Generalized Positive Energy Representations of Groups of Jets

TL;DR

This paper investigates a class of smooth projective unitary representations of jet groups with a positive energy condition, revealing restrictions that lead to their factorization through finite-dimensional structures.

Contribution

It introduces a generalized positive energy condition for representations of jet groups and establishes conditions under which these representations factor through finite-dimensional Lie groups.

Findings

Restrictions on derived representations due to the positive energy condition

Conditions for representations to factor through finite jet groups

Connections to KMS states and von Neumann algebras

Abstract

Let $V$ be a finite-dimensional real vector space and $K$ a compact simple Lie group with Lie algebra $\mathfrak{k}$. Consider the Fr\'echet-Lie group $G := J_0^\infty(V; K)$ of $\infty$-jets at $0 \in V$ of smooth maps $V \to K$, with Lie algebra $\mathfrak{g} = J_0^\infty(V; \mathfrak{k})$. Let $P$ be a Lie group and write $\mathfrak{p} := \textrm{Lie}(P)$. Let $\alpha$ be a smooth $P$-action on $G$. We study smooth projective unitary representations $\bar{\rho}$ of $G \rtimes_\alpha P$ 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 $\bar{\rho}(G)$. We show that this condition imposes severe restrictions on the derived representation $d\bar{\rho}$ of $\mathfrak{g} \rtimes \mathfrak{p}$, leading in particular to sufficient conditions for $\bar{\rho}\big|_{G}$ to factor through $J_0^2(V; K)$, or even through $K$.