Generalized positive energy representations of groups of jets

Milan Niestijl

doi:10.4171/dm/920

Generalized positive energy representations of groups of jets

Milan Niestijl^*

^*Corresponding author for this work

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Abstract

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra ř. Consider the Fréchet–Lie group G:= J0°(V; K) of i-jets at 0 e V of smooth maps V ! K, with Lie algebra g = 70° (V ; ř). Let P be a Lie group and write p:= Lie(P). Let a be a smooth P-action on G. We study smooth projective unitary representations p of G xœ P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by -ρ(G'). We show that this condition imposes severe restrictions on the derived representation dp of g ì p, leading in particular to sufficient conditions for X|g to factor through J0¡(V; K), or even through K.

Original language	English
Pages (from-to)	709-763
Number of pages	55
Journal	Documenta Mathematica
Volume	28
Issue number	3
DOIs	https://doi.org/10.4171/dm/920
Publication status	Published - 2023

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Keywords

infinite-dimensional Lie groups
KMS states
Lie algebras
positive energy representations
Unitary representations

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title = "Generalized positive energy representations of groups of jets",

abstract = "Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra {\v r}. Consider the Fr{\'e}chet–Lie group G:= J0°(V; K) of i-jets at 0 e V of smooth maps V ! K, with Lie algebra g = 70° (V ; {\v r}). Let P be a Lie group and write p:= Lie(P). Let a be a smooth P-action on G. We study smooth projective unitary representations p of G x{\oe} P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by -ρ(G'). We show that this condition imposes severe restrictions on the derived representation dp of g {\`i} p, leading in particular to sufficient conditions for X|g to factor through J0¡(V; K), or even through K.",

keywords = "infinite-dimensional Lie groups, KMS states, Lie algebras, positive energy representations, Unitary representations",

author = "Milan Niestijl",

note = "Green Open Access added to TU Delft Institutional Repository {\textquoteleft}You share, we take care!{\textquoteright} – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. ",

year = "2023",

doi = "10.4171/dm/920",

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AU - Niestijl, Milan

N1 - Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

PY - 2023

Y1 - 2023

N2 - Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra ř. Consider the Fréchet–Lie group G:= J0°(V; K) of i-jets at 0 e V of smooth maps V ! K, with Lie algebra g = 70° (V ; ř). Let P be a Lie group and write p:= Lie(P). Let a be a smooth P-action on G. We study smooth projective unitary representations p of G xœ P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by -ρ(G'). We show that this condition imposes severe restrictions on the derived representation dp of g ì p, leading in particular to sufficient conditions for X|g to factor through J0¡(V; K), or even through K.

AB - Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra ř. Consider the Fréchet–Lie group G:= J0°(V; K) of i-jets at 0 e V of smooth maps V ! K, with Lie algebra g = 70° (V ; ř). Let P be a Lie group and write p:= Lie(P). Let a be a smooth P-action on G. We study smooth projective unitary representations p of G xœ P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by -ρ(G'). We show that this condition imposes severe restrictions on the derived representation dp of g ì p, leading in particular to sufficient conditions for X|g to factor through J0¡(V; K), or even through K.

KW - infinite-dimensional Lie groups

KW - KMS states

KW - Lie algebras

KW - positive energy representations

KW - Unitary representations

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DO - 10.4171/dm/920

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SN - 1431-0635

VL - 28

SP - 709

EP - 763

JO - Documenta Mathematica

JF - Documenta Mathematica

IS - 3

ER -

Generalized positive energy representations of groups of jets

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