论文标题
喷气机组的广义正能表示
Generalized Positive Energy Representations of Groups of Jets
论文作者
论文摘要
令$ v $为有限维真实矢量空间，而$ k $一个紧凑的简单谎言组，带有lie代数$ \ mathfrak {k} $。考虑Fréchet-lie Group $ g：= j_0^\ infty（v; k）$ \ infty $ -jets的$ 0 \ in $ 0 \ in V $ of Smooth Maps $ v \ to K $，带有lie algebra $ \ mathfrak {g} = J_0^\ infty（v;令$ p $为谎言组，然后写$ \ mathfrak {p}：= \ textrm {lie}（p）$。让$α$成为$ g $的平滑$ p $ Action。我们研究$ g \rtimes_αp$的平滑投影统一表示，满足了所谓的广义正能条件。特别是，此类捕获了与$ \barρ（g）$生成的von Neumann代数上的KMS状态兼容的表示形式。我们表明，这种情况对$ \ mathfrak {g} \ rtimes \ mathfrak {p} $的派生表示$ d \barρ$施加了严格的限制，尤其是$ \barρ\ big | _ {g} $的足够条件。
Let $V$ be a finite-dimensional real vector space and $K$ a compact simple Lie group with Lie algebra $\mathfrak{k}$. Consider the Fréchet-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 $α$ be a smooth $P$-action on $G$. We study smooth projective unitary representations $\barρ$ of $G \rtimes_αP$ 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 $\barρ(G)$. We show that this condition imposes severe restrictions on the derived representation $d\barρ$ of $\mathfrak{g} \rtimes \mathfrak{p}$, leading in particular to sufficient conditions for $\barρ\big|_{G}$ to factor through $J_0^2(V; K)$, or even through $K$.
