Home Publications Research Grants Inventions & Patents Awards Additional Research Activities Faculties & Institutions Research Areas

Semibounded unitary representations of double extensions of Hilbert-Loop groups

Neeb KH (2014)

---

Publication Type: Journal article, Original article

Publication year: 2014

Journal

Publisher: Association des Annales de l'Institute Fourier; 1999

Book Volume: 64

Pages Range: 1823 - 1892

Journal Issue: 5

URI: http://eudml.org/doc/275603

Abstract

A unitary representation $\pi$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{ℒ}}_{\phi }\left(K\right)$ which are double extensions of the twisted loop group ${ℒ}_{\phi }\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi$ is a finite order automorphism of $K$ which leads to one of the $7$ irreducible locally affine root systems with their canonical $\mathbb{Z}$-grading. To achieve this goal, we extend the method of holomorphic induction to certain classes of Fréchet–Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Fréchet–BCH–Lie groups.This is the first paper dealing with global aspects of Lie groups whose Lie algebra is an infinite rank analog of an affine Kac–Moody algebra. That positive energy representations are semibounded is a new insight, even for loops in compact Lie groups.

Authors with CRIS profile

How to cite

APA:

Neeb, K.H. (2014). Semibounded unitary representations of double extensions of Hilbert-Loop groups. Annales de l'Institut Fourier, 64(5), 1823 - 1892.

MLA:

Neeb, Karl Hermann. "Semibounded unitary representations of double extensions of Hilbert-Loop groups." Annales de l'Institut Fourier 64.5 (2014): 1823 - 1892.

BibTeX: Download