Bounded and Semi-bounded Representations of Infinite Dimensional Lie Groups
Bounded and Semi-bounded Representations of Infinite Dimensional Lie Groups
复制标题
无限维李群的有界和半有界表示
DOI：
--
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
K. Neeb
中科院分区：
文献类型：
--
作者：
K. Neeb
在这篇文章中，我们描述了无限维李群的有界和半有界表示分类的最新进展。我们首先讨论半有界条件，以及如何使用平滑算子的新概念来构造 $C^*$-代数（所谓的主代数），其表示与无限维的某些半有界表示一一对应
李群$G$。这使得 $C^*$ 理论的全部力量在这种情况下可用。
然后我们讨论希尔伯特空间上几种酉群和规范群的有界表示的分类。在解释了全纯归纳法作为从有界表示到半有界表示的方法之后，我们描述了算子的 Hermitian Lie 群、循环群（具有无限维目标）、Virasoro 群和某些无限维振子群的半有界表示的分类。
In this note we describe the recent progress in the classification of bounded and semibounded representations of infinite dimensional Lie groups. We start with a discussion of the semiboundedness condition and how the new concept of a smoothing operator can be used to construct $C^*$-algebras (so called host algebras) whose representations are in one-to-one correspondence with certain semibounded representations of an infinite dimensional
Lie group $G$. This makes the full power of $C^*$-theory available in this context.
Then we discuss the classification of bounded representations of several types of unitary groups on Hilbert spaces and of gauge groups. After explaining the method of holomorphic induction as a means to pass from bounded representations to semibounded ones, we describe the classification of semibounded representations for hermitian Lie groups of operators, loop groups (with infinite dimensional targets), the Virasoro group and certain infinite dimensional oscillator groups.
扭曲希尔伯特环代数的同构
DOI：
10.4153/cjm-2016-003-x
发表时间：
2015
期刊：
Canadian Journal of Mathematics
影响因子：
--
作者：
T. Marquis;K. Neeb
通讯作者：
K. Neeb