Representations of Groups, Lie Algebras, and Algebras

长友在广义相对论中构造了恩斯特方程的无穷多个有理解。Murakami为多元亚历山大多项式构造了Turaev意义下的顶点型状态模型。利用该模型，得到了多元亚历山大多项式的一组新公理。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。这个代数可以被认为是A型lwahori Hecke代数的推广。利用这个代数，村上推广了山田多项式，它是空间图在S^3中嵌入的不变量。村上春树还调查表示的类别缠结使用Kontsevich的interated积分，并成功地给予组合描述Kontsevich的积分的结，链接和缠结。佐竹研究了E6^型的简单椭圆奇点，并明确地构造了齐藤意义下的平坦θ不变量。Yamane通过量子化单李超代数的包络代数构造了一个新的拟三角Hopf代数。他定义这个霍普代数的发电机和关系类似于德林费尔德和金宝没有简单的李代数。他的主要贡献是阐述了新的定义关系，其中没有connterpart的情况下，简单的李代数。即使对于单李超代数的包络代数，这些关系也不是以这种精确的形式已知的。Yamane的结果将在数学物理和低维拓扑中有应用。例如，他给出了他的准三角形Hopt代数的通用R-矩阵的公式，并使用这个公式，他找到了Perk和Schultz的R-矩阵的表示论解释。

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.