Brownian motion on simply connected step 3-resp. step 2-nilpotent Lie groups without using the Ventsel-Freidlin theory as in Baldi.     

Marked length rigidity for symmetric spaces

We give conditions under which a homomorphism between two Zariski dense subgroups of connected semisimple Lie groups G and G' without compact factors and with trivial center can be extended to a continuous isomorphism between G and G'. In particular we prove the marked length rigidity and the marked translation vector rigidity. This last result was motivated by a Margulis‚s question. Received: September 18, 2001     

一个(2+1)维的可积sinb-Gordon方程

利用ｍＫｄＶ方程的违推算子的遗传性，构造了一个（２＋１）维的ｓｉｎｈ－Ｇｏｒｄｏｎ型的可积模型。该模型是（１＋１）维ｓｉｎｈ－Ｇｏｒｄｏｎ方程的推广．同时给出了该模型丰富的对称性及其相应的代数结构。     

Spinless Excitons Implicit in SO(5) Theory

A pair of up-down operators are constructed explicitly for S.C.Zhang‘s SO(5) theory of high Tc superconductivity.From them two good quantum numbers are derived.The up-down operators are related to the spin-independent excitons which are not considered before.``