Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups

K. H. Neeb[1]

  • [1] Department Mathematik FAU Erlangen-Nürnberg, Cauerstrasse 11 91058 Erlangen (Germany)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 5, page 1823-1892
  • ISSN: 0373-0956

Abstract

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A unitary representation π of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators i d π ( x ) 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 ℒ ^ φ ( K ) which are double extensions of the twisted loop group ℒ φ ( K ) , where K is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and φ is a finite order automorphism of K which leads to one of the 7 irreducible locally affine root systems with their canonical 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.

How to cite

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Neeb, K. H.. "Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups." Annales de l’institut Fourier 64.5 (2014): 1823-1892. <http://eudml.org/doc/275603>.

@article{Neeb2014,
abstract = {A unitary representation $\pi $ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\{\tt d\}\pi (x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $\mathfrak\{g\}$ of $G$. We classify all irreducible semibounded representations of the groups $\widehat\{\mathcal\{L\}\}_\phi (K)$ which are double extensions of the twisted loop group $\mathcal\{L\}_\phi (K)$, 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.},
affiliation = {Department Mathematik FAU Erlangen-Nürnberg, Cauerstrasse 11 91058 Erlangen (Germany)},
author = {Neeb, K. H.},
journal = {Annales de l’institut Fourier},
keywords = {infinite dimensional Lie group; unitary representation; semibounded representation; Hilbert–Lie algebra; Hilbert–Lie group; Kac–Moody group; loop group; double extension; positive definite function; Kac-Moody group},
language = {eng},
number = {5},
pages = {1823-1892},
publisher = {Association des Annales de l’institut Fourier},
title = {Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups},
url = {http://eudml.org/doc/275603},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Neeb, K. H.
TI - Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 5
SP - 1823
EP - 1892
AB - A unitary representation $\pi $ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i{\tt d}\pi (x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $\mathfrak{g}$ of $G$. We classify all irreducible semibounded representations of the groups $\widehat{\mathcal{L}}_\phi (K)$ which are double extensions of the twisted loop group $\mathcal{L}_\phi (K)$, 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.
LA - eng
KW - infinite dimensional Lie group; unitary representation; semibounded representation; Hilbert–Lie algebra; Hilbert–Lie group; Kac–Moody group; loop group; double extension; positive definite function; Kac-Moody group
UR - http://eudml.org/doc/275603
ER -

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