# Semibounded unitary representations of double extensions of infinite dimensional pre-Hilbert-Lie groups

**Third Party Funds Group - Sub project**

**Start date :**
01.04.2009

**End date :**
30.03.2015

## Overall project details

### Overall project

SPP 1388: Representation Theory (Darstellungstheorie)

## Project details

### Scientific Abstract

Infinite dimensional Lie groups and their representations show up in all areas of mathematics and other sciences, wherever symmetries depending on infinitely many parameters arise. The goal of this project is to develop a geometric approach to the important class of semibounded unitary representations of infinite dimensional Lie groups. Typical groups arising in this context are double extensions of Hilbert Lie groups, which include oscillator groups in the abelian case, afine Kac Moody groups based on loop groups with infinite dimensional targets and a large number of groups whose Lie algebras are Z-graded. Semiboundedness of a unitary representation is a stable version of the „positive energy" condition which characterizes many representations arising in mathematical physics, resp., field theories. For a unitary representation of a Lie group it means that the selfadjoint operators from the derived representation are uniformly bounded below on some open subset of the Lie algebra. Our goal is to understand the decomposition theory and the irreducible representations in this class.The focus of the present project lies on combining algebraic, geometric and analytic aspects of the theory, such as realizations in holomorphic bundles and convexity properties of momentum maps related to spectral properties of operators to obtain classification results.

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### Funding Source