Generalized positive energy representations of groups of jets

Milan Niestijl

Documenta mathematica, Tome 28 (2023) no. 3, pp. 709-763

Résumé

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra k. Consider the Fréchet–Lie group G:=J0∞​(V;K) of ∞-jets at 0∈V of smooth maps V→K, with Lie algebra g=J0∞​(V;k). Let P be a Lie group and write p:=Lie(P). Let α be a smooth P-action on G. We study smooth projective unitary representations ρˉ​ of G⋊α​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 ρˉ​(G). We show that this condition imposes severe restrictions on the derived representation dρˉ​ of g⋊p, leading in particular to sufficient conditions for ρˉ​∣G​ to factor through J02​(V;K), or even through K.

Détail

DOI : 10.4171/dm/920

Classification : 22E66, 17B15
Mots-clés : Unitary representations, positive energy representations, KMS states, infinite-dimensional Lie groups, Lie algebras