Completely integrable systema and representation theory of infinite dimen-sional algebras

这项研究的首席研究员与台北中研院教授S.S.Roan合作，在商没有中心元素的情况下，利用其理想确定了Onsager代数商的结构。这几乎决定了Onsager代数的商的结构。特别地，通过对Kac-Moody李代数的幂零部分的理想进行分类，找到了与幂零李代数分类方案的联系，从而解决了以往研究中没有确定的情况。要找到这种关系，工作站上的计算机代数系统是非常有用和不可或缺的，特别是在计算实例中。Onsager代数可以看作是仿射李代数A^&lt；(1)&gt；_1的幂零部分的李代数变形。如果我们将A^&lt；(1)&gt；_1标识为当前代数的中心扩张，则Onsager代数是对合的不动点集。利用这一表述，我们可以研究Onsager代数的理想结构。为了解决重挖掘的问题，我们使用了A^&lt；(1)&gt；_1的主(扭)实现。我们认为这种主实现将在结合保形场理论研究其他仿射李代数的幂零部分的形变中起到重要的作用。我们正在准备关于这一结果的手稿。本研究项目中的其他研究者在研究具有随机势的一维薛定谔算符的谱、涉及Schur函数的恒等式、多线性椭圆型偏微分方程的正解、与自由费米子有关的顶点算符代数的不可约表示的分类、量子逆散射方法中的L算符与仿射量子包络代数中的Drinfeld生成子间的关系等方面取得了新的结果。

In a collaboration with Professor S.S.Roan of Academia Sinica (Taipei), the head investigator of this research project has determined the structure of quotient of the Onsager algebra by ideals of it in the case when the quotients do not have central elements. This almost determines the structure of quotients of the Onsager algebra. In particular the case not determined in the previous researches are fixed by finding a relation with the project of classifying nilpotent Lie algebras by classifying the ideals in the nilpotent part of Kac-Moody Lie algebras. To find such relationship computer algebra system on a workstation was very helpful and indispensable, especially in computing examples. The Onsager algebra can be viewed as a Lie algebra deformation of the nilpotent part of the affine Lie algebra A^<(1)>_1. If we identify A^<(1)>_1 with a central extension of the current algebra, the Onsager algebra is a fixed point set of an involution. By using this presentation we can study the structures of ideals of the Onsager algebra. In order to fix the remining case, we used the principal (twisted) realization of A^<(1)>_1. We think that this kind of principal realization will play some important role in the study of deformation of nilpotent parts of other affine Lie algebras in connection with conformal field theory. We are preparing manuscript on this result. Other investigators in this research project have obtained new results in the study of spectrum of one dimensional Schrodinger operator with a random potential, an identity involving Schur functions, positive solutions of sernilinear elliptic partial differential equations, classification of irreducible representations of vertex operator algebras related with free fermions, relationship of L operators in quantum inverse scattering method and Drinfeld's genarators in affine quantum enveloping algebras and other areas.