Representations of Groups, Lie Algebras, and Algebras

永友构造了广义相对论中恩斯特方程的无穷多个有理解。对于多变量Alexander多项式，Murakami构造了Turaev意义上的顶点型状态模型。利用该模型，得到了多变量Alexander多项式的一组新的公理。Murakami还确定了量子群U_q（gl(n, c)）的混合张量表示的中心化代数的结构。该代数可以看作是a型lwahori Hecke代数的推广。利用该代数，Murakami推广了S^3空间图嵌入的不变量Yamada多项式。Murakami还利用Kontsevich的积分研究了缠结范畴的表示，并成功地给出了Kontsevich的结、环和缠结积分的组合描述。Satake研究了E6^^^型的简单椭圆奇点，并显式构造了K.Saito意义上的平面不变量。Yamane将单李超代数的包络代数量子化，构造了一个新的拟三角Hopf代数。他通过产生子和关系来定义这个霍普代数，就像德林菲尔德和金博对简单李代数所做的那样。他的主要贡献是发现了在简单李代数中没有对应关系的新的定义关系。即使对于简单李超代数的包络代数，这些关系也不知道以这种精确的形式存在。Yamane的结果将在数学物理和低维拓扑中得到应用。例如，他给出了准三角Hopt代数的普适r矩阵的公式，并利用这个公式找到了Perk和Schultz的r矩阵的表示理论解释。

Nagatomo constructed infinitely many rational solutions of the Ernst equation in general relativity. Murakami constructed a vertex type state model in Turaev's sense for the multi-variable Alexander polynomial. By using this model, a new set of axioms for the multi-variable Alexander polynomial is obtained. Murakami also determined the structure of the centralizer algebra of the mixed tensor representations of the quantum group U_q(gl(n, c)). This algebra can be considered as a generalization of the lwahori Hecke algebra of type A.Using this algebra, Murakami generalized the Yamada polynomial, which is an invariant of embeddings of a spatial graph in S^3. Murakami also investigated representations of the category of tangles using Kontsevich's interated integral, and succeeded in giving a combinatorial description of Kontsevich's integrals of knots, links and tangles. Satake investigated the simple elliptic singularity of type E6^^^, and constructed explicitly the flat theta invariants in the sense of K.Saito. Yamane constructed a new quasi-triangular Hopf algebra by quantizing the enveloping algebras of simple Lie superalgebras. He defined this Hopt algebra by generators and relations similarly as Drinfeld and Jimbo did for simple Lie algebras. His main contribution is the descovery of new kinds of defining relations which have no connterpart in the case of simple Lie algebras. Even for the enveloping algebras of simple Lie superalgebra, these relations were not known in this precise form. Yamane's results will have applications in mathematical physics and low dimensional topology. For example, he gave a formula for the universal R-matrix of his quasi-triangular Hopt algebra, and using this formula he found a representation theoretic interpretation for the R-matrix of Perk and Schultz.