Bounded and Semi-bounded Representations of Infinite Dimensional Lie Groups

TL;DR

This paper reviews recent advances in classifying bounded and semibounded representations of infinite dimensional Lie groups, utilizing smoothing operators and host algebras to connect with $C^*$-theory, and covers various group types including gauge, loop, and Virasoro groups.

Contribution

It introduces the concept of smoothing operators and host algebras for classifying semibounded representations, extending $C^*$-theory methods to infinite dimensional Lie groups.

Findings

Classification of bounded representations for unitary and gauge groups

Application of holomorphic induction to semibounded representations

Complete classification for hermitian, loop, Virasoro, and oscillator groups

Abstract

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.