Mathematical Sciences: Representation Theory of Infinite Dimensional Classical Groups

Boyer教授将研究与算子代数相关的经典群的表示理论。重点将放在紧群的归纳极限上，紧群的复杂度仅次于有限维群。将要进行的分析将使用近似有限维C*-代数（特别是k函子及其阶结构）、动力系统及其遍历测度以及近似群的组合学的框架。一个特定的目标是对归纳极限酉群的表示进行分类，这些表示推广了众所周知的正则（反）对易关系的拟自由表示。这个项目是许多方面正在进行的努力的一部分，目的是将李群（以挪威著名数学家索夫斯·李命名）研究中长期熟悉的概念和结果扩展到更奇特的对象上。通常意义上的李群的一个例子是一个球体的所有旋转的群，其中群运算包括一个运动跟随另一个运动。旋转群的详细知识对于分析存在球对称的任何数学或物理情况都是非常有用的。这个组的一个重要特征是，我们只需要有限数量的参数（在本例中是三个）来指定它的任何成员。同样，我们可以考虑高维的旋转群。只要维数是有限的，就不会出现真正意义上的新现象，但如果我们让它无界地增加，或者说取极限，我们就得到了一种无限维的李群。许多同样的问题对于这个归纳极限旋转群和普通的三维旋转群都是有意义的。博耶教授将研究新的方法，在这个更困难的环境中回答这些问题。

Professor Boyer will investigate the representation theory of classical groups associated to operator algebras. The emphasis will be on inductive limits of compact groups, which are the next order of complexity after finite dimensional groups. The analysis to be undertaken will use the framework of approximately finite- dimensional C*-algebras (especially the K-functor and its order structure), dynamical systems and their ergodic measures, and the combinatorics of the approximating groups. One specific objective is to classify the representations of the inductive limit unitary group which generalize the well-known quasi-free representations of the canonical (anti-) commutation relations. This project is part of an effort now underway in many quarters to extend concepts and results long familiar in the study of Lie groups (named after the eminent Norwegian mathematician Sophus Lie) to objects of a much more exotic sort. An example of a Lie group in the usual sense is the group of all rotations of a sphere, where the group operation consists of following one motion by another. Detailed knowledge of the rotation group is extremely helpful in the analysis of any mathematical or physical situation where spherical symmetry is present. It is an important feature of this group that we need only a finite number of parameters (in this case three) to specify any of its members. Likewise, we can consider the rotation group in higher dimensions. No truly new phenomena occur as long as the dimension remains finite, but if we let it increase without bound and so to speak take the limit, we obtain a kind of infinite-dimensional Lie group. Many of the same questions make sense for this inductive limit rotation group as for the ordinary three-dimensional one. Professor Boyer will work on the new methods that are needed to answer them in this more difficult setting.