nxn Matrices - Representations of Infinite Groups and Invariants

Formanek 96-10118这个项目有两个部分，都与域上的nxn矩阵有关。第一部分是对某些无限群的不可约n维表示的分类，如辫子群、映射类群和Torelli群。虽然目前还不可能对所有维度进行完全分类，但低维度的分类应该是可能的。注意，许多辫子群的表示都是由数学家和物理学家构造的，特别是与杨-巴克斯特方程有关的。第二部分研究了nxn矩阵的不变量和恒等式。这是对环论和群论这两个代数分支的研究。这些问题和攻击它们的方法是组合的，涉及特定的环和群，而不是一般的理论。如上所述，其中一些问题也是物理学家感兴趣的。

FORMANEK 96-10118 This project has two parts, both concerned with nxn matrices over a field. The first part is the classification of the irreducible n-dimensional representations of certain infinite groups, such as braid groups, mapping class groups, and Torelli groups. While there is presently no possibility of a complete classification in all dimensions, a classification in low dimensions should be possible. Note that many representations of braid groups have been constructed by both mathematicians and physicists, especially in connection with the Yang-Baxter equation. The second part is the study of the invariants and identities of nxn matrices. This is research in ring theory and group theory, two branches of algebra. The problems and methods to attack them are combinatorial, and deal with particular rings and groups, rather than general theory. As noted above, some of these questions are also of interest to physicists.